Quantum	Gravity	and	Taoist	Cosmology: Exploring	the	Ancient	Origins	of	Phenomenological	String Theory Steven	M.	Rosen [This	article	is	slated	for	Progress	in	Biophysics	and	Molecular	Biology, Vol.	131	(2017),	pp.	34−60.] ABSTRACT In	the	author's	previous	contribution	to	this	journal	(Rosen	2015),	a phenomenological	string	theory	was	proposed	based	on	qualitative	topology	and hypercomplex	numbers.	The	current	paper	takes	this	further	by	delving	into	the ancient	Chinese	origin	of	phenomenological	string	theory.	First,	we	discover	a connection	between	the	Klein	bottle,	which	is	crucial	to	the	theory,	and	the	Ho-t'u,	a Chinese	number	archetype	central	to	Taoist	cosmology.	The	two	structures	are	seen to	mirror	each	other	in	expressing	the	psychophysical	(phenomenological)	action pattern	at	the	heart	of	microphysics.	But	tackling	the	question	of	quantum	gravity requires	that	a	whole	family	of	topological	dimensions	be	brought	into	play.	What we	find	in	engaging	with	these	structures	is	a	closely	related	family	of	Taoist forebears	that,	in	concert	with	their	successors,	provide	a	blueprint	for	cosmic evolution.	Whereas	conventional	string	theory	accounts	for	the	generation	of nature's	fundamental	forces	via	a	notion	of	symmetry	breaking	that	is	essentially static	and	thus	unable	to	explain	cosmogony	successfully,	phenomenological/Taoist string	theory	entails	the	dialectical	interplay	of	symmetry	and	asymmetry	inherent in	the	principle	of	synsymmetry.	This	dynamic	concept	of	cosmic	change	is elaborated	on	in	the	three	concluding	sections	of	the	paper.	Here,	a	detailed	analysis of	cosmogony	is	offered,	first	in	terms	of	the	theory	of	dimensional	development	and its	Taoist	(yin-yang)	counterpart,	then	in	terms	of	the	evolution	of	the	elemental force	particles	through	cycles	of	expansion	and	contraction	in	a	spiraling	universe. The	paper	closes	by	considering	the	role	of	the	analyst	per	se	in	the	further evolution	of	the	cosmos. KEYWORDS:	quantum	gravity;	Taoism;	yin	and	yang;	topology;	phenomenological philosophy;	physics;	cosmogony;	string	theory;	symmetry;	dimension 1.	CONTEMPORARY	PHYSICS,	ARCHETYPAL	NUMBERS,	AND	THE	HO-T'U In	recent	decades,	there	has	been	little	meaningful	progress	in	mainstream	physics toward	a	theory	that	would	effectively	unify	all	the	forces	of	nature-the	quantum 2 mechanical	forces	as	well	as	gravitation.	Musing	ironically	on	this,	physicist	Lee Smolin	(2006)	observed	that,	"for	more	than	two	centuries...our	understanding	of the	laws	of	nature	expanded	rapidly....	[yet]	today,	despite	our	best	efforts,	what	we know	for	certain	about	these	laws	is	no	more	than	what	we	knew	back	in	the	1970s" (viii).	What	I	demonstrated	in	my	previous	contribution	to	this	journal	(Rosen 2015)	is	that	the	crux	of	the	problem	lay	in	quantum	mechanics'	most	basic	term:	!, the	quantum	of	action.	Although	this	indivisible	atom	of	process	found	at	the	submicroscopic	Planck	length	of	10–35	m	constitutes	the	cornerstone	of	more	than	a century	of	successful	experimentation	in	physics,	it	also	marks	the	end	of	the	road for	classical	analysis,	since	here	the	analytical	continuity	of	space	gives	way	to	a "graininess"	or	discreteness	that	admits	of	no	further	quantitative	determination. Moreover,	the	indivisibility	of	quantized	action	is	closely	tied	to	an	indivisibility	of observer	and	observed,	and	this	implicit	inseparability	of	subject	and	object	is anathema	to	the	program	of	objective	analysis	that	had	gone	unquestioned	for centuries.	While	physicists	have	been	able	to	use	probabilistic	methods	to	work around	the	underlying	difficulty	when	their	investigations	are	limited	to	the quantum	mechanical	forces,	the	problem	has	proven	intractable	when	gravitation must	be	included	in	order	to	realize	a	comprehensive	account	of	nature. Reflecting	on	the	persistent	failure	of	physics	to	arrive	at	a	workable	theory of	quantum	gravity,	Smolin	called	for	a	different	style	of	doing	physics,	a	"more reflective,	risky,	and	philosophical	style"	(2006,	294)	that	confronts	"the	deep philosophical	and	foundational	issues	in	physics"	(290).	I	applaud	this	call	for	a more	philosophically	oriented	physics,	and	I	propose	that	the	stalemate	in	physics suggests	it	will	no	longer	be	possible	for	us	to	rely	on	the	old	philosophical foundation.	That	is	because	the	fusion	of	subject	and	object	encountered	at	the	heart of	quantum	physics	profoundly	conflicts	with	the	currently	dominant	philosophy	of science,	presupposing	as	it	does	a	sharp	division	of	subject	and	object.	In	my	2015 essay,	I	described	in	detail	a	new	foundation	for	contemporary	physics: phenomenological	philosophy,	an	approach	in	which	the	intimate	interaction	of subject	and	object	is	accepted	as	fundamental.	Although	phenomenology	remains important	in	the	present	paper,	it	shares	the	spotlight	with	a	much	older philosophical	tradition,	one	that	in	fact	influenced	Martin	Heidegger	and	Maurice Merleau-Ponty,	two	of	the	leading	figures	in	the	phenomenological	movement.	As philosopher	Stephen	Priest	put	it: Much	of	what	we	have	thought	of	as	Heidegger's	originality	is	essentially Taoist	and	Zen	and	the	ramifications	of	this	through	Heidegger's	influence	on Merleau-Ponty's	corpus	have	yet	to	be	identified....Heideggerian	'Modern Continental	Philosophy'	should	no	longer	be	studied	in	abstraction	from	its East	Asian	ground.	(2003,	243)1 The	picture	of	nature	advanced	by	Taoism	some	twenty-five	hundred	years ago	was	of	a	cosmos	in	constant	cyclical	flux.	Its	ebb	and	flow	of	energies	was 1	In	making	this	assertion,	Priest	was	commenting	on	Reinhard	May's	book,	Heidegger's	Hidden Sources:	East	Asian	Influences	on	His	Work	(1996). 3 expressed	by	the	paradoxical	dynamics	of	two	opposing	but	interpenetrating	forces: yang	is	the	active	force	related	to	brightness,	positivity,	expansion	and	masculinity; yin	is	the	passive	medium	associated	with	darkness,	negativity,	contraction,	and femininity.	Operating	in	close	harmony,	yin	and	yang	generate	patterns	of	change that	are	specified	by	the	trigrams	and	hexagrams	depicted	in	the	oldest	text	of classical	China:	the	I	Ching	or	Book	of	Changes.	And	the	system	of	transformations set	forth	in	the	I	Ching	is	closely	related	to	a	primal	pattern	of	action	known	as	the Ho-t'u.	In	the	next	section,	we	will	see	the	significance	of	the	Ho-t'u	for contemporary	physics,	but	before	I	proceed	to	examine	this	archetypal configuration,	let	me	clarify	my	use	of	the	term	"archetype,"	and	discuss	the	general meaning	of	archetypal	numbers. In	the	psychology	of	C.	G.	Jung,	archetypes	are	structures	of	the	"collective unconscious"	imprinted	in	nature	at	the	deepest	level.	This	was	Jung's	way	of speaking	of	that	which	goes	beyond	the	egoic	realm	of	arbitrarily	fashioned,	finite forms	to	participate	in	something	more	timeless	and	universal. The	prefix	arch- comes	from	the	Greek	archos,	a	ruler,	and	appears	in	such	words	as	"archbishop" and	"architect"	(master	builder),	signifying	what	is	chief	or	principal.	Archaeo- means	"ancient"	and	derives	from	the	Greek	arché,	the	beginning,	the	first.	Thus arche	denotes	what	comes	first,	both	in	time	and	importance. What	is	the	meaning	of	type?	The	word	stems	from	the	Greek	typos,	"a	blow, the	mark	of	a	blow,	figure,	outline	...	from	typtein,	to	beat,	strike"	(Webster's Unabridged	Dictionary).	The	etymology	of	"type"	therefore	suggests	action.	Putting this	together	with	arche,	we	can	say	that	an	archetype	is	a	primary	action	pattern, one	that	is	both	fundamental	and	originary.	What	I	am	emphasizing	here	is	the dynamic	character	of	the	archetype.	Therefore,	unlike	the	traditional	Platonic	sense of	archetype	as	eidos,	an	absolute,	changeless	idea	or	form,	I	give	precedence	to viewing	the	archetype	as	an	elemental	process.	While	it	is	true	that	the	word	"type" has	assumed	a	predominantly	static	meaning	for	us-"a	symbol;	an	emblem;	a	token ...	[a]	general	form,	structure,	plan	...	a	model"	(Webster's	Unabridged	Dictionary)- my	own	use	of	the	term	archetype	retains	the	decidedly	more	active	sense	of	the original	Greek	typos:	"from	typtein,	to	beat,	strike." Now,	before	his	death,	Jung	suggested	to	his	colleague,	Marie-Louise	von Franz,	that	the	profoundest	expression	of	archetypal	truth	may	be	embodied	in certain	primordial	number	patterns.	This	provided	the	impetus	for	Number	and Time	(1974),	von	Franz's	account	of	number	as	primal	quality,	as	the	underlying, energetic	organizing	principle	of	the	unus	mundus,	the	cosmic	field	from	which	less fundamental	archetypes	become	manifest. Drawing	from	a	wealth	of	ancient	and	medieval	Western	and	non-Western sources,	von	Franz	demonstrates	that	the	contemporary	Western	notion	of	number as	empty	quantity	in	fact	grew	out	of	a	very	different	meaning	of	number.	In building	on	Jung's	characterization	of	number	as	the	most	basic	and	primitive element	of	matter	and	psyche	alike,	von	Franz	portrays	numbers	as	"psychophysical patterns	of	motion"	(1974,	74).	"We	are	dealing	here,"	says	von	Franz, with	the	principle	according	to	which	number,	taken	qualitatively,	is understood	to	function	as	a	preconscious	...	principle	of	activity;	each 4 number	must	be	thought	of	as	containing	a	specific	activity	that	streams	forth like	a	field	of	force.	From	this	standpoint	numbers	signify	different	rhythmic configurations	of	the	[cosmic	ground].	(1974,	74–75) Here	the	meaning	of	number	differs	significantly	from	the	common	understanding prevalent	today.	According	to	the	latter,	numbers	are	generally	taken	as mathematical	values	representing	particular	quantities	used	for	counting	and making	calculations.	While	numbers	in	this	sense	are	certainly	indispensable	in measuring	extrinsic	motions,	they	themselves	are	completely	inert.	In	contrast,	von Franz's	archetypal	numbers	have	their	own	internal	rhythms	or	pulsations,	each one	constituting	a	distinctive	"phase	of	transformation"	in	time	(1974,	42). Moreover,	assembled	into	certain	field	configurations	or	matrices,	these	numbers constitute	nothing	less	than	"cosmic	plans"	(22).	"In	the	ancient	Chinese	view,"	says von	Franz,	"the	universe	was	'organized'	according	to	numerical	patterns	of	this kind"	(24).	Echoing	von	Franz,	let	me	emphasize	that	the	numbers	in	question	are not	mere	abstractions	but	are	"psychophysical"	in	nature.	Rather	than	just	being objective,	they	express	the	intimate	interplay	of	object	and	subject	also	found	at	the heart	of	modern	physics,	with	its	quantum	of	action. Von	Franz	devotes	considerable	attention	to	a	particular	numerical	array, one	that	is	among	the	most	primordial	number	configurations	of	archaic	China:	the "sequence	of	older	heaven"	or	Ho-t'u	(Fig.	1).2	This	model	provides	a	blueprint	for the	generation	of	the	trigrams	featured	in	Taoism's	I	Ching. Figure	1.	The	Ho-t'u	number	archetype	depicted	as	an	array	of	black	and	white	circles	(left)	and	as	a sequence	of	Arabic	numerals	(right).	(Adapted	from	von	Franz	1974,	23.	Source:	Granet,	Pensée chinoise,	©	Albin	Michel	1936,	1968,	130.) 2	In	considering	the	Ho-t'u,	Von	Franz	actually	includes	an	accompanying	number	array,	the	Lo-shu or	"sequence	of	younger	heaven." This	structure	is	less	primordial	than	the	Ho-t'u	and	thought	to derive	from	it	(see	235–241). For	our	purposes,	it	will	suffice	to	focus	our	attention	exclusively	on the	older	configuration. 5 The	Ho-t'u	sequence	consists	essentially	of	a	double	cycle	of	numbers,	each	of	which constitutes	a	rhythmic	action	unto	itself.	In	the	first	cycle,	there	are	four	such pulsations	making	up	an	inner	number	orbit:	1,	2,	3,	4.	The	gestural	pattern suggested	by	this	is	cross-like,	and	can	be	taken	as	following	the	four	cardinal	points of	the	compass	-	North,	South,	East,	and	West,	respectively	(contra	Western convention,	the	original	Chinese	diagrams	placed	North	below	and	South	above). Cycle	1	is	succeeded	by	a	movement	at	the	center,	5,	after	which	we	shift	to	the outer	number	orbit	for	the	second	cycle	of	actions:	6,	7,	8,	9.	These	latter	pulsations can	be	seen	to	correspond	to	the	intercardinal	compass	points	Northwest, Southeast,	Northeast,	and	Southwest,	respectively.	The	sequence	ends	with	a	second central	pulsation,	10. The	relationship	of	the	Ho-t'u	sequence	to	the	Taoist	trigrams	is	indicated	by von	Franz	as	shown	in	Figure	2. Figure	2.	Ho-t'u	number	archetype	(right)	and	associated	trigrams	(left),	with	trigram	names displayed	in	center.	(Adapted	from	von	Franz	1974,	25.	Source:	Granet,	Pensée	chinoise, © Albin Michel	1936,	1968,	148.) The	movement	from	K'un	(N)	to	Ch'ien	(S)	to	Li	(E)	to	K'an	(W)	corresponds	to	the first	Ho-t'u	cycle	(1–4).	Von	Franz's	diagram	implies	that	an	action	in	the	unmarked position	at	the	center	of	the	trigram	wheel	would	come	next,	this	being	equivalent	to position	number	5	on	the	Ho-t'u	cross.	Then	there	is	the	second	set	of	trigram actions-Ken	(NW)àTui	(SE)àChen	(NE)àSun	(SW)-corresponding	to	Ho-t'u cycle	2.	The	trigram	sequence	concludes	with	a	final	pulsation	at	the	center	of	the wheel,	matching	Ho-t'u	number	10	(see	Fig.	1,	right).3 Speaking	of	the	Ho-t'u	as	involving	an	"internal	double	movement"	(236), Von	Franz	cites	Wilhelm's	explanation	of	a	passage	from	Taoism's	I	Ching: "When	the	trigrams	[...]	are	in	motion,	a	double	movement	is	observable: first,	the	usual	clockwise	movement,	cumulative	and	expanding	as	time	goes 3	Commentators	have	not	always	been	unanimous	on	the	question	of	how	to	assign	trigrams	and their	associated	compass	directions	to	the	numbers	of	the	Ho-t'u.	But	pursuing	debate	on	this	matter is	beyond	the	scope	of	the	present	paper. 6 on,	and	determining	the	events	that	are	passing;	second,	an	opposite, backward	movement,	folding	up	and	contracting	as	time	goes	by,	through which	the	seeds	of	the	future	take	form	[...]	If	we	understand	how	a	tree	is contracted	into	a	seed,	we	understand	the	future	unfolding	of	the	seed	into	a tree."	(Wilhelm	quoted	in	von	Franz,	1974,	236–237) A	little	later	in	her	book,	referring	to	the	"Pre-World	heavenly	order" (another	term	associated	with	the	Ho-t'u)	as	comprising	a	"double	internal	pattern of	motion,"	von	Franz	again	cites	Wilhelm's	translation	of	the	I	Ching: "Thunder	brings	about	movement,	wind	brings	about	dispersion,	rain	brings about	moisture,	the	sun	brings	about	warmth."	[...]	The	text	continues: "Keeping	Still	brings	about	standstill,	the	Joyous	brings	about	pleasure,	the Creative	brings	about	rulership,	the	Receptive	brings	about	shelter."	As Wilhelm	points	out,	this	[latter]	passage	describes	a	retrograde	movement.	In it,	the	forces	of	the	coming	year	are	unrolled.	Pursuance	of	this	line,	says Wilhelm,	leads	to	knowledge	of	the	future,	the	effect	of	which	is	prepared	in advance	by	its	causes	(i.e.,	seeds),	which	take	shape	by	contracting.	This "retrograde"	movement	is	analogous	to	the	retrograde	number	steps	which,	I have	postulated,	lie	behind	the	qualitative	number	series.	(Von	Franz	1974, 243–244) Von	Franz	summarizes	the	Ho-t'u-related	"Pre-World"	sequence	of	trigrams	as comprising	a	"rhythmical	exchange	of	powers"	in	which	the	"first	four	inner rhythms	are	physical	(movement,	dissolution,	moistening,	heating)	and	the	second [retrograde]	four	are	psychic	(maintenance,	rejoicing,	mastery,	recovery)"	(footnote 16,	243–244).	For	von	Franz,	this	double-cycle	"represents	an	attempt	to	visualize the	whole	of	psychological	and	physical	existence,	the	unus	mundus,	by	antithetical time	movements	and	internal	rhythms"	(245). What	relevance	does	this	ancient	cosmology	have	for	the	current	crisis	in modern	cosmology,	with	its	problem	of	quantum	gravity?	In	my	2015	contribution to	this	journal,	I	outlined	the	approach	to	quantum	gravity	offered	in	The	SelfEvolving	Cosmos	(Rosen	2008a).	Presently,	I	am	going	to	build	on	what	I	said	in 2015,	eventually	expanding	it	to	bring	out	more	fully	the	diachronic	aspect,	that	of cosmic	evolution	per	se.	In	the	course	of	doing	this,	we	will	see	how	the	account	of quantum	gravity	I	have	proposed	is	aligned	with	the	"cosmic	plans"	(von	Franz 1974,	22)	implicit	in	the	Ho-t'u	and	related	archetypal	structures	of	Taoist philosophy. 2.	THE	TOPOLOGY	OF	PSYCHOPHYSICAL	ACTION I	mentioned	at	the	outset	of	the	previous	section	that	the	sub-microscopic dynamism	fundamental	to	quantum	mechanics	is	!,	the	quantum	of	action.	Let	us now	consider	this	concept	in	greater	detail.	The	formula	for	!	tells	us	something about	its	nature:	!	=	h/2Π.	In	quantum	theory,	h	is	the	constant	of	proportionality 7 that	relates	the	energy	(E)	of	a	quantum	of	radiation	to	the	frequency	(v)	of	the oscillation	that	produced	it:	E	=	hv.	Now,	if	we	rewrite	this	equation	by	replacing frequency	(v)	with	its	inverse,	namely,	time,	we	then	have	E	=	h/T	or	h	=	ET,	and,	in physics,	energy	multiplied	by	time	is	a	measure	of	action.	Thus	h,	Planck's	constant, gives	us	quantized	action.	The	angularity	of	said	action,	its	internal	"spin,"	is expressed	by	the	application	of	phase,	as	we	see	in	the	formula	!	=	h/2Π.	Here	h	is operated	upon	by	a	phase	of	2Π	radians,	equivalent	to	a	turn	of	360°. I	also	mentioned	earlier	that	in	quantum	mechanics,	!	is	regarded	as	an indivisible	"atom	of	process,"	for	!	is	not	reducible	to	smaller	units	that	could	be applied	in	its	quantitative	analysis.	This	indivisibility	of	the	quantum	domain	is closely	related	to	its	basic	indeterminacy	or	uncertainty.	According	to	Heisenberg's uncertainty	principle,	there	is	a	built-in	limit	to	the	information	we	can	obtain	about the	physical	properties	of	quantum	systems.	This	limitation	can	be	stated	in	terms	of Planck's	constant:	ΔpΔq	≈	!,	where	p	and	q	are	variables	such	as	position	and momentum,	or	time	and	energy	(variables	that	are	paired	or	conjugated	so	as	to	be essentially	indivisible	from	each	other).	The	formula	says	that	the	product	of	the uncertainties	(Δs)	of	such	paired	terms	approximately	equals	(cannot	be	less	than) the	value	of	Planck's	constant.	Clearly	then,	the	phasic	indivisibility	(h/2Π)	of Planck-level	action	is	equivalent	to	its	uncertainty	(ΔpΔq). There	is	another	way	to	look	at	the	quantum	uncertainty.	Nearing	the	submicroscopic	Planck	length	of	10–35	m,	it	appears	that	precise	objective	measurement is	thwarted	by	the	fact	that	the	energy	that	must	be	transferred	to	a	system	in	order to	observe	it	disturbs	that	system	significantly.	This	well-known	"problem	of measurement"	in	quantum	mechanics	expresses	quantum	indivisibility	in	terms	of the	indivisibility	of	the	observer	and	the	observed.	It	seems	that	in	QM,	the	observer no	longer	can	maintain	the	classical	posture	of	detached	objectivity;	unavoidably, s/he	will	be	an	active	participant.	Evidently	this	means	that	quantum	mechanical action	cannot	be	regarded	merely	as	objective	but	must	be	seen	as	entailing	an intimate	merging	of	object	and	subject,	of	matter	and	psyche,	that	defies	Newtonian order.	And	the	psychophysical	character	of	!	brings	back	to	mind	the	kind	of	action characteristic	of	the	Ho-t'u	number	archetype. Consider	further	the	quantized	action	of	!.	It	takes	the	form	of	an	odd spinning	that	Wolfgang	Pauli	modeled	by	using	complex	numbers.	Musès	(1976) suggested	that	Pauli's	spin	matrices	for	the	electron	are	actually	based	on	a	kind	of complex	number	or	"hypernumber"	that	goes	beyond	Pauli's	imaginary	i:	the hypernumber	ε	(defined	as	ε2	=	+1,	but	ε	≠	±	1).	What	I	demonstrate	in	Cosmos	is that	the	geometric	counterpart	of	ε	is	a	topological	curiosity	known	as	the	Klein bottle.	This	structure	plays	a	highly	significant	role	in	Cosmos	and	in	my	2015	paper, and	I	would	now	like	to	ponder	it	once	more	in	the	present	context,	bringing	out	its inherently	psychophysical	nature. Let	us	begin	with	the	simple	illustration	I	offered	in	2015. 8 Figure	3.	Parts	of	the	Klein	bottle	(after	Ryan	1993,	98) Figure	3	is	my	adaptation	of	communication	theorist	Paul	Ryan's	linear schemata	for	the	Klein	bottle	(1993,	98).	According	to	Ryan,	the	three	basic	features of	the	Klein	bottle	are	"part	contained,"	"part	uncontained,"	and	"part	containing." Here	we	see	how	the	part	contained	opens	out	(at	the	bottom	of	the	figure)	to	form the	perimeter	of	the	container,	and	how	this,	in	turn,	passes	over	into	the uncontained	aspect	(in	the	upper	portion	of	Fig.	3).	The	three	parts	of	this	structure thus	flow	into	one	another	in	a	continuous,	self-containing	movement	that	flies	in the	face	of	the	Cartesian	division	of	contained,	containing,	and	uncontained- symbolically,	of	object,	space,	and	subject.	We	can	also	see	an	aspect	of	discontinuity in	the	diagram.	At	the	juncture	where	the	part	uncontained	passes	into	the	part contained,	the	structure	must	intersect	itself.	Would	this	not	break	the	figure	open, rendering	it	simply	discontinuous?	While	this	is	indeed	the	case	for	a	Klein	bottle conceived	as	an	object	in	ordinary	space,	properly	seen	the	Klein	bottle	actually enacts	a	dialectic	of	continuity	and	discontinuity,	as	will	become	clearer	in	further exploring	this	peculiar	structure.	We	can	say	then	that,	in	its	highly	schematic	way, the	one-dimensional	diagram	lays	out	symbolically	the	basic	terms	of	a "continuously	discontinuous"	dialectic	involving	not	only	subject	and	object	(psyche and	matter),	but	the	mediating	space	as	well.	Depicted	here	is	the	process	by	which the	three-dimensional	object	of	the	sub-microscopic	realm,	in	the	act	of	containing itself,	is	transformed	into	the	subject.	This	blueprint	for	psychophysical interrelatedness	gives	us	a	graphic	indication	of	how	the	mutually	exclusive categories	of	classical	thought	are	surpassed	by	a	relation	of	mutual	inclusion. The	paradoxical	nature	of	the	Klein	bottle	finds	expression	in	its	surprising property	of	one-sidedness.	More	commonplace	topological	figures	such	as	the sphere	and	the	torus	are	two-sided;	their	opposing	sides	can	be	identified	in	a straightforward,	unambiguous	fashion.	Therefore,	they	meet	the	classical expectation	of	being	closed	structures,	structures	whose	interior	regions	("parts contained")	remain	interior.	In	the	contrasting	case	of	the	Klein	bottle,	inside	and outside	are	freely	reversible.	Thus,	while	the	Klein	bottle	is	not	simply	an	open structure,	neither	is	it	simply	closed,	as	are	the	sphere	and	the	torus.	In	studying	the properties	of	the	Klein	bottle,	we	are	led	to	a	conclusion	that	is	paradoxical	from	the classical	viewpoint:	this	structure	is	both	open	and	closed.	The	Klein	bottle therefore	helps	to	convey	to	us	the	fluid	relationships	between	inside	and	outside, 9 closure	and	openness,	continuity	and	discontinuity	that	are	lost	to	us	in	Cartesian experience. However,	must	the	self-containing	one-sidedness	of	the	Klein	bottle	be	seen as	involving	the	spatial	container?	Granting	the	Klein	bottle's	symbolic	value,	could we	not	view	its	inside-out	flow	from	"part	contained"	to	"part	containing"	merely	as a	characteristic	of	an	object	that	itself	is	simply	"inside"	of	space,	with	space continuing	to	play	the	classical	role	of	that	which	contains	without	being	contained? In	other	words,	despite	its	suggestive	quality,	does	the	Klein	bottle	not	lend	itself	to classical	idealization	as	a	mere	object	in	space	just	as	much	as	any	other	structure? The	schematic	representation	of	the	Klein	bottle	provided	by	Figure	3	shows that	it	possesses	the	curious	property	of	passing	through	itself.	When	we	consider the	actual	construction	of	a	Klein	bottle	in	three-dimensional	space	(by	joining	one boundary	circle	of	a	cylinder	to	the	other	from	the	inside),	we	are	confronted	with the	fact	that	no	structure	can	penetrate	itself	without	cutting	a	hole	in	its	surface,	an act	that	would	render	the	model	topologically	imperfect	(simply	discontinuous).	So the	Klein	bottle	cannot	be	assembled	effectively	when	one	is	limited	to	three dimensions. Mathematicians	observe	that	a	form	that	penetrates	itself	in	a	given	number of	dimensions	can	be	produced	without	cutting	a	hole	if	an	added	dimension	is available.	The	point	is	imaginatively	illustrated	by	Rudolf	Rucker	(1977).	He	asks	us to	picture	a	species	of	"Flatlanders"	attempting	to	assemble	a	Moebius	strip,	which is	a	lower-dimensional	analogue	of	the	Klein	bottle.	Rucker	shows	that,	since	the reality	of	these	creatures	would	be	limited	to	two	dimensions,	when	they	would	try to	make	an	actual	model	of	the	Moebius,	they	would	be	forced	to	cut	a	hole	in	it.	Of course,	no	such	problem	with	Moebius	construction	arises	for	us	human	beings,	who have	full	access	to	three	external	dimensions.	It	is	the	making	of	the	Klein	bottle	that is	problematic	for	us,	requiring	as	it	would	a	fourth	dimension.	Try	as	we	might	we find	no	fourth	dimension	in	which	to	execute	this	operation. In	contemporary	mathematics	however,	the	fact	that	we	cannot	create	a proper	model	of	the	Klein	bottle	in	three-dimensional	space	is	not	seen	as	an obstacle.	The	modern	mathematician	does	not	limit	himor	herself	to	the	concrete reality	of	space	but	feels	free	to	invoke	any	number	of	higher	dimensions.	Notice though,	that	in	summoning	into	being	these	extra	dimensions,	the	mathematician	is extrapolating	from	the	known	three-dimensionality	of	the	concrete	world.	This procedure	of	dimensional	proliferation	is	an	act	of	abstraction	that	presupposes	that the	nature	of	dimensionality	itself	is	not	altered.	In	the	case	of	the	Klein	bottle,	the "fourth	dimension"	required	to	complete	its	formation	remains	an	extensive continuum:	an	infinitely	dense,	structureless	substrate	that	itself	does	not	change but	that	mediates	changes	in	the	structured	objects	it	contains	(the	properties	of classical	space	are	examined	in	depth	in	Rosen	2008a	and	2015).	To	be	sure,	the "fourth	dimension"	is	acknowledged	as	but	a	formal	construct	and	the	Klein	bottle	is regarded	as	an	abstract	mathematical	object	simply	contained	in	this	hyperspace (whereas	the	sphere,	torus,	and	Moebius	strip	are	relatively	concrete	mathematical objects,	since	tangibly	perceptible	models	of	them	may	be	successfully	fashioned	in three	dimensions).	We	see	here	how	the	conventional	analysis	of	the	Klein	bottle unswervingly	adheres	to	the	classical	formulation	of	objects	in	space.	Moreover, 10 whether	a	mathematical	object	must	be	approached	through	hyperdimensional abstraction	or	it	is	concretizable,	the	mathematician's	attention	is	always	directed outward	toward	an	object,	toward	that	which	is	cast	before	his	or	her	subjectivity. This	is	the	aspect	of	the	classical	stance	that	takes	subjectivity	as	the	detached position	from	which	all	objects	are	viewed;	here,	never	is	subjectivity	as	such opened	to	view.	Thus	the	posture	of	contemporary	mathematics	is	faithfully	aligned with	that	of	Descartes	and	Newton	in	whatever	topic	it	may	be	addressing.	Always, there	is	the	mathematical	object	(a	geometric	form	or	algebraic	function),	the analytically	continuous	space	in	which	the	object	is	contained,	and	the	seldomacknowledged	uncontained	subjectivity	of	the	mathematician	who	is	carrying	out the	analysis.	And	there	is	never	any	doubt	about	the	strict	separation	of	these	three basic	categories. Now,	in	his	study	of	topology,	Barr	advised	that	we	should	not	be	intimidated by	the	"higher	mathematician....	We	must	not	be	put	off	because	he	is	interested only	in	the	higher	abstractions:	we	have	an	equal	right	to	be	interested	in	the tangible"	(1964,	20).	The	tangible	fact	about	the	Klein	bottle	that	is	glossed	over	in the	higher	abstractions	of	modern	mathematics	is	its	hole.	Because	the	standard approach	has	always	presupposed	extensive	continuity,	it	cannot	come	to	terms with	the	inherent	discontinuity	of	the	Klein	bottle	created	by	its	self-intersection. Therefore,	all	too	quickly,	"higher"	mathematics	circumvents	this	concrete	hole	by an	act	of	abstraction	in	which	the	Klein	bottle	is	treated	as	a	properly	closed	object embedded	in	a	hyper-dimensional	continuum.	Also	implicit	in	the	mainstream approach	is	the	detached	subjectivity	of	the	mathematician	before	whom	the	object is	cast.	I	suggest	that,	by	staying	with	the	hole,	we	may	bring	into	question	the classical	intuition	of	object-in-space-before-subject. Let	us	look	more	closely	at	the	hole	in	the	Klein	bottle. What	kind	of	hole	is it?	This	loss	of	continuity	is	necessary.	One	certainly	could	make	a	hole	in	the Moebius	strip,	torus,	or	any	other	object	in	three-dimensional	space,	but	such discontinuities	would	not	be	necessary	inasmuch	as	these	objects	can	be	properly assembled	in	space	without	rupturing	them.	It	is	clear	that	whether	such	objects	are cut	open	or	left	intact,	the	closure	of	the	space	containing	them	will	not	be	brought into	question;	in	rendering	these	objects	discontinuous,	we	do	not	affect	the assumption	that	the	space	in	which	they	are	embedded	is	simply	continuous.	With the	Klein	bottle	it	is	different.	Its	discontinuity	does	speak	to	the	supposed continuity	of	three-dimensional	space	itself,	for	the	necessity	of	the	hole	in	the bottle	indicates	that	space	is	unable	to	contain	the	bottle	the	way	ordinary	objects appear	containable.	We	know	that	if	the	Kleinian	"object"	is	properly	to	be	closed, assembled	without	merely	tearing	a	hole	in	it,	an	"added	dimension"	is	needed. Thus,	for	the	Klein	bottle	to	be	accommodated,	it	seems	the	three-dimensional continuum	itself	must	in	some	way	be	opened	up,	its	continuity	opened	to	challenge. Of	course,	we	could	attempt	to	sidestep	the	challenge	by	a	continuity-maintaining act	of	abstraction,	as	in	the	standard	mathematical	analysis	of	the	Klein	bottle. Assuming	we	do	not	employ	this	stratagem,	what	conclusion	are	we	led	to	regarding the	"higher"	dimension	that	is	required	for	completing	the	Klein	bottle?	If	it	is	not	an extensive	continuum,	what	sort	of	space	or	dimension	is	it? 11 The	question	can	be	addressed	by	looking	again	at	Figure	3.	At	first	glance, we	see	a	schematic	line	drawing	of	a	Klein	bottle	contained	within	the	twodimensional	frame	of	the	illustration.	But	when	we	consider	the	actual	content	of the	diagram,	we	see	that,	instead	of	showing	the	Klein	bottle	as	an	object	enclosed	in space,	the	diagram	portrays	space	("part	containing")	as	an	aspect	of	the	Klein bottle!	What	is	symbolized	schematically	in	Figure	3	is	a	dimension	of	a	Kleinian kind,	one	in	which	space	itself	is	but	one	of	three	ontological	terms	all	of	which	flow together	rather	than	being	split	off	from	one	another	as	happens	with	the	Cartesian paradigm	of	object-in-space-before-subject.	It	is	clear	from	this	that	the	Kleinian dimension	is	not	physical	but	psychophysical,	for	here	subject	and	object,	psyche and	physical	matter,	are	mediated	by	a	space	in	which	they	permeate	each	other.	So I	am	proposing	that,	when	the	Klein	bottle	is	regarded	in	concrete	terms	rather	than as	a	mathematical	abstraction,	we	actually	cannot	complete	its	construction	as	an object	in	a	spatial	continuum	but	are	obliged	to	reformulate	dimensionality	itself along	Kleinian	lines.	Note	that,	in	this	reformulation,	the	hole	in	the	Klein	bottle	is neither	lightly	dismissed	in	the	interest	of	maintaining	continuity,	nor	does continuity	give	way	to	sheer	discontinuity.	Instead,	the	very	splitting	of	continuity and	discontinuity	is	called	into	question	and	we	entertain	the	proposition	that,	in completing	the	Klein	bottle,	in	making	it	whole,	it	does	not	lose	its	hole.	Grasped	in this	dialectical	way,	we	may	say	in	fact	that	the	Klein	bottle	constitutes	a	"(w)hole,"	a paradoxical	structure	in	which	continuity	and	discontinuity	are	entwined. Now,	recall	our	association	of	the	Klein	bottle	with	the	microphysical spinning	action	that	lies	at	the	heart	of	quantum	mechanics.	This	points	to	the dynamic	character	of	the	Kleinian	configuration	when	it	is	understood	in	the microworld	context.	Here,	more	than	a	fixed	structure,	it	is	an	action,	a psychophysical	atom	of	process.	And	when	we	remember	the	correlation	of	Kleinian topology	with	the	hypernumber	ε,	what	comes	to	mind	are	the	archetypal	numbers of	the	Ho-t'u	described	by	von	Franz.4	For,	as	noted	earlier,	such	numbers	are "psychophysical	patterns	of	motion"	(1974,	74)	and	each	number	is	"thought	of	as containing	a	specific	activity	that	streams	forth	like	a	field	of	force"	(75).	But	how specific	is	the	correspondence	between	the	dynamic	structure	of	the	Ho-t'u	number field	and	that	of	the	Klein	bottle? In	the	previous	section	we	found	that	the	Ho-t'u	consists	of	a	double	cycle	of numbers,	with	each	number	embodying	a	rhythmic	action	unto	itself,	an	internal "phase	of	transformation,"	as	von	Franz	termed	it	(1974,	42).	The	first	cycle contains	four	such	phases.	This	is	followed	by	a	return	to	center	(marked	by	the number	5	in	Fig.	1),	after	which	we	enter	the	second	cycle	of	four	phases.	Here	the forward	or	"clockwise"	orientation	of	the	number	rhythms	in	cycle	1	has	been reversed,	with	phases	now	being	characterized	as	having	a	backward	or	retrograde action.	And,	whereas	the	"first	four	inner	rhythms	are	physical...the	second	four	are psychic"	(footnote	16,	243–244).	Are	we	to	take	these	two	cycles	as	related	in	a merely	sequential	fashion?	When	the	second	cycle	is	enacted	is	the	first	cycle	simply 4	We	know	that	the	Ho-t'u	plays	a	crucial	role	in	the	I	Ching.	Jungian	theorist	Nathan	Schwartz-Salant (2017)	relates	the	I	Ching	to	the	Klein	bottle	by	suggesting	that	the	latter	is	the	structure	that	best expresses	the	dynamic	balance	between	order	and	disorder	evidenced	in	the	former. 12 left	behind,	which	would	suggest	that	the	cycles	are	essentially	external	to	each other?	It	seems	that	von	Franz	cannot	be	suggesting	this	in	view	of	her	emphasis	on the	thoroughly	psychophysical	nature	of	the	Ho-t'u	number	field.	At	one	point	she asserts	that	"number	is	bound	up	with	the	latent	material	aspect	of	the	psyche	and with	the	latent	psychic	aspect	of	matter"	(157).	Somehow	then,	latent	within	the physical	circulation	of	the	Ho-t'u	(cycle	1)	is	an	aspect	of	the	psychical	circulation (cycle	2),	and	vice	versa.	Von	Franz	offers	an	enigmatic	clue	for	the	interconnection of	psyche	and	matter	in	her	discussion	of	a	related	mandala,	the	double	wheel	of Ezekiel,	wherein	two	wheels	are	depicted	as	intersecting	each	other	at	right	angles (see	Fig.	4). Figure	4.	Ezekiel's	wheels The	wheels,	says	von	Franz,	"do	not	work	in	unison	but	are	contiguous	at	the	center, which	is	a	technical	impossibility.	The	two	systems	are	incommensurable"	(262). Commenting	further,	she	notes	that	the	"mysterious	point	of	contact	between	the two	systems	appears	to	be	the	center	or	a	sort	of	pivot	where	psyche	and	matter meet"	(263).	But	the	iconic	image	of	Ezekiel's	wheels	does	not	in	itself	effectively indicate	the	mutual	latency	of	psyche	and	matter,	since	the	wheels	are	shown	as separate	except	for	the	aforementioned	point	of	contact.	If	the	images	of	the	Ho-t'u (Fig.	1)	and	Ezekiel's	wheels	(Fig.	4)	express	the	intimate	relationship	between psyche	and	matter	in	merely	abstract	symbolic	terms,	is	there	a	deeper,	more concretely	embodied	way	of	delivering	this	conjunction? I	submit	that	the	hidden topology	of	the	Ho-t'u	is	none	other	than	that	of	the	Klein	bottle.	To	confirm	this,	we begin	by	working	with	the	lower-dimensional	counterpart	of	the	Klein	bottle:	the Moebius	strip. Let	us	compare	the	curiously	configured	Moebius	with	its	more	conventional analogue,	a	cylindrical	ring. 13 Figure	5.	Cylindrical	ring	(a)	and	Moebius	strip	(b) The	ordinary	ring	(Fig.	5a)	is	constructed	by	cutting	out	a	narrow	strip	of	paper	and joining	the	ends.	To	produce	the	surface	of	Moebius	(Fig.	5b),	give	one	end	of	the paper	strip	a	half	twist	(through	an	angle	of	180°)	before	linking	it	with	the	other. Now,	imagine	yourself	circling	about	the	cylindrical	ring.	Positioned	on	the	inner surface	of	this	two-sided	structure,	you	move	360°	around	to	complete	one revolution,	returning	to	your	point	of	departure.	Naturally,	your	passage	around	the inside	of	the	ring	never	takes	you	to	the	outside.	Throughout	the	journey,	you remain	on	the	side	on	which	you	began. It	is	not	like	this	with	movement	around	the	one-sided	Moebius	surface. While	360°	of	revolution	do	seem	to	bring	you	back	to	your	point	of	origin,	at	the same	time,	it	is	not	your	point	of	origin,	since	you	are	now	on	the	other	side	of	the surface.	Notice	the	way	ordinary	circular	revolution	maintains	the	simple	dichotomy between	point	of	origin	and	displacement	from	that	point:	by	180°	of	movement	on the	cylindrical	ring,	you	are	furthest	removed	from	where	you	began,	and	by	360°, you	are	back	where	you	started,	the	displacement	being	simply	and	completely reversed.	On	the	other	hand,	with	360°	of	Moebius	revolution,	there	is	a	circling back	to	the	point	of	origin	that	is	at	once	the	point	most	remote	from	said	origin, since-when	sidedness	is	taken	into	account-it	is	the	point	on	the	other	side	of	the surface	that	is	diametrically	opposed	to	the	starting	point.	What	happens	if	you continue	revolving	beyond	this	360°	point	of	quasi-return?	After	an	additional	360°, you	find	that	you	have	now	truly	returned,	since	you	have	come	back	to	your	point of	departure	on	the	same	side	of	the	surface	from	which	you	first	set	out. Consider	the	general	resemblance	of	Moebius	action	to	the	action	pattern	of the	Ho-t'u.	Both	are	double	cycles	involving	an	initial	return	to	origin	(the	phase-5 return	to	center,	in	the	case	of	the	Ho-t'u;	see	Fig.	1),	followed	by	a	second	circuit that	repeats	the	process,	now	completing	it	in	earnest	with	a	second	return	to	origin (Ho-t'u	phase	10).	However,	instead	of	showing	the	Ho-t'u	as	consisting	simply	of separate	cycles	connected	by	a	separate	thus	external	inter-cyclical	linking	phase (phase	5),	the	one-sidedness	of	the	Moebius	superimposes	the	cycles	upon	each other,	concretely	conveying	the	paradox	that	these	two	circulations-the	physical and	the	psychical-also	are	one.	We	can	say	that	the	cycles	of	the	Moebius	are	so intimately	entwined	that	the	transition	to	cycle	2,	the	new	cycle,	is	just	as	much	a literal	repetition	of	the	same	cycle. 14 A	related	feature	of	the	Moebius	strip	further	confirms	its	intimate correspondence	with	the	Ho-t'u.	For	the	latter,	we	have	found	that,	whereas	cycle	1 entails	clockwise	movement,	cycle	2	is	counterclockwise.	A	shift	in	clock	sense	is just	what	takes	place	in	revolving	around	the	Moebius,	though	the	manner	of transformation	is	more	subtle	than	a	simple	association	of	one	cycle	with	clockwise action	and	the	other	with	counterclockwise	action.	To	demonstrate	this	property,	I incorporate	a	test	body	into	the	model:	a	face	in	profile	(Fig.	6). Figure	6.	Revolution	of	left-facing	profile	on	cylindrical	ring	(a)	and	on	Moebius	strip	(b) Figure	6a	shows	a	left-facing	profile	revolving	in	a	counterclockwise direction	around	a	cylindrical	ring	(see	arrows). It	is	clear	that	action	on	the	simple ring	will	continue	indefinitely	in	this	manner,	with	the	orientation	of	the	profile never	changing	(though	the	half-face	is	turned	upside	down).	In	Figure	6b,	we	see the	profile	moving	counterclockwise	about	the	Moebius	strip. Entering	the	twist, the	left-facing	form	is	changed	into	its	right-facing	counterpart,	the	transformation being	completed	after	360°	have	been	traversed.	The	transformation	of	left	into right	therefore	coincides	with	the	occurrence	of	one	full	cycle	of	Moebius	action. This	change	in	orientation	can	be	seen	to	reflect	a	change	in	clock	sense,	for,	what	is counterclockwise	to	a	left-facing	profile	will	be	clockwise	to	one	that	faces	right. Thus,	like	the	action	of	the	Ho-t'u,	Moebius	action	involves	a	reversal	of	clock	sense. However,	the	structure	of	the	Moebius	reveals	a	process	of	orientation reversal	that	is	more	nuanced	than	what	von	Franz's	text	indicates	for	the	Ho-t'u. With	the	Moebius,	it	is	obvious	that	cycle	1	action	is	not	uniformly	clockwise	from beginning	to	end,	this	being	followed	by	a	shift	in	gears	that	gives	a	second,	purely 15 counterclockwise	cycle.	Rather,	the	clockwise	orientation	of	cycle	1	is	fully	realized only	at	the	end	of	the	cycle	as	the	culmination	of	an	ongoing	transformation	from	an initial	counterclockwise	orientation.	Then,	in	entering	cycle	2,	the	direction	of	the gear-shift	has	itself	shifted	so	that	the	momentum	is	now	from	clockwise	to counterclockwise.	The	fact	that	change	in	clock	sense	is	inherent	to	Moebius	action means	that	no	separate	switching	of	gears	is	required	to	bring	it	about.	The	gears are	shifting	throughout	each	cycle,	with	the	quasi-return	to	origin	marking	the completion	of	the	shift	in	one	direction,	and	the	readiness	to	begin	shifting	in	the other.	The	broader	implications	of	these	transformations	of	clock	sense	will	soon	be discussed. Now,	while	the	Moebius	strip	does	generally	provide	us	with	an appropriately	dynamic	and	paradoxical	model	of	the	Ho-t'u	double	cycle,	the specific	phase	structure	of	the	Moebius	does	not	meet	the	requirements	of	the	Hot'u. This	observation	ties	in	with	the	proposition	I	offered	above:	it	is	the	Klein bottle	that	is	required	for	the	full	embodiment	of	the	Ho-t'u	archetypal	field. What we	will	eventually	see	is	that	the	lower-dimensional	Moebius	structure	properly embodies	a	lower-order	archetypal	field. The	limitation	of	the	Moebius	with	respect	to	the	Ho-t'u	can	readily	be demonstrated.	Again,	a	Moebius	strip	is	produced	simply	by	twisting	one	end	of	a narrow	band	of	paper	before	joining	it	to	the	other	end.	If	the	continuously	curved structure	thus	created	(shown	in	Figs.	5b	and	6b)	is	flattened	out,	we	obtain	the triangular	version	of	the	Moebius	illustrated	in	Figure	7. Figure	7.	Flattened	Moebius	strip The	flattening	of	the	Moebius	band	gives	its	quantized	structure.	Instead	of each	cycle	consisting	of	an	unbroken	revolution	through	360°,	now	each	is composed	of	distinct	phases	set	off	from	one	another	by	the	creation	of	edges.	For full	correspondence	to	the	Ho-t'u,	we	require	two	cycles	of	four	phases	each.	What we	have	instead	is	a	structure	whose	two	cycles	each	consist	of	only	three	phases. To	bring	out	the	phase	structure	of	the	Klein	bottle,	we	begin	with	a comparison	(see	Fig.	8	and	Fig.	9). 16 Figure	8.	Construction	of	torus	(upper	row)	and	Klein	bottle	(lower	row) Both	rows	of	Figure	8	depict	the	progressive	closing	of	a	tubular	surface	that initially	is	open.	In	the	upper	row,	the	end	circles	of	the	tube	are	joined	in	the conventional	way,	brought	together	through	the	three-dimensional	space	outside the	body	of	the	tube	to	produce	a	torus.	By	contrast,	the	end	circles	in	the	lower	row are	superimposed	from	inside	the	body	of	the	tube,	an	operation	requiring	the	tube to	pass	through	itself.	This	results	in	the	formation	of	the	Klein	bottle. By	comparing	Figure	8	with	Figure	9,	we	start	to	see	how	a	quantized	version of	the	Klein	bottle	can	be	produced. Figure	9.	Construction	of	flat	torus	(after	Barr	1964,	23)	(©	Courtesy	of	Dover	Publications) In	Figure	9,	Barr	shows	how	a	long	paper	cylinder	(pictured	at	the	top)	can	be flattened	out	along	the	dashed	line,	then	folded	over	so	that	the	ends	are	facing	each other.	The	ends	can	then	be	glued	together	to	produce	a	flat	version	of	the	torus. 17 Figure	10.	Construction	of	flat	Klein	bottle	(adapted	from	Barr	1964,	23) Figure	10	illustrates	the	creation	of	a	flat	Klein	bottle.	Once	again	the	paper cylinder	is	flattened	but,	this	time,	we	cut	a	slit	near	the	left	end	of	the	cylinder, indicated	in	the	diagram	by	the	dashed	line	(see	middle	step	of	Fig.	10).	To	complete the	construction,	we	insert	the	right	end	into	the	slit,	drawing	it	through	so	that	it can	be	brought	together	with	the	left	end	from	inside	the	body	of	the	tube,	similar	to what	happens	with	the	unflattened	Klein	bottle	of	Figure	8	when	its	end	circles	are superimposed	via	self-intersection.	(The	end	to	be	inserted	in	the	slit	can	be narrowed	to	allow	it	to	fit.) By	flattening	and	folding	the	paper	cylinder	as	we	have,	eight	distinct surfaces	have	been	created.	These	surfaces,	distinguished	from	one	another	by	the edges	created	in	the	folding,	give	the	phases	of	the	quantized	Klein	bottle,	which correspond	in	their	number	to	the	trigram	phases	of	the	Ho-t'u.	But	the	one-sided Kleinian	configuration	is	structured	more	subtly	than	its	two-sided	toroidal counterpart,	whose	four	inside	surfaces	are	categorically	separated	from	the	four surfaces	on	the	outside.	Let	us	attempt	to	clarify	the	Kleinian	structure	by	opening up	the	folded	paper	model	(Fig.	11). 18 Figure 11. Opened Klein bottle. The folded paper model is opened up to expose its eight subsurfaces. Dashed lines indicate the placement of the slit connecting the two sides of the bottle. Accompanying the	Ho-t'u	number	values	are the	correlated	compass	directions	and trigram	names given	in	the	I	Ching	(See	Fig.	2). The	movement	from	one	surface	to	another	in	cycle	1	marks	the	passage from	inside	to	outside.	These	quantized	transitions	are	shown	as	following	the pattern	of	the	Ho-t'u	cross.	With	the	Ho-t'u	(Fig.	2),	action	is	directed	from	below (North;	K'un)	to	above	(South;	Ch'ien),	thereby	establishing	the	cross'	vertical	axis. Then,	after	moving	down	and	to	the	left	(East;	Li),	there	is	a	transition	to	the	right (West;	K'an)	that	lays	down	the	horizontal	axis,	a	movement	that	intersects	the vertical.	The	self-intersection	of	the	Ho-t'u	cross	that	climaxes	in	passing	from	the third	to	fourth	stages	of	action	is	mirrored	on	the	Klein	bottle.	Its	first	three	phases are	followed	by	a	fourth	that	brings	its	self-intersection,	as	indicated	by	the	dashed line	signifying	the	slit	where	the	surface	passes	through	itself.5	And	in	moving through	the	slit,	we	"return	to	center"	(phase	5),	whereupon	we	find	ourselves ready	to	begin	again	in	cycle	2	with	phase	6,	though	we	are	now	beginning	on	the outside	of	the	Klein	bottle.	Note	that,	unlike	the	other	phases	of	cycle	1,	phase	5	is associated	neither	with	a	compass	direction	nor	a	trigram.	The	fifth	phase	merely marks	the	completion	of	cycle	1	by	returning	us	to	our	point	of	departure,	though now	on	the	opposite	side	of	the	surface,	and	it	is	this	quasi-return	that	prepares	us for	initiating	the	phase	actions	of	cycle	2. I	want	to	emphasize	that	the	Klein	bottle	is	indeed	one-sided.	This	means	that inside	and	outside	are	not	strictly	separated	from	each	other	as	with	the	two-sided 5	Along	with	the	cross,	the	pelican	came	to	symbolize	Christianity	in	medieval	Europe.	Like	the	cross, the	pelican	was	seen	as	a	symbol	of	self-sacrifice,	for	the	mother	of	this	species	was	believed	capable of	feeding	her	young	with	her	own	blood	by	pressing	her	bill	into	her	chest.	Elsewhere	I	explored	the symbolic	relationship	between	the	self-intersecting	pelican	and	the	Klein	bottle	(see	Rosen	1995, 2014). 19 torus	but	flow	into	each	other	in	accordance	with	the	bottle's	own	internal structure.	So	cycle	1	does	not	consist	of	transitions	between	sub-surfaces	all	of which	simply	lie	on	the	outside	of	the	Klein	bottle.	Instead,	movement	from	surface to	surface	entails	a	progressive	passage	from	inside	to	outside	that	takes	place	quite naturally,	reaching	its	culmination	with	the	phase-5	completion	of	cycle	1.	It	is perhaps	easier	to	see	the	effect	of	one-sidedness	with	the	continuous	action	on	the Moebius	strip	shown	in	Figures	5b	and	6b.	Starting	from	any	point	on	the	Moebius,	a revolution	of	360°	will	bring	us	back	to	the	place	where	we	began	(we	"return	to center")	but	on	the	opposite	side	of	the	strip. We	have	observed	as	well	that	Moebius	revolution	changes	clock	sense, transforming	counterclockwise	action	into	clockwise	action	and	vice	versa	(leftright	orientation	is	correspondingly	transformed;	see	Fig.	6b).	The	same	applies	to action	on	the	Klein	bottle.	However,	even	though	clock	sense	changes	in	cycle	1,	we can	still	associate	the	overall	direction	of	movement	in	this	cycle	with	clockwise, given	that	the	momentum	is	shifting	throughout	from	an	initial	counterclockwise spinning	to	a	clockwise	one.	Cycle	1	can	thus	be	said	to	involve	the	generation	of clockwise	action.	In	relating	the	quantized	Klein	bottle	to	the	I	Ching,	we	identify	the four	specific	phases	of	the	cycle-1	shift	to	clockwise	as	the	cardinal	compass directions	of	the	Ho-t'u:	North,	South,	East,	and	West. Cycle	2	reverses	the	momentum	established	in	cycle	1.	Beginning	with	phase 6,	our	trajectory	now	takes	us	from	outside	to	inside	and	from	clockwise	to counterclockwise.	In	view	of	the	way	clock	sense	shifts	over	the	course	of	this	cycle, we	can	say	that	the	cycle	involves	the	generation	of	counterclockwise	action. Relating	this	to	the	I	Ching,	the	four	phases	of	cycle	2	are	identified	as	the intercardinal	Ho-t'u	directions:	Northwest,	Southeast,	Northeast,	and	Southwest. (Note,	however,	that,	in	section	8,	an	alternative	interpretation	of	the	phases	of	cycle 2	is	offered	that	departs	from	I	Ching	tradition.) Before	going	any	further,	I	want	to	clarify	the	nature	of	the	rotational	action we	are	dealing	with.	First	of	all,	Figure	6	shows	the	relative	simplicity	of	cylindrical action	(6a)	when	compared	with	action	on	the	Moebius	(6b).	In	the	former,	the	leftfacing	profile	merely	revolves	around	the	length	of	the	ring	whereas,	in	the	latter,	it also	rotates	about	its	own	axis.	The	same	combination	of	rotational	components (longitudinal	and	transverse)	is	found	with	Kleinian	rotation.	It	is	the	transverse component	of	this	topological	action	that	transforms	left-right	orientation	and	clock sense.	In	the	case	of	the	action	portrayed	in	Figure	11,	jumping	from	one	numbered phase	surface	to	another	around	the	cross	is	the	quantized	counterpart	of continuous	longitudinal	revolution	on	the	Klein	bottle.	The	corresponding	Ho-t'u diagram	shown	only	in	terms	of	its	numerical	sequence	(Fig.	2,	right)	shows	only this	longitudinal	component.	But	when	we	consider	the	accompanying	trigrams	and their	compass	directions	(Fig.	2,	left),	the	transverse	component	of	Kleinian	action becomes	implicated,	with	the	four	directions	symbolizing	four	discrete	states	in	the transition	from	counterclockwise	to	clockwise.	Thus,	coming	to	understand	the hidden	topology	of	the	Ho-t'u	helps	us	see	more	clearly	how	the	Ho-t'u	diagram	per se	and	the	pulsating	trigrams	of	the	I	Ching	are	aspects	of	the	same	dynamic structure. 20 What	are	the	broader	implications	of	all	this?	In	the	present	section,	I	have employed	topology	to	relate	the	microworld	action	described	in	contemporary physics	to	the	action	given	in	ancient	China's	Ho-t'u	number	archetype.	Although	I have	presented	tangible	paper	models	to	detail	the	manner	in	which	this	action takes	place,	the	action	itself	in	fact	cannot	be	realized	as	an	objective	event	in physical	space	because	it	is	essentially	psychophysical.	And	as	an	archetypal	action (the	prefix	archaeoderives	from	the	Greek	arché,	the	beginning,	the	first),	we	may take	it	as	the	pre-objective	antecedent	from	which	the	experience	of	objective physical	reality	first	arises.	This	is	consistent	with	the	viewpoint	expressed	by	von Franz. Above	I	noted	von	Franz's	comments	that	number	archetypes	like	the	Ho-t'u are	"cosmic	plans"	(22),	and	that	"the	universe	was	'organized'	according	to numerical	patterns	of	this	kind"	(24).	Considering	also	that	the	"first	four	inner rhythms	[of	the	Ho-t'u]	are	physical"	(footnote	16,	243–244),	the	Kleinian generation	of	clockwise	action	in	cycle	1	can	be	related	to	the	creation	of	the objective	material	world.	Interesting	in	this	regard	is	that	the	first	cycle	of	the	Ho-t'u is	seen	as	"cumulative	and	expanding	as	time	goes	on"	(236).	In	cosmogonic	terms, this	can	be	associated	with	the	expansion	of	the	physical	universe.	Here	we	have "the	usual	clockwise	movement"	(236),	that	in	which	time	flows	forward	in	its natural	sequence	(Smith	et	al	1990,	118)	following	time's	arrow	of	increasing entropy	associated	with	cosmic	dilation. After	the	genesis	of	clockwise	action	from	an	initial	counterclockwise orientation	in	the	first	Ho-t'u	cycle,	the	momentum	shifts	back	to	counterclockwise in	the	second.	Interpreted	cosmogonically,	this	is	the	backward	or	retrograde	cycle wherein	preoccupation	with	the	physical	universe	projected	"out	there"	in	cycle	1	is now	overcome	and	an	inward	awareness	of	the	psyche	is	regained-"the	second four	[inner	rhythms]	are	psychic,"	says	von	Franz.	But	it	is	not	that	psyche	is	favored one-sidedly	in	this	cycle.	Instead,	the	psychophysical	nature	of	the	cosmos	is	realized in	full.	What	had	been	projected	as	purely	physical	or	objective	is	thus	presently drawn	back	in	to	disclose	the	archetypal	source	of	said	projection,	a	source	that	is itself	more	primordial	than	the	splitting	of	matter	and	psyche,	object	and	subject.	In drawing	inward,	the	retrograde	action	of	cycle	2	reverses	the	outthrust	of	the expanding	universe	to	bring	"an	opposite,	backward	movement,	folding	up	and contracting	as	time	goes	by"	(Wilhelm	quoted	in	von	Franz,	236–237).	Smith	et	al, citing	a	passage	from	the	Shuo-Kua	of	the	I	Ching,	put	it	this	way:	"'going	along	with ...	[time's	ordinary]	sequence'	indicates	that	one	comes	to	know	things	in	the	order that	they	occur	naturally...[whereas]	'going	against	the	direction	of	time,'	means reaching	with	the	mind	into	the	future,	against	the	flow,	and	knowing	things	before the	stream	of	time	has	brought	them	to	us"	(1990,	118). I	acknowledge	the	speculative	nature	of	this	take	on	cosmic	evolution.	In	the pages	to	follow,	I	will	elaborate	on	my	approach,	clarifying	it,	and	providing	further substantiation.	This	will	involve	reprising	my	demonstration	in	The	Self-Evolving Cosmos	(2008a)	that	an	effective	rendering	of	cosmogony	and	the	related	issue	of quantum	gravity	requires	us	to	recognize	that	the	Klein	bottle	is	actually	but	one member	of	a	whole	family	of	interrelated	topological	structures.	In	bringing	this topological	family	into	play,	we	will	engage	with	a	corresponding	family	of	Taoist 21 structures	that	goes	beyond	the	system	of	trigrams.	In	this	way,	the	link	between modern	Western	cosmology	and	ancient	Eastern	cosmology	will	be	brought	into sharper	focus. 3.	QUANTUM	GRAVITY	AND	STRING	THEORY I	opened	this	paper	by	noting	the	lack	of	progress	in	theoretical	physics	toward	an account	that	would	unify	the	four	fundamental	forces	of	nature.	We	found	that reconciling	the	quantum	mechanical	forces	with	gravitation	requires	a	theory	that can	work	effectively	around	the	sub-microscopic	Planck	length	of	10–35	m. Indispensable	to	such	a	theory	is	!,	the	quantum	of	action.	This	indivisible	atom	of process	constitutes	the	core	of	quantum	mechanics	and	can	be	successfully employed	in	relatively	low	energy	and	low	magnification	regimes,	but	dealing	with it	below	the	Planck	length	involves	energies	so	tumultuous	that	the	quantum properties	of	discontinuity	and	observer-observed	interaction	can	no	longer	be managed	via	probabilistic	analysis. It	was	in	the	1970s,	following	the	progress	achieved	in	unifying	the	quantum mechanical	forces	(strong,	weak,	and	electromagnetic),	that	work	on	a	theory	of quantum	gravity	began	in	earnest.	And	this	is	when	confrontation	with	the	chaos	of the	Planck	realm	could	no	longer	be	avoided.	The	equations	that	would	unify	the four	forces	were	now	completely	unable	to	contain	the	wildly	fluctuating	Planckian energies,	as	manifested	by	the	infinite	probabilities	that	turned	up	to	render	those equations	useless.	Movement	toward	an	effective	theory	of	quantum	gravity	has thus	come	to	a	standstill	over	the	past	forty	years.	Commenting	on	the	impasse, Smolin	notes	that	advances	have	been	blocked	"despite	our	best	efforts"	(2006,	viii). What	"best	efforts"	is	he	referring	to? Since	the	1970s,	the	quest	for	a	mathematical	unification	of	nature	has largely	been	dominated	by	an	approach	known	as	string	theory.	In	this	endeavor,	the attempt	is	made	to	avoid	probing	below	the	Planck	threshold	simply	by	assuming that	the	smallest	constituents	of	nature	are	not	indefinitely	miniscule	point-particles as	previous	theory	had	assumed,	but	string-like	vibrating	elements	of	finite extension	conveniently	scaled	at	the	Planck	length.	It	is	because	this	stratagem	has been	able	to	eliminate	infinite	terms	from	quantum	gravitational	equations	that	it has	become	the	preferred	approach.	But	the	price	paid	for	this	positivistic	ploy	has come	to	be	acknowledged	(Smolin	2006,	Woit	2006).	In	my	own	explorations	of	the matter	(Rosen	2004,	2008a,	2008b,	2015),	I	have	identified	two	problems	with string	theory. First,	while	it	is	true	that	string	theory	serves	the	classical	ontology	by sidestepping	sub-Planckian	ambiguity,	an	epistemic	ambiguity	takes	its	place.	String theory's	general	equations	may	be	free	of	unmanageable	infinities,	but	theorists must	be	able	to	solve	these	highly	abstract	equations	in	a	manner	that	produces	a specific	description	of	the	world	as	we	know	it.	As	things	now	stand,	the	equations yield	a	vast	array	of	possible	solutions	with	no	guiding	principle	by	means	of	which the	field	can	be	narrowed	in	unique	correspondence	with	known	physical	reality.	A second	limitation	of	the	theory	is	that	it	seems	to	contradict	itself	in	its	assumption 22 of	fundamental	particles	with	finite	extension.	"Strings	are	truly	fundamental,"	says physicist	Brian	Greene,	"they	are	'atoms,'	uncuttable	constituents"	of	nature.	So, "even	though	strings	have	spatial	extent,	the	question	of	their	composition	is without	any	content"	(1999,	141).	But	isn't	this	a	contradiction?	For-at	least according	to	the	classical	concept	of	the	continuum	not	explicitly	challenged	by string	theory,	to	be	spatially	extended	is	to	be	cuttable,	in	fact,	infinitely	divisible. How	then	could	a	string	be	a	fundamental	particle,	an	atomic	or	indivisible ingredient	of	nature,	when	it	is	spatially	extended?	In	sum,	string	theory	is	both ambiguous	and	it	appears	to	contradict	itself	when	seen	in	classical	terms. But	I	suggest	that	despite	the	limitations	of	conventional	string	theory,	if	we take	the	vibratory	pattern	of	the	fundamental	strings	as	essentially	Kleinian	in nature-with	Kleinian	spin	not	objectified	but	understood	phenomenologically,	in psychophysical	terms	that	bridge	the	gap	between	subject	and	object-string	theory can	gain	greater	coherence.	In	fact,	I	demonstrate	in	Cosmos	that	by	reformulating the	theory	in	the	context	of	the	psychophysical	discipline	I	call	topological phenomenology,	it	can	be	cast	in	a	form	that	provides	a	definitive	(albeit	qualitative) account	of	quantum	gravity,	one	that	unambiguously	yields	the	fundamental particles	of	the	standard	model.	We	will	explore	the	basis	for	this	conclusion	and review	its	details	in	coming	sections.	In	the	course	of	doing	this,	the	work	of	Cosmos will	be	carried	forward	and	the	ancient	Chinese	roots	of	phenomenological	string theory	will	be	brought	to	light. 4.	THE	TOPODIMENSIONAL	FAMILY We	know	that	the	primary	element	of	action	in	microphysics	is	!,	the	quantum	of action	associated	with	the	emission	of	radiant	energy.	We	have	found	as	well	that the	curious	spinning	accompanying	this	quantized	dynamic	is	modeled	by	the hypernumber	ε,	and	that	the	geometric	counterpart	of	ε	is	the	Klein	bottle.	In	the form	of	ε!,	the	Klein	bottle	is	thus	seen	to	implicitly	embody	the	angular	action	that lies	at	the	heart	of	quantum	mechanics.	Bearing	in	mind	that	the	Klein	bottle	cannot be	properly	understood	as	a	Cartesian	object	in	physical	space	but	must	be	grasped as	inherently	psychophysical,	I	propose	we	view	Kleinian	spin	as	the	basic	building block	of	a	psychophysical	account	of	quantum	gravity. In	Pauli's	matrices,	!/2	is	taken	as	the	fundamental	unit	of	electron	spin.	In fact,	!/2	is	the	basis	for	determining	the	spin	of	all	subatomic	particles,	fermions and	bosons	alike.	Given	the	essential	role	played	by	spin	in	quantum	mechanics	and the	underlying	significance	of	the	Klein	bottle	in	said	spin,	I	suggested	in	Cosmos	that all	microworld	dynamics	arise	from	spin	of	the	Kleinian	kind:	ε!/2. Now,	in	his	further	exploration	of	the	hypernumber	ε,	Musès	indicated	a "higher	epsilon-algebra"	wherein	"√εn	involves	in,	the	subscripts	of	course	referring to	the	(n	+	1)th	dimension	since	i≡i1	already	refers	to	D2"	(1968,	42).	In	view	of	the intimate	relationship	between	ε	and	the	Klein	bottle,	can	Musès'	implication	of	a dimensional	hierarchy	of	hypernumber	values	be	given	topological	expression?	The Klein	bottle	does	lend	itself	to	such	a	generalization. 23 Mathematicians	have	investigated	the	transformations	that	result	from bisecting	topological	surfaces.	If	the	Klein	bottle	is	bisected,	cut	down	the	middle,	it will	fall	into	a	pair	of	oppositely-oriented	Moebius	strips.	Next,	bisecting	the	onesided	Moebius	strip,	a	two-sided	lemniscatory	surface	will	be	produced,	its	sides being	related	enantiomorphically	(i.e.,	as	mirror	opposites).	Finally,	cutting	the lemniscate	down	the	middle	yields	interlocking	lemniscates.	The	transformation brought	about	by	this	bisection	is	clearly	the	last	one	of	any	significance,	since additional	bisections-being	bisections	of	lemniscates,	can	only	produce	the	same result:	interlocking	lemniscates.	The	bisection	series	is	completed	then	when	we obtain	interlocking	lemniscates,	a	structure	termed	the	sub-lemniscate.	By experimenting	with	the	bisection	of	the	Klein	bottle	in	this	way,	a	family	of	four nested	topological	structures	is	discovered	(Fig.	12). Figure	12.	Topological	bisection	series.	From	top	to	bottom:	Klein	bottle,	Moebius	strip,	lemniscate, sub-lemniscate In	Cosmos,	dimensional	differences	among	the	four	members	of	the	bisection series	are	studied	phenomenologically.	While	to	ordinary	observation	each	member appears	as	but	a	two-dimensional	surface	in	three-dimensional	space, phenomenological	reflection	leads	to	the	insight	that	each	actually	constitutes	a dimensional	lifeworld	unto	itself	("lifeworld"	is	a	phenomenological	term	for	the non-classical,	intrinsically	psychophysical	realm	wherein	subject	and	object	are intimately	entwined).	Whereas	the	Klein	bottle	is	three-dimensional,	its	nested correlates	are	of	progressively	lower	dimension:	the	Moebius	is	two-dimensional, the	lemniscate	is	one-dimensional,	and	the	sub-lemniscate	is	zero-dimensional.	And, like	the	Klein	bottle,	each	of	these	lower-dimensional	worlds	is	a	psychophysical realm	of	action	that	surpasses	the	Cartesian	division	of	psyche	and	matter,	subject 24 and	object.	This	account	of	several	different	topodimensional	lifeworlds	embedded within	each	other	is	consistent	with	the	hierarchy	of	ε-like	spin	structures	suggested by	Musès. εD0 εD0/εD1 εD0/εD2 εD0/εD3 εD1/εD0 εD1 εD1/εD2 εD1/εD3 εD2/εD0 εD2/εD1 εD2 εD2/εD3 εD3/εD0 εD3/εD1 εD3/εD2 εD3 Table	1. Interrelational	matrix	of	topodimensional	spin	structures Table	1,	the	topodimensional	spin	matrix,	gives	the	ε-based	counterpart	of the	topological	bisection	series.	The	three-dimensional	Kleinian	spinor	is	written εD3,	with	lower-dimensional	members	of	the	tightly	knit	spin	family	designated	εD2, εD1,	and	εD0	(corresponding	to	the	Moebial,	lemniscatory,	and	sub-lemniscatory circulations,	respectively).	These	terms	are	arrayed	on	the	principal	diagonal	of	the matrix	(extending	from	upper	left	to	lower	right).	The	interrelationships	among	the four	principal	matrix	elements,	taken	two	at	a	time,	are	reflected	in	the	elements appearing	off	the	main	diagonal. Generally	speaking,	Table	1	unpacks	the	dialectical	structure	of topodimensional	interrelations.	In	keeping	with	the	"musical"	implications	of	string theory,	we	may	regard	topodimensional	action	as	inherently	vibratory	in	nature. The	principal	diagonal	of	the	Table	contains	a	dimensional	series	of	fundamental vibrations	or	tones,	and	these	four	principal	terms	are	coupled	to	each	other	two	at a	time	by	six	pairs	of	overtone-undertone	intervals	related	to	each	other	in	the mirror-opposed	fashion	of	enantiomorphs.	The	dimensional	overtone	ratios	are	the values	extending	below	the	fundamental	tones,	whereas	the	undertone	ratios	are the	values	appearing	to	the	right	of	the	fundamentals.	(In	Cosmos,	the topodimensional	action	matrix	is	seen	as	analogous	to	the	old	Pythagorean	table, which	is	portrayed	as	an	expanding	series	of	musical	intervals,	with	fundamental tones	on	the	principal	diagonal,	flanked	by	overtones	and	undertones.	For	more	on this,	see	section	5.) Consider	in	Table	1	the	two	principal	tones	of	highest	dimensionality:	εD2	and εD3.	These	matrix	elements	are	linked	by	the	overtone	and	undertone	given	in	the two	corresponding	non-principal	cells,	εD3/εD2	and	εD2/εD3	(respectively).	The enantiomorphically-related	coupling	cells	in	question	are	the	hypernumber counterparts	of	the	concretely	observable,	oppositely	oriented	Moebius	strips which,	when	glued	together,	form	the	Klein	bottle.	Taken	strictly	as	a	principal matrix	element,	the	hypernumber	Moebius	vibration	is	the	spin	structure	that constitutes	the	two-dimensional	lifeworld	(εD2).	But	when	we	shift	our	view	of	the Moebius,	consider	it	in	relation	to	higher,	Kleinian	dimensionality,	a	kind	of "doubling"	takes	place	in	which	the	εD2	singular	Moebius	spin	structure	becomes	a pair	of	asymmetric,	mirror	opposed	twins,	εD3/εD2	and	εD2/εD3.	It	is	through	the fusion	of	these	dimensional	enantiomorphs	that	Kleinian	dimensionality	is crystallized.	Since	the	Table-1	matrix	indicates	that	all	four	principal dimensionalities	or	fundamental	tones	are	interrelated	by	accompanying	off- 25 diagonal	overtone-undertone	pairs,	we	can	draw	the	general	conclusion	that	higher dimensions	emerge	through	processes	of	enantiomorphic	fusion	(this	will	be	spelled out	completely	in	due	course). The	process	of	dimensional	generation	can	be	clarified	in	broad	terms	by relating	it	to	a	reverse	movement	through	the	bisection	series	wherein	topological structures	are	not	divided	but	glued	together.	To	begin,	we	imagine	the	fusion	of interlocking	lemniscates	that	yields	the	single	lemniscate.	This	corresponds	to	the generation	of	the	one-dimensional	lifeworld	(εD1).	Next,	we	picture	the enantiomorphically-related	sides	of	the	two-sided	lemniscate	merging	to	form	the one-sided	Moebius	structure,	this	being	associated	with	the	genesis	of	the	twodimensional	lifeworld	(εD2).	Finally,	we	imagine	Moebius	enantiomorphs	fusing	to produce	the	Klein	bottle,	which	corresponds	to	the	evolution	of	our	threedimensional	lifeworld	(εD3).	With	each	fusion,	a	lower-dimensional	lifeworld	is absorbed	by	a	world	of	higher	dimension,	taken	into	it	in	such	a	way	that	the	lower dimension	is	concealed.	In	the	end,	we	have	three	lower-dimensional	vibratory structures	concealed	within	the	three-dimensional	Kleinian	vibration,	much	as lower	dimensions	are	hidden	by	becoming	"curled	up"	within	visible	3	+	1dimensional	space-time	in	the	conventional	string	theoretic	account	of	dimensional cosmogony.	It	turns	out,	in	fact,	that	the	phenomenological	approach	arrives	at	the same	total	number	of	dimensions	as	does	the	conventional	theory. What	I	demonstrate	in	Cosmos	is	that	the	Kleinian	spinor,	εD3,	is	not	itself	an extended	three-dimensional	space,	but	is	a	quantized	three-dimensional	blend	of space	and	time	that	first	gives	birth	to	our	familiar	3	+	1-dimensional	space-time.6	In like	manner,	the	two-dimensional	Moebius	spinor	(εD2)	spins	out	a	2	+	1dimensional	space-time,	the	lemniscatory	spinor	(εD1)	sends	forth	a	1	+	1dimensional	space-time,	and	the	sub-lemniscatory	spinor	(εD0)	projects	a	0	+	1dimensional	space-time.	A	simple	summation	of	projected	space-time	dimensions gives	us	a	total	of	ten,	with	the	six	lower	dimensions-(2	+	1)	+	(1	+	1)	+	(0	+	1)- being	hidden	within	the	larger	3	+	1-dimensional	space-time.	This	picture	of	overall ten-dimensionality,	with	six	dimensions	concealed,	accords	with	the	basic	account provided	by	orthodox	string	theory.	Thus	we	may	say	that	our	four phenomenological	spinors	spin	out	the	ten	space-time	dimensions	of	string	theory.	7 Yet	despite	the	general	agreement	on	the	number	of	space-time	dimensions, important	differences	exist	between	conventional	and	phenomenological interpretations	of	string	theory.	Mainstream	theorists	have	approached	cosmogony by	adopting	the	concept	of	symmetry	breaking.	In	this	narrative,	the	four	forces	of nature	are	conceived	as	vibrating	strings	that	initially	existed	in	a	purely	symmetric ten-dimensional	space	scaled	around	the	Planck	length.	Subsequently,	the	perfect primordial	symmetry	was	spontaneously	broken	by	a	dimensional	bifurcation	in which	four	of	the	original	dimensions	expanded	to	produce	the	visible	universe	we 6	The	Kleinian	spinor	is	a	"natal	space,"	a	"matrix	[for...]	existing	space,"	to	use	Merleau-Ponty's	way of	describing	the	generative	role	played	by	phenomenological	dimensionality	(1964,	176). 7	With	the	extension	of	string	theory	known	as	M-theory,	eleven	dimensions	are	actually	entailed, though	the	eleventh	dimension	is	not	like	the	other	ten.	This	"extra"	dimension	in	fact	may	be interpreted	in	phenomenological	terms.	See	The	Self-Evolving	Cosmos. 26 know	today,	with	the	other	dimensions	remaining	hidden.	Coupled	with	this	was	the breaking	of	force-field	symmetry	to	create	the	appearance	of	irreconcilable differences	among	the	forces. However,	while	the	foregoing	account	of	cosmogony	incorporates	both dimensional	and	force-field	symmetry	breaking,	the	two	are	not	precisely	aligned with	each	other	in	the	theoretical	reckoning.	This	reflects	the	fact	that	contemporary theorists	have	been	unable	to	articulate	a	complete	geometric	rendering	of	cosmic evolution.	For	the	geometric	program	fully	to	be	realized,	the	physical	events described	in	the	standard	and	inflationary	models	of	cosmic	development	would need	to	be	specifically	expressible	as	dimensional	events.	What	Heinz	Pagels	noted twenty	years	ago	in	discussing	the	extra-dimensional	(Kaluza-Klein)	interpretation of	cosmogony	remains	true	today:	"No	one	has	yet	been	able	to	find	a	realistic Kaluza-Klein	theory	which	yields	the	standard	model"	(1985,	328).	In	the	stringtheoretic	application	of	Kaluza-Klein	theory,	one	obvious	reason	for	this	limitation	is the	absence	of	a	conceptual	principle	that	could	guide	the	analyst	to	unambiguous solutions	of	the	ten-dimensional	general	equations,	solutions	specifying	the	exact shapes	of	the	hidden	dimensions	that	would	correspond	to	the	physical	facts	of	the standard	model.	Of	course,	if	the	prevailing	theory	cannot	tell	us	what	the dimensional	structures	are	that	correspond	to	physical	reality,	it	can	hardly	inform us	on	how	these	dimensions	develop.	In	point	of	fact,	there	is	really	no	positive feature	intrinsic	to	the	theory	that	provides	for	the	evolution	of	dimensions.	From what	I	can	tell,	the	only	reason	dimensional	bifurcation	is	assumed	to	have	taken place	at	all	is	that	theorists	must	somehow	account	for	the	present	inability	to observe	six	of	the	ten	dimensions	needed	for	a	consistent	rendering	of	quantum gravity	(one	that	avoids	untenable	probability	values). Smolin	seems	to	put	his	finger	on	the	underlying	problem	in	calling	attention to	the	"wrong	assumption"	physicists	"are	all	making"	when	they	present	the	"whole history	of	constant	motion	and	change...as	something	static	and	unchanging"	(2006, 256–57).	When	authentic	change	is	thus	denied,	it	is	not	surprising	that	no	natural, parsimonious	way	of	accounting	for	cosmogony	is	forthcoming.	Conventional	string theory	well	exemplifies	this	adherence	to	the	classical	intuition	of	changelessness	in the	primacy	it	gives	to	the	notion	of	symmetry.	It	is	in	assuming	an	initial	state	of "perfect	symmetry"	that	theorists	must	resort	to	the	artifice	of	"spontaneous symmetry	breaking,"	an	alleged	event	that-far	from	being	a	natural	consequence	of the	purely	symmetric	theory-is	gratuitously	invoked	without	a	compelling explanation	of	its	basis. The	inherent	dynamism	of	phenomenological	string	theory	affords	a	way	out of	the	impasse.	Instead	of	artificially	appending	asymmetry	to	a	primordially	perfect symmetry,	a	dialectic	of	symmetry	and	asymmetry	is	offered	that	permits	an unequivocal,	intrinsically	meaningful	account	of	the	evolving	forces	of	nature.	This principle	of	"synsymmetry"	(Rosen	1975,	1994,	2006,	2008a)	is	implicit	in	the topological	bisection	series	and	its	associated	topodimensional	spin	matrix	(Table 1). For	a	simple	illustration,	consider	the	Moebius	strip.	It	arises	from	the	fusion of	mirror-opposed,	asymmetrically-related	sides	of	the	lemniscate.	We	can	say	that, through	this	union	of	opposites,	the	asymmetry	of	lemniscatory	sides	is	rendered 27 symmetric.	However,	while	the	Moebius	can	be	deemed	symmetric	vis-à-vis	the fused	lemniscatory	sides	that	constitute	it,	at	the	same	time	it	is	itself	a	member	of an	enantiomorphically	asymmetric	pair	whose	own	fusion	produces	the	Klein	bottle. We	may	generally	conclude	that	the	members	of	our	topodimensional	family	are neither	simply	asymmetric	nor	simply	symmetric,	but	synsymmetric:	a	given member	combines	symmetry	and	asymmetry	in	such	a	way	that	it	is	symmetric	in relation	to	its	lower-dimensional	counterpart	and	asymmetric	in	relation	to	its higher	one	(the	sub-lemniscate	is	an	exception	to	this,	since	it	has	no	lowerdimensional	counterpart).	I	propose	that	the	synsymmetry	concept,	viewed dynamically	in	terms	of	enantiomorphic	fusion	events,	constitutes	a	guiding principle	for	cosmogony.	The	forces	and	particles	of	nature	evolve	by	a	general process	wherein	asymmetric	dimensional	enantiomorphs	fuse	to	create	a dimensional	symmetry	that	at	once	inherently	gives	way	to	new	asymmetry.	My topo-phenomenological	interpretation	of	cosmogony	is	elaborated	in	sections	6–9, where	a	lot	more	detail	is	given	on	dimensional	development,	cosmogony	by enantiomorphic	fusion,	and	quantum	gravity.	Indeed,	a	lot	more	detail	will	be needed. Recall	that,	in	the	course	of	examining	the	relationship	between	the	Klein bottle	and	the	Ho-t'u	number	archetype	in	section	2,	I	offered	a	preliminary	glimpse of	the	cosmogonic	implications	of	my	account.	Since	the	topodimensional explanation	of	cosmogony	just	given	does	not	address	those	implications completely,	it	will	be	necessary	for	me	to	expand	on	what	I	have	said	here.	Only when	further	detail	has	been	disclosed	on	the	specific	nature	of	cosmic	evolution will	we	be	able	to	fully	appreciate	the	connection	between	what	was	intimated	in section	2	and	what	has	been	set	forth	in	the	present	section.	And	this,	in	turn,	will allow	us	to	grasp	the	full	relationship	between	topodimensional	and	Taoist approaches	to	cosmogony.	However,	before	carrying	out	the	requisite	elaboration,	I want	to	demonstrate	that	the	topological	family	forming	the	core	of phenomenological	string	theory	has	its	counterpart	in	ancient	China. 5.	THE	WORLD'S	OLDEST	STRING	THEORY Music	has	long	...	provided	the	metaphors	of	choice	for	those	puzzling	over questions	of	cosmic	concern.	From	the	ancient	Pythagorean	"music	of	the spheres"	to	the	"harmonies	of	nature"	that	have	guided	inquiry	through	the ages,	we	have	collectively	sought	the	song	of	nature	in	the	gentle	wanderings	of celestial	bodies	and	the	riotous	fulminations	of	subatomic	particles.	With	the discovery	of	superstring	theory,	musical	metaphors	take	on	a	startling	reality, for	the	theory	suggests	that	the	microscopic	landscape	is	suffused	with	tiny strings	whose	vibrational	patterns	orchestrate	the	evolution	of	the	cosmos. (Greene	1999,	135) I	demonstrated	in	section	2	that	the	hidden	topology	of	ancient	China's	Ho-t'u number	archetype	is	Kleinian	in	nature.	In	the	previous	section,	we	found	that	the Klein	bottle	does	not	stand	alone	but	is	a	member	of	a	closely-knit	family	of	four topological	structures	constituting	the	core	of	phenomenological	string	theory. 28 What	we	are	presently	going	to	see	is	that	the	Ho-t'u	and	its	associated	trigram system	also	participate	in	a	larger	fourfold	grouping,	one	that	comprises	the	"string theory"	from	which	phenomenological	string	theory	can	be	said	to	have	originated. I	noted	earlier	that	the	vibrations	or	tones	of	the	topodimensional	action	matrix (Table	1)	bear	a	resemblance	to	the	tonal	system	set	forth	in	the	old	Pythagorean table.	This	table	is	also	known	as	the	Lambdoma	(since	it	is	often	shown	in	a configuration	suggestive	of	the	Greek	letter	lambda),	and	it	is	connected	to	the	idea of	the	"music	of	the	spheres."	The	first	four	intervals	of	the	Pythagorean	table	are displayed	in	Table	2. Table	2.	Section	of	the	Pythagorean	table The	table	is	usually	portrayed	as	an	indefinitely	expanding	series	of	musical intervals,	but	there	are	many	variations	of	it	in	the	literature,	with	some	excluding the	0/0	interval	entirely	or	placing	it	outside	the	body	of	the	table	(see,	for	example, Godwin	1989,	95).	What	is	shown	in	the	representation	of	the	table	I	have	employed is	a	set	of	relationships	that	mirrors	our	topodimensional	action	matrix	(Table	1). There	is	a	principal	diagonal	the	first	tone	of	which	is	0/0,	seen	in	the	upper	lefthand	corner;	this	corresponds	to	εD0,	the	zero-dimensional	term	of	Table	1.	I	will have	more	to	say	about	0/0	below.	The	0/0	fundamental	tone	is	followed	on	the main	diagonal	by	a	series	of	non-zero	fundamentals	constituting	reflexive,	selfdivisive	"forms	of	unity"	(Musès,	1968,	32).	That	is,	each	non-zero	fundamental	can be	taken	as	a	self-division	that	by	ordinary	arithmetic	would	equal	1	but	that	is nevertheless	distinctive	in	the	Pythagorean	context:	1/1	≠	2/2	≠	3/3.	This	series	of self-divisions	may	be	related	to	the	topological	series	of	self-intersecting	forms pictured	in	Figure	12,	and	it	is	implicit	as	well	in	the	values	of	Table	1.	Note	also that,	in	both	Tables	1	and	2,	the	four	principal	terms	are	coupled	to	each	other	two at	a	time	by	six	enantiomorphically	related	pairs	of	overtone-undertone	values.	We may	therefore	entertain	the	proposition	that	the	values	provided	in	Table	1	are	the topodimensional	counterparts	of	the	Pythagorean	musical	intervals	appearing	in Table	2.	Accordingly,	these	rhythmical	relationships	can	be	said	to	give	us	the "music	of	the	dimensional	spheres." 0/0 0/1 0/2 0/3 1/0 1/1 1/2 1/3 2/0 2/1 2/2 2/3 3/0 3/1 3/2 3/3 29 Musicologist	Joscelyn	Godwin	asserts	that	the	Pythagorean	table	is	"an	image of	the	universe"	(1992,	191)	and	"a	means	toward	symbolic	explanation	and possible	illumination	concerning	cosmic	and	metaphysical	realities"	(190). Musicologist	Rudolf	Haase	similarly	emphasizes	the	spiritual	and	cosmogonic implications	of	the	Lambdoma,	intimating	that	it	may	be	nothing	less	than	an archetypal	ground	plan	for	the	creation	of	all	that	is: It	is	well	known	that	the	construction	of	the	world	on	the	basis	of	two antithetical	principles	is	an	age-old	concept:	it	is	best	known	as	the	yang	and yin	of	Chinese	tradition,	but	also	forms	an	important	element	of	Pythagorean philosophy,	which	is	why	Kayser	[Haase's	teacher]	revived	this	dualism	in referring	to	the	similarly	dual	structure	of	the	Lambdoma,	regarding	it	as	the foundation	of	the	latter.	(Haase,	1989,	103) For	her	part,	von	Franz	observes	that	the	"ancient	Chinese...divided	their whole	psychophysical	cosmos	into	a	periodic	twofold	rhythm,	a	reciprocal...YinYang	motion.	Yin	and	Yang	are...a	symphony	of	alternating	rhythms	in	which	spatial elements	(in	front–behind)	and	temporal	elements	(before–after)	are	not separable." Von	Franz	goes	on	to	cite	Marcel	Granet's	musical	simile	wherein	"Yin and	Yang	play	in	concert	(tiao)	and	harmonize	(ho).	'The	whole	universe	has	a rhythmical	basic	structure'"	(1974,	95).	In	a	footnote	on	the	same	page,	von	Franz suggests	that	"Ho	means	the	harmony	of	a	piece	of	music."	And	in	that	same	context, she	speaks	of	the	"rhythmical	internal	movement"	found	in	the	"older	heavenly order,"	the	order	related	to	the	Ho-t'u.	Later,	after	referring	to	field-like	Chinese number	matrices	"which	served	as	rhythmically	organized...images	of	the	aspects	of the	cosmic	whole,"	she	notes	the	possibility	that	"the	famous	Pythagorean lambdoma...also	was	originally	a	field	arrangement	in	circular	form"	(1974,	147). The	foregoing	reflections	on	the	Pythagorean	table	and	the	harmonies	of	yin and	yang	offer	an	intimation	of	the	primordial	"string	theory"	on	which	our	topophenomenological/psychophysical	string	theory	is	based.	The	Ho-t'u	number archetype	is	of	course	part	of	this	ancient	symphony.	But	we	have	seen	that	the trigram	system	associated	with	the	Ho-t'u	maps	onto	the	Klein	bottle	and	the	latter is	just	one	member	of	a	whole	topological	family.	We	are	now	prepared	to	look	at the	corresponding	yin-yang	family	of	which	the	Ho-t'u	trigram	system	is	but	one member. In	section	1,	I	introduced	the	cosmology	of	Taoism	by	describing	its	vision	of	a universe	in	constant	flux.	Philosopher	Jeaneane	Fowler	develops	this	theme	at length,	beginning	with	the	observation	that	"that	which	informed	all	the transformations	in	the	cosmos	was	posited	as	Tao"	(2005,	46).	Citing	the	I	Ching, Fowler	further	asserts	that	the	Tao	is "the	immutable,	eternal	law	at	work	in	all	change...the	principle	of	the	one	in the	many."	...	[Tao]	is	that	which	generates	the	tension	between	opposites, that	which	makes	changes	and	transformations	possible,	and	the	power	that renews	that	tension	from	moment	to	moment.	It	is	the	quiet,	spontaneous 30 power	that	eternally	gives	energy	to	the	cosmos,	to	the	rhythmic	composition of	the	stars	and	planets	as	much	as	to	the	energy	that	a	tiny	seed	needs	for germination.	(46) Later	in	her	text,	Fowler	corrects	the	earlier	impression	that	the	Tao	is associated	exclusively	with	the	One,	for	"Tao	is	beyond	even	the	One"	(76).	Indeed, we	really	cannot	say	what	the	Tao	is.	Classical	Taoist	texts	such	as	the	I	Ching	and the	Tao	Te	Ching	characterize	the	Tao	as	unnamable,	incapable	of	being	expressed	in language.	The	Tao	has	thus	typically	been	linked	to	negative	concepts	such	as emptiness,	limitlessness,	or	nothingness	(Ozaki	2001,	Chia	and	Wei	2009,	Grigg 1994).	If	the	Tao	is	indeed	associated	with	nothingness,	and	if	Taoism	forms	the foundation	of	the	Pythagorean	Lambdoma	as	Haase	suggests,	we	might	expect	the zero-dimensional	interval	of	the	Lambdoma,	0/0,	to	reflect	nothingness.	In confirming	this,	Godwin	first	underscores	the	special	status	of	0/0	as	that	which "sounds	no	tone	but	is	the	silence	toward	which	all	tones	tend"	(1992,	192).	Then, working	with	the	analogy	between	silence	and	Non-Being,	Godwin	suggests	that, just	as	the	positive	tones	are	rooted	in	the	silence	of	0/0,	"all	Being	culminates	in	a Nothingness	beyond	Being	that	...	is	paradoxically	its	sole	support	and	positive origin"	(193).	We	may	thus	take	0/0	as	the	Pythagorean	expression	of	the	Tao. Taoist	thinkers,	drawing	from	the	earlier	Yin-Yang	School	of	philosophy	(the School	of	Naturalists),	posited	the	idea	that,	from	the	nothingness	of	the	Tao	there arises	the	paradoxical	duality	of	yin	and	yang,	the	opposed	yet	interpenetrating forces	that	drive	the	dynamics	of	nature.	But	the	I	Ching	goes	further,	identifying two	additional	levels	of	evolution	from	the	original	Tao.	As	theologian	Whalen	Lai puts	it:	"The	I	Ching	says:	From	the	Great	Ultimate	[i.e.,	the	Tao]	come	the	two	poles [yin	and	yang];	from	the	two	poles	come	the	four	forms	[...	and]	from	the	four	forms [bigrams]	come	the	eight	trigrams"	(1980,	254).	Lai	provides	a	diagram	of	this	(Fig. 13). Figure	13.	The	yin-yang	"family	tree":	four	levels	of	cosmogony	(adapted	from	Lai	1980,	254). Copyright	©	2008	by	John	Wiley	&	Sons,	Inc.	Reprinted	by	permission	of	John	Wiley	&	Sons,	Inc. Having	considered	above	the	proposition	that	the	values	of	the topodimensional	action	matrix	(Table	1)	are	correlated	with	the	values	of	the Pythagorean	table	(Table	2),	I	would	now	like	to	extend	this	to	include	the	four members	of	the	yin-yang	family	depicted	in	Figure	13.	We	have	already	seen	the involvement	of	two	family	members:	(1)	the	Tao,	associated	with	the	0/0 31 Pythagorean	interval	and	the	zero-dimensional,	sub-lemniscatory	member	of	the topodimensional	family;	and	(2)	the	trigrams,	linked	to	the	3/3	Pythagorean interval	and	the	three-dimensional,	Kleinian	member	of	the	topodimensional	family. In	due	course,	we	will	see	how	the	other	two	branches	of	the	yin-yang	family-the unigram	mode	of	yin	and	yang	and	the	bigrams-are	correlated	with	the	oneand two-dimensional	members	of	the	topodimensional	family,	respectively.	For	the moment,	I	will	just	observe	that	the	numbers	of	yin-yang	lines	correspond	directly to	our	topodimensional	numbers.	The	Tao	itself	is	devoid	of	yin-yang	lines	and	this matches	the	zerodimensional	term	of	Table	1,	εD0.	The	single	yin-yang	lines emerging	from	the	Tao	parallel	the	one-dimensional	term,	εD1.	The	yin-yang counterparts	of	the	two-dimensional	lifeworld	(εD2)	are	the	double	lines	constituting the	bigrams,	and	the	triple	lines	comprising	the	trigrams	relate	to	the	threedimensional	realm	(εD3).	Evidently	then,	the	number	of	yin-yang	lines	bears	a simple,	one-to-one	relationship	to	dimension	number. Now,	let	us	focus	again	on	the	trigrams.	In	the	first	two	sections	of	this	paper, we	investigated	the	I	Ching's	Ho-t'u	number	archetype	and	found	that	its	associated trigrams	are	organized	into	two	cycles	of	four	phases	each,	with	phase	action following	the	underlying	topology	of	the	Klein	bottle.	The	cosmogonic	implications of	this	were	explored	and	we	were	led	to	the	provisional	conclusion	that	cycle	1 involves	the	generation	of	forward	or	clockwise	action	entailing	an	expansion	of	the universe	that	projects	the	appearance	of	an	objective	physical	world.	Then,	in Kleinian	cycle	2,	the	gears	shift	to	backward	and	the	projection	is	withdrawn,	the universe	contracting	to	bring	to	awareness	the	psychophysical	nature	of	reality. What	we	have	come	to	realize	in	subsequent	sections	is	that	the	Klein	bottle	and	the Ho-t'u,	besides	being	related	to	one	another,	are	each	members	of	whole	families	of interrelated	structures-one	family	topodimensional,	the	other	Taoist.	For	a	full understanding	of	cosmogony,	the	entire	family	must	be	taken	into	account	in	each case,	so	we	clearly	will	need	to	work	out	the	relationships	among	all	the	members	of both	cosmic	families.	Moreover,	while	the	overview	of	dimensional	generation offered	in	section	4	provides	a	general	idea	of	the	pattern	of	cosmic	evolution	giving rise	to	the	fundamental	forces	of	nature,	it	accounts	only	for	the	expansion	of	the universe,	not	its	subsequent	contraction	as	adumbrated	in	the	Kleinian	Ho-t'u.	The aspect	of	contraction	surely	must	be	included. In	the	next	section,	I	offer	a	general	interpretation	of	dimensional development	that	addresses	the	issue	of	lifeworld	expansion	and	contraction.	Then, in	section	7,	the	specific	stages	of	cosmogony	are	laid	out	for	the	topodimensional family,	and	this	is	followed	by	a	section	showing	that	the	family	of	Tao	lends	itself	to essentially	the	same	analysis.	What	comes	through	in	these	sections	is	the	deep affinity	and	underlying	harmony	of	Western	and	Eastern	"string	theories." 6.	GENERAL	STAGES	OF	DIMENSIONAL	DEVELOPMENT Phenomenological	thinking	provides	us	with	an	insight	into	the	embodied dialectical	interplay	of	object,	space,	and	subject	lying	behind	the	Cartesian	facade	of a	changeless	space	wherein	objects	are	cast	before	detached	subjects. Phenomenologically,	space	is	not	a	static	context	for	the	mechanistic	transformation 32 of	externally	related	objects;	rather,	the	lifeworld	dimension	is	a	"spatio-subobjective"	being	(what	Merleau-Ponty	called	"an	'element'	of	Being";	1968,	139)	that transforms	itself	organically.	We	can	see	from	this	that	the	phenomenological approach	lends	itself	to	the	idea	of	dimensional	generation	in	a	way	that	classical thinking	does	not.	The	lifeworld	dimension	does	not	just	contain	earthly	matter	but is	itself	of	the	earth.	Like	earthly	matter,	this	dimension	is	in	process.	And	we	are about	to	see	that,	like	living	matter,	it	passes	through	stages	of	biogenesis.	Thus	the phenomenological	dimension	possesses	the	character	of	a	living	organism-though not	a	finite	particular	organism,	to	be	sure.	Instead	the	dimension	in	question	is	a generic	organicity,	a	whole	dimension	of	life.	And	that	dimension	develops. Dimensional	development	deals	not	with	changes	taking	place	in	an individual	or	subject	per	se,	but	with	how	the	very	relationship	of	subject,	object, and	space	changes.	In	the	The	Self-Evolving	Cosmos	we	discover	that	the	Cartesian framework	of	object-in-space-before-subject	that	seems	to	preclude	an	organismic basis	for	dimensional	change	in	fact	itself	arises	in	the	intermediary	stage	of	a developmental	process	that	is	indeed	organically	dimensional.	In	the	opening	stage, object,	space,	and	subject	are	largely	undifferentiated.	Rather	than	constituting	welldefined	ontological	categories,	they	comprise	only	an	incipient	flux	of	embryonic possibilities.	In	the	earliest	fragment	of	Western	philosophy,	Anaximander	referred to	this	inchoate	condition	as	the	apeiron	(see	Rosen	2004).	Literally	meaning "without	measure,"	the	old	Greek	word	was	variously	interpreted	as	"limitless," "boundless,"	"indeterminate,"	or	"unintelligible"	(Angeles	1981,	14).	In	the	protoscientific	discipline	of	alchemy,	the	initial	state	of	affairs	was	termed	prime	matter: "prima	materia,	which	is	the	original	chaos	and	the	sea"	(Jung	1970,	9).	Then,	from the	primordial	flux,	the	first	differentiation	of	subject	and	object	emerges	within	the medium	of	a	nascently	differentiated	space.	This	marks	the	beginning	of	stage	2	and, as	it	unfolds,	distinctions	among	subject,	object,	and	the	space	that	contains	them harden	into	categorical	divisions	that	are	now	assumed	to	have	been	there	from	the start.	Completely	overshadowed	is	the	dialectical	process	that	gave	rise	to	these divisions.	In	thus	expanding	from	the	"black	hole"	of	the	primeval	lifeworld,	an extensive	universe	is	opened	up	and	the	original	connection	with	the	psyche	is obscured.	What	began	as	a	dense	psychophysical	"soup"	is	now	projected	as	a purely	physical	world	stretching	"out	there"	before	us,	an	objective	realm	from which	the	lived	subject	has	been	dropped.	(Earlier	I	gave	an	example	of	this objective	stance	in	discussing	the	posture	of	conventional	mathematics	wherein attention	is	always	focused	on	the	mathematical	object,	with	the	subjectivity	of	the mathematician	seldom	being	acknowledged.) The	first	two	general	stages	of	dimensional	transformation	have	their counterparts	in	the	Kleinian	action	of	the	Ho-t'u	described	in	section	2.	The	initial stage	corresponds	to	the	original	"counterclockwise"	action	from	which	clockwise action	is	generated	in	Ho-t'u	cycle	1.	Since	the	paper	model	of	the	Klein	bottle	giving the	topology	of	the	Ho-t'u	is	itself	but	an	object	in	three-dimensional	space,	it	cannot show	that	the	opening	phase	of	cycle	1	actually	entails	neither	clockwise	nor counterclockwise	action,	for	this	undifferentiated	state	of	affairs	affords	no	definite orientation.	The	three	subsequent	phases	of	Ho-t'u	cycle	1	parallel	the	second general	stage	of	dimensional	development:	"the	usual	clockwise	movement, 33 cumulative	and	expanding	as	time	goes	on."	It	is	here	that	an	objective	world	is projected	and	time	flows	forward	in	its	ordinary	sequence	according	to	time's	arrow of	increasing	entropy	in	an	expanding	physical	universe. What	of	the	third	general	stage	of	dimensional	development?	This	coincides with	the	second	cycle	of	the	Kleinian	Ho-t'u.	We	have	already	considered	the proposition	that,	in	entering	cycle	2,	the	gears	shift	from	clockwise	to counterclockwise,	forward	to	backward,	and	the	universe	contracts.	As	this	process is	enacted,	the	projection	of	a	purely	physical	three-dimensional	universe	is withdrawn.	It	is	by	overcoming	our	fixation	on	the	external	world	of	objects	and drawing	our	attention	inward	that	we	become	aware	of	the	psychophysical	nature	of the	cosmos.	We	are	going	to	see	that,	in	thus	moving	backward,	lower-dimensional lifeworlds	concealed	in	earlier	stages	of	cosmogony	are	brought	to	light.	The	I	Ching implies	that,	through	the	cosmic	contraction	and	reversal	of	time	that	takes	place	in cycle	2,	"the	seeds	of	the	future	take	form."	In	this	regard,	Wilhelm	says	in	the	I Ching	commentary	I	cited	section	1:	"If	we	understand	how	a	tree	is	contracted	into a	seed,	we	understand	the	future	unfolding	of	the	seed	into	a	tree"	(Wilhelm	quoted in	von	Franz	1974,	236–37).	I	will	shed	light	on	the	meaning	of	this	enigmatic passage	in	the	next	section. 7.	THE	COSMOGONIC	SPIRAL The	description	of	dimensional	development	just	given	does	not	take	into	account the	fact	that	each	member	of	our	topodimensional	family	has	its	own	distinctive pattern	of	change,	its	own	unique	phase	structure.	We	have	learned	that dimensional	transformation	generally	entails	stages	of	expansion	and	contraction, forward-directed	stages	in	which	a	world	is	projected	and	retrograde	stages	in which	the	projection	is	withdrawn.	But	we	have	not	yet	seen	how	this	is	specifically orchestrated	for	the	four	fundamental	lifeworld	"tones"	that	sound	the	music	of	the dimensional	spheres.	So	the	time	has	come	to	bring	our	picture	of	cosmogony	into sharper	focus. Consider	again	Table	1,	the	interrelational	matrix	of	topodimensional	spin structures.	Taken	by	itself,	this	table	affords	but	a	static	picture	of	dimensional associations,	one	that	is	"averaged	over,"	i.e.	abstracted	from,	the	actual	facts	of dimensional	change.	Therefore,	to	fill	in	the	concrete	details	of	how	the	several dimensional	spinors	evolve	in	relation	to	one	another,	we	must	set	the	matrix	in motion.	This	is	achieved	in	Table	3. 34 35 Table	3a	displays	the	full	course	of	development	of	all	orders	of topodimensional	action.	In	effect,	the	Table	expands	Table	1	diachronically,	now showing	the	stages	of	contraction	as	well	as	those	of	expansion.	While	the hypernumber	values	given	in	the	cells	of	these	matrices	are	the	same	as	those exhibited	in	Table	1,	with	the	new	table	we	can	study	the	specific	lines	of	evolution of	the	spin	structures	that	the	hypernumbers	represent.	Employing	the	principle	of synsymmetry	set	forth	in	section	4,	we	will	track	these	interwoven	cosmogonic pathways. Generally	speaking,	each	matrix	of	Table	3a	features	on	its	main	diagonal (that	arrayed	from	upper	left	to	lower	right)	one	of	our	four	fundamental	spinors. These	principal	terms-εD0,	εD1,	εD2,	and	εD3-correspond	to	the	four	reflexive	selfdivisions	(0/0,	1/1,	2/2,	3/3)	constituting	the	series	of	fundamental	tones	of	the Pythagorean	table	(Table	2),	which,	in	turn,	are	aligned	with	the	topological	series	of self-intersecting	forms	(Fig.	12).	The	hypernumber	ratios	appearing	off	the	main diagonal	of	each	matrix	are	members	of	enantiomorphic	pairings	that	pertain	to developmental	relationships	between	the	different	dimensional	spinors	(as	will soon	be	discussed). Table	3a	is	basically	to	be	read	in	a	circular	fashion:	we	begin	by	reading	it from	bottom	to	top,	then	reverse	course	to	read	it	from	the	top	down.	Starting	from the	matrix	at	the	bottom,	the	upward	movement	through	the	four	matrices	gives	the clockwise,	forward,	projective	stages	found	in	cycle	1	of	dimensional	generation, indexed	by	the	stage	values	appearing	to	the	left	of	the	matrices.	We	then	switch	to the	counterclockwise	or	retrograde	stages	of	cycle	2	wherein	projections	are withdrawn,	now	reading	back	down	through	those	same	matrices,	with	stage numbers	presently	displayed	to	the	right.	The	parenthetic	hypernumber	terms accompanying	each	stage	number	indicate	the	topodimensional	spinor	or	spinors	to which	that	stage	number	applies;	since	the	zero-dimensional	spinor	does	not undergo	development,	it	does	not	appear	here.	If	the	sequence	of	stages	for	each topodimensional	spinor	is	considered	separately	from	that	of	the	other	spinors,	we see	that	Table	3a	in	fact	does	not	describe	the	action	of	a	single	circle	(upward through	the	clockwise	stages,	downward	through	the	counterclockwise	ones)	but	of circles	nested	within	circles,	so	that	the	overall	pattern	is	actually	that	of	a	spiral. To	preserve	the	thoroughly	interwoven,	nonlinear	character	of	dimensional interrelatedness,	Table	3a	displays	the	several	windings	of	the	dimensional	spiral	as overlapping	one	another.	However,	this	makes	the	Table	somewhat	difficult	to	read. To	facilitate	understanding,	I	offer	Table	3b	as	a	visual	aid.	Here	the	circulations	of the	dimensional	spiral	have	been	parsed,	teased	apart	for	easier	identification. 36 37 The	driving	force	behind	the	generation	of	lifeworld	dimensions	depicted	in Table	3	was	already	identified	in	section	4:	an	n-dimensional	lifeworld	is crystallized	through	the	fusion	of	n−1-dimensional	enantiomorphs.	Involved	here	is the	dialectical	interplay	of	symmetry	and	asymmetry	that	I	have	termed synsymmetry.	Prior	to	their	fusion,	enantiomorphs	are	non-identical	mirror opposites,	thus	asymmetric	with	respect	to	each	other.	This	condition	of enantiomorphic	asymmetry	prevails	at	the	beginning	of	a	winding.	Then,	as	cycle	1 of	the	winding	progresses,	the	merger	of	n−1-dimensional	enantiomorphs	gives	rise to	n-dimensional	symmetry	(an	example	from	section	4	is	the	fusion	of	opposing sides	of	the	lemniscate	to	yield	the	Moebius	strip).	The	synsymmetry	concept further	implies	that	the	establishment	of	symmetry	must	be	followed	by	a	new order	of	asymmetry.	This	is	initiated	in	cycle	2	of	the	winding.	Here	enantiomorphs are	incubated	for	the	n-dimensional	structure	(the	Moebius	strip,	for	example)	that meet	the	necessary	conditions	for	opening	up	the	next	winding,	that	wherein	ndimensional	enantiomorphs	will	fuse	to	generate	n+1-dimensional	symmetry	(e.g., the	symmetry	of	the	Klein	bottle).	It	is	this	synsymmetric	process	of	dimensional fusion	and	"diffusion"	(i.e.,	the	creation	of	asymmetric	enantiomorphs)	that	fuels	the movement	from	one	winding	of	the	cosmogonic	spiral	to	another.	I	suggest, moreover,	that	the	synsymmetry	idea	helps	us	comprehend	the	meaning	of	the	I Ching	passage	cited	earlier:	"If	we	understand	how	a	tree	is	contracted	into	a	seed, we	understand	the	future	unfolding	of	the	seed	into	a	tree."	In	the	cosmic contraction	and	time	reversal	occurring	in	cycle	2	of	a	given	winding,	"the	seeds	of the	future	take	form"	by	propagating	the	enantiomorphs	required	for	generation	of new	symmetry	(a	new	"tree")	in	the	next	winding. In	Table	3b,	we	see	the	progressive	increase	in	the	number	of	stages	through which	dimensional	development	occurs	as	we	go	from	lowerto	higher-dimensional windings	of	the	dimensional	spiral	via	the	process	of	synsymmetry.	The	first winding	of	the	spiral	is	that	of	the	sub-lemniscatory	matrix,	which	consists	of	but	a single	cell.	In	effect,	the	"radius"	of	this	circulation	is	"zero,"	for	it	entails	no transformation	whatsoever.8	No	stages	of	development	can	be	found	for	εD0	nor	are any	necessary,	since	this	zero-dimensional	sphere	is	comprised	only	of	the	seeds	of higher-dimensional	structures	whose	generation	will	be	facilitated	in	subsequent windings	by	the	support	they	receive	through	εD0's	enantiomorphic	overtones (εD1/εD0,	εD2/εD0,	εD3/εD0)	and	undertones	(εD0/εD1,	εD0/εD2,	εD0/εD3). Advancing	to	the	second	winding	of	the	spiral,	the	matrix	expands	to	the	2	× 2	structure	associated	with	the	generation	of	the	one-dimensional	lemniscatory lifeworld.	The	opening	stage	finds	a	primordial	matrix	in	which	the	one-dimensional spinor	resides	in	but	an	embryonic	form.	In	our	psychophysical	string	theory,	where spinning	particles	can	be	seen	as	vibrating	strings	creating	waves	that	possess 8	Of	course,	the	"radii"	of	all	dimensional	circulations	are	"zero"	in	the	sense	that	stage	transitions	are not	actually	displacements	in	an	extensive	spatial	continuum,	though	they	appear	as	such	in	Table	3. The	several	dimensional	circulations	portrayed	in	the	Table	cannot	really	have	finite	radii	because- instead	of	taking	place	within	space,	they	constitute	the	quantized	pre-spatial	actions	from	which space	first	arises. 38 overtones	and	undertones,	we	may	characterize	the	zero-dimensional	sublemniscatory	spinor	as	a	"carrier	wave"	whose	εD1/εD0	overtone	and	εD0/εD1 undertone	carry	the	fledgling	one-dimensional	spinor	(see	"Waves	Carrying	Waves," chapter	7	of	Cosmos).	One-dimensional	action	is	but	nascently	oriented	here;	the differentiation	of	the	lemniscatory	lifeworld	has	not	yet	taken	place. Moving	upward	now	to	the	stage-2	matrix	of	the	lemniscatory	winding,	we see	that	εD1	has	gained	maturity.	This	entails	the	fusion	of	the	asymmetric	sublemniscatory	enantiomorphs,	εD1/εD0	and	εD0/εD1,	wherein	they	are	"annihilated," being	absorbed	into	the	emergent	symmetric	structure	of	the	expanding lemniscatory	universe	(to	be	more	exact,	εD1	is	the	mature	quantized	spinor	that projects	an	expanded	universe).	However,	with	the	transition	to	stage	2,	the primordial	state	of	affairs	is	not	simply	left	behind;	rather,	it	is	relegated	to	the background.	And	while	the	primal	potency	of	εD0	is	in	eclipse,	a	depotentiated, objectified	version	of	it	is	projected	as	a	point-like	lower	dimension	in	the	onedimensional	world	(concrete	examples	of	this	will	be	given	in	section	9	when	we	are working	with	particle	evolution).	The	attenuation	of	εD0	is	represented	in	Table	3b by	placing	this	value	in	parentheses.	At	the	same	time	that	the	sub-lemniscate	is projected	as	a	zero-dimensional	object	in	the	one-dimensional	world,	the lemniscatory	world	is	itself	projected	as	something	objectively	"out	there."	So forward-oriented	projective	activity	is	occurring	in	earnest	in	this	second	stage	of cycle	1.	Whereas	the	lemniscate's	action	is	but	incipiently	oriented	in	stage	1,	it presently	assumes	a	definite	clockwise	direction. The	completion	of	stage	2	brings	the	first	cycle	of	lemniscatory	archetypal action	to	a	close. A	critical	turning	point	is	now	reached	where	the	gears	shift	from forward	to	backward,	clockwise	to	counterclockwise.	Entering	the	second	cycle,	the expansion	of	the	one-dimensional	world	is	reversed	and	its	projection	is withdrawn.9 Let	us	look	more	closely	at	the	nature	of	this	act	of	withdrawal,	or	what	we may	call	retrojection.	It	does	not	imply	that	the	forward	movement	of	cycle	1	simply ceases	in	cycle	2.	Projective	action	continues,	and	yet	it	is	transformed	by	being counter-acted.	What	exactly	does	this	mean?	Suppose	you	were	handling	a	textured piece	of	fabric,	one	whose	fibers	were	arranged	in	a	certain	direction.	Is	it	not	by running	your	fingers	against	the	grain	of	the	material	that	its	direction	becomes more	clearly	discernable?	Similarly,	in	the	cycle-two	backward	movement	against the	grain	of	cycle	one,	awareness	is	gained	of	the	very	process	of	forward-directed projection	that	transpires	in	that	first	cycle.	It	is	in	this	way	that	the	projection	is retracted,	consciously	taken	back,	in	the	midst	of	its	ongoing	occurrence. The	retrograde	orientation	is	established	in	stage	3	of	the	lemniscatory winding.	What	was	projected	as	an	objective,	"purely	physical"	one-dimensional 9	In	the	interest	of	manageable	exposition,	action	cycles	are	not	shown	topologically	but	appear	as linear	sequences	of	matrix	cells	in	Table	3.	Thus,	in	the	lemniscatory	winding	of	Table	3b,	cycle	1	is comprised	of	stages	1	and	2,	and	cycle	2	consists	of	stages	3	and	4.	But	these	phases	of	action	are indeed	topological	and	a	more	accurate	way	of	indicating	the	double	cyclicity	of	the	lemniscate	would be	via	the	emblem	of	infinity	that	tangibly	displays	the	lemniscate's	two	cycles	of	opposing	clock orientations:	∞. 39 world	in	stage	2	is	presently	recognized	as	first	arising	from	the	symmetry-creating psychophysical	act	of	fusing	the	sub-lemniscatory	enantiomorphs	of	stage	1.	This basically	accords	with	von	Franz's	assessment	of	the	cycles	of	the	Ho-t'u,	where	she identifies	Ho-t'u	cycle	1	as	"physical"	and	cycle	2	as	"psychic"	(see	section	1).	It should	be	clear	however,	that	the	one-dimensional	realm	of	the	lemniscate	is	far simpler	than	the	three-dimensional	world	of	the	Kleinian	Ho-t'u	that	von	Franz	was dealing	with.	The	greater	number	of	stages	in	the	Kleinian	winding	shown	in	Table 3b	reflects	the	fact	that	development	goes	further	here,	with	a	sharper	division emerging	between	subject	and	object,	psyche	and	matter.	We	can	say	in	general	that the	progressively	increasing	number	of	stages	found	in	passing	from	one	winding	of the	cosmogonic	spiral	to	another	indicates	the	growing	complexity	of	the	lifeworlds generated. Note	that	in	stage	3	of	the	lemniscatory	winding,	the	awareness	gained	of	the stage-1	source	of	projection	is	somewhat	limited	inasmuch	as	this	cognizance	of	the zero-dimensional	realm	itself	remains	strictly	one-dimensional.	It	is	lemniscatory consciousness	that	prevails	in	stage	3	as	it	did	in	stage	2	and	the	lemniscate's abstract	perception	of	the	sub-lemniscatory	sphere	does	not	return	it	to	that primeval	setting	in	a	concrete	way.	Although	the	projection	of	the	one-dimensional world	has	been	effectively	withdrawn	in	stage	3,	that	of	the	zero-dimensional	world has	not	(in	the	Table,	εD0	remains	enclosed	in	parentheses,	indicating	its	continuing presence	as	a	depotentiated	projection). It	is	through	the	retrojection	occurring	in	stage	4	that	the	repression	of	the sub-lemniscate	is	lifted	and	its	potency	restored.	Here,	rather	than	merely apprehending	the	sub-lemniscate	from	a	lemniscatory	perspective,	the	lemniscate	is carried	back	to	its	embryonic	origin	and	the	zero-dimensional	structure	itself	gains full	presence.	This	return	to	the	beginning	is	no	mere	regression	in	which	the mature	lemniscate	utterly	reverts	to	its	incipient	form.	Despite	Table	3b's	linear sequencing	of	stages,	they	are	in	fact	not	simply	separated	from	each	other	but overlap.	Therefore,	just	as	the	zero-dimensional	structure	of	stage	1	is	not	left behind	in	passing	to	stage	2	but	continues	in	the	background,	the	mature	onedimensional	structure	of	stage	3	maintains	its	presence	in	stage	4.10 With	the	retrojection	enacted	in	stage	4,	closure	is	brought	to	the lemniscatory	winding	of	the	cosmogonic	spiral.	The	fully	developed	lemniscate	and revitalized	sub-lemniscate	are	now	both	on	the	scene.	They	are	wholly	present	to each	other	and	can	enter	into	resonance	to	sound	the	"music	of	the	dimensional spheres."	Because	the	relationship	is	no	longer	that	of	a	mature	spinor	carrying	a spinor	that	is	still	undeveloped,	because	both	dimensional	spinors	are	now	mature, we	can	reinterpret	the	enantiomorphic	ratios	as	expressing	the	intimate	reciprocity of	archetypal	players. The	second	cycle	of	the	lemniscatory	winding	not	only	completes	the development	of	the	lemniscate	but	also	paves	the	way	for	entry	into	the	Moebius winding.	For	it	is	here,	by	the	synsymmetric	process	of	dimensional	diffusion,	that 10	It	is	true,	however,	that	while	stages	overlap	one	another	in	both	cycles	of	dimensional development,	the	quality	of	overlap	in	cycle	2	differs	from	that	in	cycle	1.	See	Cosmos,	Chapter	10,	for a	discussion	of	"transparent"	and	"translucent"	forms	of	stage	overlap. 40 the	lemniscatory	"tree"	contracts	into	a	"seed"	required	for	the	growth	of	the Moebius	"tree."	The	other	"seed"	necessary	for	subsequent	Moebius	development	is of	course	the	de-repressed	sub-lemniscate.	The	seeds	synchronously	sown	in	cycle	2 of	the	lemniscatory	winding	become	the	asymmetric	carrier	waves	whose enantiomorphic	fusions	generate	Moebius	symmetry	in	the	next	topodimensional epoch. When	the	lemniscatory	winding	closes,	the	cosmogonic	spiral	opens	out	into the	Moebius,	consisting	of	3	×	3	matrices	transformed	over	six	stages.	The	compass has	shifted	so	that	initial	action	is	once	again	unoriented.	In	stage	1	of	this	twodimensional	circulation,	we	find	that-as	a	result	of	the	germinal	activity	in	the second	cycle	of	the	last	epoch-the	carrying	capacity	of	the	zero-dimensional	sublemniscate	has	been	enlarged.	Through	its	new	pair	of	enantiomorphs,	εD2/εD0	and εD0/εD2,	the	sub-lemniscate	serves	as	carrier	wave	for	the	embryonic	Moebius spinor.	The	fusion	of	these	asymmetric	enantiomorphs	on	behalf	of	the	developing Moebius	then	brings	us	into	stage	2.	With	the	sub-lemniscatory	carrier	having	been absorbed	into	the	Moebius	wave,	the	original	zero-dimensional	spinor	is	eclipsed and	an	attenuated	version	of	it	is	projected	(as	indicated	by	the	appearance	of	εD0	in parentheses).	For	its	part,	the	Moebius	gains	its	measure	of	symmetry,	becoming more	mature.	A	clockwise	orientation	is	assumed	in	this	stage,	and	we	have	the	first projection	of	an	objective	two-dimensional	world	stretching	before	an	emergent two-dimensional	subjectivity.	However,	whereas	the	one-dimensional	lemniscate required	the	fusion	of	but	a	single	pair	of	enantiomorphs	to	complete	its	cycle-1 development,	the	more	complex	Moebius	pattern	calls	for	two	such	fusions. This	is	where	the	lemniscatory	contraction	of	the	previous	winding	bears fruit.	Its	effect	is	that	the	lemniscate	can	currently	play	the	role	of	a	carrier	wave, expressed	as	the	enantiomorphic	overtone-undertone	coupling,	εD2/εD1	and	εD1/εD2. It	is	the	fusion	of	these	enantiomorphs	that	brings	us	into	the	third	stage	of	Moebius cosmogony,	the	conclusive	stage	in	its	forward-directed	unfoldment.	With	the merger	of	lemniscatory	enantiomorphs,	the	lemniscatory	carrier	wave	dissolves into	εD2	and	a	depotentiated	form	of	it	is	projected	(denoted	in	stage	3	by	its enclosure	in	brackets).	This	attenuating	projection	is	accompanied	by	a	second	and final	projective	expansion	of	the	two-dimensional	world.	The	now-mature	Moebius spinor	thus	reaches	full	symmetry. Next	comes	the	transition	from	cycle	1	to	cycle	2	of	the	Moebius	winding. Entering	stage	4,	the	gears	are	put	into	reverse,	expansion	switches	to	contraction, and	two-dimensional	spin	is	reoriented,	becoming	counterclockwise.	In	this	first retrograde	movement,	there	is	an	initial	recognition	that	the	cycle-1	preoccupation with	an	objective	physical	reality	external	to	the	perceiving	subject	has	actually arisen	from	sub-objective,	psychophysical	acts	of	projection.	(Do	note	that	subjectobject	relations	in	the	Moebius	winding	are	simpler,	less	differentiated,	than	those in	the	Kleinian	winding;	see	Rosen	2006	and	2008a	for	an	intensive	examination	of this	distinction.)	But	the	stage-4	withdrawal	of	the	two-dimensional	projection leaves	untouched	the	depotentiating	lower-dimensional	projections.	Then,	moving concretely	backward	in	stage	5,	two-dimensional	retrojection	is	taken	further.	The earlier	(stage-2)	projection	of	εD2	is	counteracted	and	the	repression	of	the	onedimensional	lemniscatory	carrier	wave	is	lifted.	The	Moebius	spinor	now	enters	into 41 harmony	with	its	lemniscatory	counterpart,	and	we	interpret	the	enantiomorphic ratios	of	stage	5	as	expressing	that	harmony. Going	still	further	back	in	stage	6,	the	zero-dimensional	sub-lemniscatory carrier	emerges	from	obscurity	to	become	harmonically	attuned	to	the	Moebius.	The projective	actions	of	cycle	1	have	now	been	wholly	counteracted	and	the	Moebius spinor	has	achieved	synchrony	with	its	lower-dimensional	relatives	in	the topodimensional	family.	And	since	the	synchrony	is	realized	amidst	the	process	of contraction,	these	three	spinors	come	to	constitute	the	seed-structures	manifested as	carrier	waves	in	the	next	epoch. With	the	completion	of	the	Moebius	winding,	the	dimensional	spiral	dilates once	again	and	we	find	ourselves	in	the	opening	stage	of	the	Kleinian	winding	of	4	× 4	matrices	evolving	over	eight	stages.	Beyond	its	incipiently	oriented	stage,	the three-dimensional	Kleinian	organism	gains	symmetry	and	matures	through	three clockwise	phases.	This	follows	the	pattern	of	the	earlier	windings.	In	each	stage	a different	fusion	of	enantiomorphs	occurs	entailing	the	absorption	of	a	lowerdimensional	carrier	wave (sub-lemniscatory,	lemniscatory,	and	Moebial)	that facilitates	the	growth	of	the	Kleinian	wave.	And	with	each	new	fusion	in	the	cycle-1 expansion	of	Kleinian	dimensionality,	there	is	a	new	and	more	differentiated projection	of	an	objective	three-dimensional	world. Subsequently,	the	gears	are	reversed	once	more	and	we	pass	into	the	second cycle	of	Kleinian	evolution,	where	expansion	becomes	contraction	and	clockwise action	becomes	counterclockwise.	Stage	5	brings	the	Kleinian	organism's	first realization	that	the	"objective	physical	universe"	is	in	fact	a	psychophysical projection.	In	the	three	ensuing	stages	of	cycle	2,	the	cycle-1	repressions	imposed	on the	lower-dimensional	carrier	waves	are	successively	lifted	and	each	enters	into resonance	with	the	Kleinian	dimension.	With	all	the	members	of	the topodimensional	family	resounding	in	harmony,	the	"symphony	of	dimensional spheres"	is	heard	in	its	fullness. But	what	happens	next?	Does	the	cosmogonic	spiral	now	close	into	a	circle that	brings	us	to	the	end	of	cosmogony?	What	of	the	logic	of	synsymmetry,	of dimensional	fusion	and	diffusion?	In	cycle	2	of	the	Kleinian	winding,	must	the Kleinian	"tree"	not	contract	into	a	"seed"	for	a	new,	meta-Kleinian	"tree,"	a	new round	of	topodimensional	transformations	in	a	novel	winding	of	the	spiral?	This implication	is	inescapable	if	we	are	to	avoid	abandoning	arbitrarily	the synsymmetry	principle	so	critical	to	our	analysis.	Indeed,	when	the	analysis	was related	to	the	Pythagorean	table	in	section	5,	I	noted	that	this	table	is	usually portrayed	as	an	indefinitely	expanding	series	of	musical	intervals	that	goes	beyond the	3/3	interval	correlated	with	Kleinian	three-dimensionality.	Nevertheless,	with the	prospect	of	surpassing	the	Kleinian	dimension,	it	appears	we	have	reached	a watershed,	for	our	analysis	of	cosmic	evolution	has	come	up	against	the	limits	of	our own	analytical	capability.	While	we	can	abstractly	anticipate	higher-dimensional windings	of	the	cosmogonic	spiral,	we	cannot	gain	concrete	knowledge	of	them because	they	lie	beyond	the	conceptual	framework	we	employ	as	three-dimensional analysts. I	do	not	hold	with	the	Kantian	view	that	the	tangible	experience	of	a dimension	transcending	our	cognitive	frame	is	forever	inaccessible	to	us.	Instead	I 42 suggest	that	we	can	indeed	come	to	palpably	realize	the	higher	dimension	but	that doing	so	requires	more	than	just	changing	the	content	of	our	analysis.	Our	analytic framework	itself	must	evolve,	stretching	to	accommodate	the	new	dimension. Before	I	am	finished,	I	will	attempt	to	clarify	what	this	means	and	what	it	may require. To	sum	up	this	section,	the	foregoing	analysis	brings	to	light	the	stages	of dimensional	development	that	serve	to	distinguish	one	topodimensional	spin structure	from	another.	While	the	several	windings	of	the	dimensional	spiral overlap,	each	lifeworld	circulation	runs	its	own	course,	with	distinct	circulations being	marked	by	differences	in	the	number	of	stages	that	each	requires	to	carry	out its	clockwise	projections	and	counterclockwise	retrojections.	These	differences reflect,	in	turn,	differing	degrees	of	lifeworld	complexity,	which	show	up	in	the capacity	for	discriminating	subject	and	object,	psyche	and	matter. 8.	THE	COSMOGONIC	SPIRAL	IN	LIGHT	OF	THE	TAO In	completing	our	study	of	how	dimensions	evolve,	we	have	reached	another crossroad	between	West	and	East.	For	we	are	now	prepared	to	grasp	the relationship	between	our	section-7	treatment	of	the	Klein	bottle	in	the	context	of dimensional	generation	and	our	approach	to	it	in	section	2,	where	it	is	taken	in conjunction	with	Taoism's	Ho-t'u	and	associated	trigrams. In	the	earlier	section,	the	quantized	phase	structure	of	the	Klein	bottle	was seen	to	correspond	to	that	of	the	Ho-t'u:	both	consist	of	two	oppositely	directed cycles	of	four	phases	each.	This	is	of	course	the	same	structure	manifested	in	the Kleinian	winding	of	the	cosmogonic	spiral.	What	we	could	not	see	in	section	2	is	the thoroughly	interrelational	nature	of	Kleinian	development.	This	only	comes	to	light when	we	take	into	account	the	Klein	bottle's	membership	in	its	topological	family. Table	3	shows	how	each	stage	in	the	unfoldment	of	the	Klein	bottle	is	defined	by	its relationship	to	another	family	member.	And	what	we	see	in	Table	4	is	not	only	the correspondence	of	the	Kleinian	winding	of	the	cosmogonic	spiral	to	the	trigrams	of the	Ho-t'u,	but	the	correlation	of	all	members	of	the	topodimensional	family	with	the family	of	Tao. 43 44 For	Table	4,	I	have	replaced	the	topodimensional	values	of	Table	3b	with their	intimately	related	Taoist	counterparts.	The	four	orders	of	the	Tao	are	given	in the	four	windings	of	the	cosmogonic	spiral.	The	first	winding	consists	of	the	Tao itself,	shown	in	the	upper	left-hand	quadrant	of	the	Table	as	the	single	value,	0. Whereas	all	other	windings	comprise	extended	developmental	matrices,	the	Tao stands	alone.	As	was	said	of	the	zero-dimensional	sub-lemniscate,	the	Tao	per	se does	not	evolve	through	stages.	Acting	via	its	seed	structures,	it	functions	solely	as	a catalyst	for	the	matrical	generation	of	higher	orders	of	itself:	the	unigrams, consisting	of	the	basic	modes	of	yin	(-	-)	and	yang	(-);	the	bigrams,	entailing	the doubling	of	the	yin-yang	lines;	and	the	trigrams.	Paralleling	the	generation	of	the topodimensional	spinors,	Taoist	cosmogony	is	seen	as	driven	by	the	synsymmetric process	of	enantiomorphic	fusion	and	diffusion.	Within	each	winding	of	Table	4,	the unbracketed	terms	on	the	main	diagonals	of	the	matrices	show	the	stages	of development	of	the	fundamental	yin-yang	spinors.	The	coupled	enantiomorphs located	off	the	main	diagonals	are	the	yin-yang	ratio	pairs,	each	of	which	expresses	a relationship	between	members	of	two	different	orders	of	the	yin-yang	family. Cosmic	evolution	is	facilitated	by	the	symmetry-generating	enantiomorphic	fusion occurring	in	the	first	cycle	of	a	given	winding,	followed	by	the	cycle-2	diffusion	that sows	the	seeds	for	the	asymmetric	enantiomorphs	of	the	next	winding. Every	winding	of	the	Taoist	cosmogonic	spiral	is	to	be	read	in	the	same circular	manner	as	its	topodimensional	counterpart	in	Table	3.	Reading	through	the stages	of	a	given	winding	from	the	bottom	matrix	to	the	top,	we	have	the	clockwise, projective	stages	of	cycle	1.	Then	switching	gears,	we	read	downward	from	the	top matrix,	and	this	gives	the	counterclockwise	stages	of	retrojection	that	constitute cycle	2. Let	us	begin	our	reading	of	Table	4	by	reversing	the	sequence	in	which	the windings	were	studied	in	Table	3b.	We	will	not	proceed	from	the	Tao	and	track	the cosmogonic	spiral's	expansion	through	the	unigrams,	to	the	bigrams	and	the trigrams.	Instead	we	will	start	with	the	trigram	winding	since	this	is	where	we	can best	articulate	a	principle	that	will	involve	a	significant	departure	from	the	classical treatment	of	the	yin-yang	family. The	trigram	winding	of	Table	4	displays	four	4	×	4	matrices	evolving	over two	cycles	of	four	stages	each	(à	la	the	Kleinian	winding	of	Table	3b).	In	the	first cycle,	we	see	the	cosmogonic	process	by	which	the	psychophysical	trigram	spinor progressively	develops	from	an	initially	embryonic	origin;	as	it	matures	and	gains symmetry,	it	spins	out	an	objective	physical	world.	The	trigram	sequence	follows that	of	the	Ho-t'u	cross.	In	the	first	stage,	the	embryonic	spinor	is	represented	by	the trigram	K'un,	consisting	of	three	broken	lines;	its	compass	direction	is	North.	We observe	in	the	bottom	matrix	of	the	winding	the	enantiomorphic	support	K'un receives	from	the	Tao	carrier	wave	(denoted	by	0).	Then,	via	the	fusion	of	these enantiomorphs,	we	move	upward	to	the	next	stage,	where	the	spinor	is	now directed	South	and	is	manifested	as	Ch'ien,	expressed	by	three	solid	lines.	Not	yet mature,	Ch'ien	is	carried	by	the	enantiomorphs	of	the	mature	yin-yang	spinor originating	in	the	unigram	winding.	Repeating	the	process	of	fusion,	the	trigram spinor	advances	to	stage	3,	assuming	the	structure	of	the	Easterly-directed	Li:	a 45 broken	line	between	two	solid	lines.	Li	is	carried	by	the	enantiomorphs	of	"Young Yin,"	the	mature	spinor	of	the	bigram	winding.	(Bear	in	mind	that	a	carrier	wave always	involves	the	mature	form	of	a	lower-order	spinor,	since	a	spinor	cannot	play the	supportive	role	of	carrier	until	its	development	is	completed	in	its	own winding.)	The	last	projective	stage	of	the	trigram	winding	arises	from	the	fusion	of bigram	enantiomorphs.	In	this	phase,	the	trigram	spinor	reaches	full	maturity	as K'an,	a	solid	line	between	two	broken	lines.	Note	that	the	depotentiating	projections previously	discussed	for	Kleinian	evolution	are	mirrored	in	trigram	development: with	each	enantiomorphic	fusion	of	cycle	1,	the	lower-order	spinor	that	had supported	the	trigram's	projective	potentiation	is	itself	subjected	to	a depotentiating	projection	that	narrows	it	down	and	objectifies	it,	occluding	its	true nature.	In	Table	4,	spinors	thus	attenuated	are	enclosed	in	brackets. We	know	what	happens	in	cycle	2.	The	gears	shift	from	forward	to	backward, projection	to	retrojection,	and	the	retrograde	stages	of	the	cycle	unfold.	The projections	of	cycle	1	are	presently	withdrawn,	with	lower-order	spinors	emerging from	eclipse	to	enter	into	synchrony	with	the	trigram	spinor.	And	in	the	contraction of	the	trigram	cosmos,	these	spinors	become	seeds	for	the	next	winding.	Notice	that, in	Table	4,	the	stages	of	cycle	2	are	associated	with	no	new	trigrams	and accompanying	compass	directions,	as	they	are	in	the	classical	rendition	of	the trigrams	(see	Figure	11).	Why	have	I	dropped	four	trigrams? In	the	traditional	approach	to	the	I	Ching,	all	eight	permutations	of	the trigrams	are	employed	and	from	this,	64	hexagrams	are	built	for	the	purposes	of personal	guidance	and	divination,	with	philosophical	and	cosmological	commentary added.	What	I	am	proposing	is	an	alternative	way	of	working	with	the	trigrams	that uses	only	the	ones	specified	in	cycle	1.	Instead	of	adhering	to	the	custom	of introducing	additional	trigrams	and	compass	directions	for	cycle	2,	I	suggest	that the	very	same	trigrams	be	employed,	now	in	the	retrograde	orientation. There	is	a	sense	in	which	the	trigrams	of	cycle	1	and	their	affiliated	compass points	can	be	considered	more	fundamental	than	their	cycle-2	counterparts.	First	of all,	we	observe	that	the	compass	points	conventionally	featured	in	cycle	2	do	not constitute	directions	that	are	uniquely	different	from	the	four	cardinal	directions arising	in	cycle	1.	The	cycle-2	directions	are	deemed	"intercardinal"	in	that	they combine	the	already	established	primary	compass	directions	without	introducing anything	qualitatively	new.	We	see	from	Figure	2	that	the	intercardinal	directions- Northwest,	Southeast,	Northeast,	Southwest-are	obtained	simply	by	rotating	the Ho-t'u	cross	by	an	angle	of	45°	to	give	the	compass	markings	that	lie	halfway between	the	cardinals. Perhaps	more	importantly,	while	the	classical	trigrams	of	cycle	2-Ken,	Tui, Chen,	and	Sun-provide	novel	permutations	of	the	triple	yin-yang	lines	and	these are	subject	to	new	interpretations,	there	is	no	change	in	the	underlying combinatorial	structure	of	the	trigrams.	The	four	trigrams	of	cycle	1	exhaust	the possible	combinations	of	broken	and	solid	lines	taken	three	at	a	time:	all	solid;	all broken;	two	solid,	one	broken;	two	broken,	one	solid.	What	we	have	in	the	second cycle	are	but	repetitions	of	those	primary	structures.	The	permutations	are	different but	the	more	basic	combinations	stay	the	same.	So,	just	as	the	classical	compass points	of	cycle	2	provide	no	fundamentally	new	compass	directions,	the	trigrams	of 46 this	cycle	offer	no	innovations	in	basic	combinatorial	structure.	What	happens	when we	limit	the	trigrams	to	the	primary	ones	given	in	cycle	1	and	take	cycle	2	as	a backward	movement	through	those	very	same	trigrams?	We	can	then	see	clearly	the isomorphic	alignment	of	the	trigram	order	of	the	yin-yang	family	with	its	Kleinian equivalent	in	the	topodimensional	family. Moving	backward	now	in	the	cosmogonic	spiral	of	Table	4,	let	us	consider	the yin-yang	winding	that	corresponds	to	the	Moebius	winding	of	Table	3b.	The counterpart	of	the	Moebius	is	the	archetypal	field	that	involves	the	generation	of	the bigram	spin	structure	over	two	cycles	of	three	stages	each.	In	the	classical	literature, the	set	of	bigrams	is	known	as	the	Four	Emblems	or	Four	Symbols	(Hulse	2002). The	traditional	bigrams	are	exhibited	in	Figure	13,	along	with	their	eight	associated trigrams,	only	four	of	which	we	are	employing	in	our	combinatorial	approach. Comparing	Figure	13	with	Table	4,	we	see	the	connection	given	in	the	former between	the	stage-1	trigram	K'un	with	its	three	broken	lines	and	the	bigram composed	of	two	broken	lines,	called	"Old	Yin"	or	"Greater	Yin"	(Hulse	2002,	379). Aligning	bigram	with	trigram	development,	Old	Yin	is	taken	to	signify	the	first	stage of	the	bigrams.	By	the	same	token,	the	linking	in	Figure	13	of	the	stage-2	trigram Ch'ien	(three	solid	lines)	with	the	"Old	Yang"	bigram	composed	of	two	solid	lines suggests	that	Old	Yang	be	associated	with	the	second	stage	of	the	bigram	winding.	In Table	4,	the	third	and	final	stage	in	cycle	1	of	this	winding	corresponds	to	the bigram	designated	"Young	Yin,"	consisting	of	a	broken	line	atop	a	solid	line.	But what	of	the	fourth	bigram	shown	in	Figure	13,	made	up	of	a	solid	line	above	a broken	line?	This	bigram,	named	"Young	Yang,"	is	a	permutation	of	the	same combination	of	lines	that	constitutes	Young	Yin:	a	solid	line	and	a	broken	line.	In keeping	with	the	principle	of	working	only	with	basic	combinations,	one	of	these permutations	must	be	excluded.	Why	has	Young	Yang	been	chosen	for	elimination and	not	Young	Yin?	Figure	13	shows	that	the	former	is	connected	with	the	stage-4 trigram	K'an	whereas	the	latter	is	linked	to	the	stage-3	trigram	Li.	Since	Table	4 maintains	consistency	in	developmental	sequencing	by	presenting	the	bigrams	in the	same	stage	order	as	the	trigrams	with	which	they	are	coupled,	Young	Yin	must correspond	to	stage	3	of	the	bigram	winding,	leaving	Young	Yang	as	the	extraneous permutation	to	be	dropped.	The	compass	directions	for	the	three	stages	of	bigram cycle	1	follow	the	sequence	of	the	trigrams-North	à	South	à	East,	with	West excluded. As	with	the	trigram	winding,	that	of	the	bigrams	displayed	in	Table	4	shows the	generation	of	the	spinor	through	two	cycles,	the	first	expanding	projectively	to spin	out	an	objective	world	(though	one	less	differentiated	and	complex	than	the trigram	world),	the	second	contracting,	moving	backward,	and	withdrawing	the projections	of	the	first.	Again	we	see	the	process	of	development	facilitated	in	cycle 1	by	mature	lower-order	spinors	acting	as	carrier	waves	that	give	enantiomorphic support	to	the	evolving	spinor.	In	stage	1,	Old	Yin,	the	embryonic	bigram,	is supported	by	the	enantiomorphs	of	the	Tao.	Enantiomorphic	fusion	then	brings	us to	Old	Yang	in	stage	2.	Not	yet	fully	developed,	this	bigram	is	carried	by	the enantiomorphs	of	the	yin-yang	unigram.	Next,	when	unigram	enantiomorphs	fuse, we	encounter	Young	Yin,	the	mature	bigram	spin	structure	of	the	third	and	final stage	of	projection.	This	is	followed	of	course	by	the	switching	of	gears.	Passing 47 backward	through	the	bigram	stages	of	cycle	1,	the	cycle-2	spinor	withdraws	its projections	and	enters	into	harmony	with	the	lower-order	spinors	that	had	been repressed	on	its	behalf.	In	this	contraction	of	the	bigram	universe,	the	spinors condense	into	germinating	seeds,	to	bear	fruit	as	the	enantiomorphs	that	facilitate trigram	generation. We	now	go	still	further	back	on	the	cosmogonic	spiral	to	the	yin-yang equivalent	of	the	lemniscatory	winding.	This	lower-order	archetypal	field	comprises a	double	cycle	of	spin	transformations	linked	to	the	unigrams.	Cycle	1	involves	but two	stages.	In	the	first	of	these,	we	have	the	embryonic	spin	structure	represented by	a	single	broken	line,	designated	"Primal	Yin"	(associated	with	the	compass	point, North).	This	spinor	is	carried	by	the	enantiomorphs	of	the	Tao.	Just	one	fusion	is needed	to	bring	forth	the	mature	structure	of	stage	2,	denoted	by	single	yin	and yang	lines	(and	oriented	to	the	South).	With	the	movement	backward	that	ensues	in cycle	2,	there	is	initial	recognition	in	stage	3	that	a	world	has	been	projected, followed	by	the	more	concrete	retrojection	enacted	in	stage	4	through	which	the Tao	emerges	from	its	stage-2	eclipse	to	converge	harmonically	with	yin	and	yang. The	concomitant	cosmic	contraction	forms	the	seeds	for	the	bigram	winding. Let	me	emphasize	the	relative	simplicity	of	the	cosmogonic	process	in	the unigram	winding.	As	already	noted,	lower-order	spin	structures	project	less complex	lifeworlds,	with	a	lesser	capacity	for	differentiating	subject	and	object.	We can	say	accordingly	that	the	"objective	physical	world"	projected	in	the	unigram winding	is	but	a	weakly	differentiated	one	involving	a	form	of	consciousness	that	is not	as	sharply	focused	and	discriminating	as	the	awareness	achieved	in	higher dimensional	windings. Continuing	back	through	the	spiral	to	its	source,	we	reach	"that	which inform[s]	all	the	transformations	in	the	cosmos"	(Fowler	2005,	46):	the	Tao	itself. The	Tao,	says	Fowler,	is	what	"generates	the	tension	between	opposites	[i.e.	yin	and yang]....	It	is	the	quiet,	spontaneous	power	that	eternally	gives	energy	to	the	cosmos, to	the	rhythmic	composition	of	the	stars	and	planets	as	much	as	to	the	energy	that	a tiny	seed	needs	for	germination"	(46).	The	Tao	is	a	primordial	matrix.	While	this term	is	broadly	defined	in	mathematics	as	an	array	of	numbers,	its	general dictionary	definition	indicates	an	environment	or	material	context	in	which something	develops.	Tao	appears	on	its	own	in	Table	4	as	the	single-celled	matrix from	which	all	develops.	It	is	because	the	Tao	is	associated	with	silence, nothingness,	and	zero-dimensionality	(see	section	5)	that	we	have	assigned	to	it	the value	of	0.	Recalling	Godwin's	coupling	of	the	Pythagorean	0/0	with	that	which "sounds	no	tone	but	is	the	silence	toward	which	all	tones	tend"	(1992,	192),	we	now turn	our	attention	back	to	the	psychophysical	"tones"	of	quantum	gravity	for	a	string theoretic	specification	of	the	"music	of	the	dimensional	spheres." 9.	THE	SPIRAL	OF	QUANTUM	GRAVITY In	the	past	two	sections,	the	evolution	of	the	primary	spinors	was	explained	in highly	theoretical	terms.	I	would	now	like	to	flesh	this	out	by	showing	its correspondence	with	the	elementary	forces	of	string	theory.	The	Self-Evolving Cosmos	describes	in	considerable	detail	the	phenomenological	approach	to	the 48 standard	model	of	particle	physics.	I	offered	a	synoptic	sketch	of	this	in	my	2015 article	and	must	limit	myself	to	an	abbreviated	rendition	in	the	current	paper	as well.	But	the	latter	does	go	further	than	the	former,	and,	in	some	important	respects, the	present	analysis	advances	the	work	done	in	Cosmos	itself. What	Cosmos	suggests	is	that	a	full	account	of	the	fundamental	particles	of string	theory	may	be	provided	by	embedding	the	theory	in	the	matrix	of	primordial spin	structures	given	in	Table	1.	This	matrix	constitutes	a	special	application	of	the hypernumber	idea,	one	that	delivers	a	highly	specific	reckoning	of	primordial	spin action.	The	topodimensional	array	of	four	fundamental	spinors	(shown	on	the principal	diagonal	of	the	matrix)	can	be	directly	associated	with	the	four	types	of gauge	bosons	found	in	nature.	The	gauge-boson	correlates	of	Table	1	are	displayed in	Table	5.	What	is	the	basis	of	these	correlations? G G/g G/(W,	Z) G/γ g/G g g/(W,	Z) g/γ (W,	Z)/G (W,	Z)/g W,	Z (W,	Z)/γ γ/G γ/g γ/(W,	Z) γ Table	5.	Spin	matrix	of	gauge	bosons.	G	is	the	graviton;	g	is	the	strong	gauge	boson;	W,Z	is	the	weak gauge	boson	particle	pair;	and	γ	is	the	photon We	know	that	Table	1	signifies	a	process	of	generation	in	which	higher topological	dimensions	evolve	from	lower	ones.	The	facts	of	particle	evolution	lend themselves	to	straightforward,	one-to-one	correlation	with	this	process.	The	first force	particle	to	"freeze	out"	of	the	Big	Bang's	hot	primordial	soup	is	the hypothesized	graviton,	G.	The	graviton	of	Table	5	is	associated	with	εD0,	the	zerodimensional	sub-lemniscatory	action	of	Table	1,	which	can	be	written	εD0(!/2)	to give	expression	to	subatomic	particle	spin;	thus,	G	≡	εD0(!/2).	Next	to	separate	itself from	the	primordial	chaos	is	the	strong	gauge	boson,	g,	and	we	relate	it	to	εD1 lemniscatory	action,	writing	g	≡	εD1(!/2).	Then	the	weak	force	emerges,	given	by the	boson	pair	W	and	Z,	which	we	identify	with	εD2(!/2).	When	the	three	orders	of lower-dimensional	gauge	bosons	have	"frozen	out,"	what	remains	is	γ,	the	photon, topodimensionally	expressed	as	εD3(!/2). If	the	principal	terms	or	"fundamental	tones"	of	the	Table-5	matrix	give	the four	gauge	bosons,	what	is	the	significance	of	the	"overtone-undertone"	couplings appearing	off	the	principal	diagonal?	In	Table	1,	these	are	the	topodimensional enantiomorphs	whose	synsymmetric	fusions	drive	the	process	of	dimensional generation.	The	overtone-undertone	couplings	appear	in	Table	5	as enantiomorphically	related	boson	ratios.	It	is	from	their	interactions	that	the primary	gauge	bosons	emerge.	In	phenomenological	string	theory,	boson-ratio interaction	not	only	accounts	for	the	generation	of	the	four	kinds	of	gauge	bosons, but	also	for	the	production	of	the	12	fermions	of	the	standard	model.	The	six	pairs	of ratios	involved	in	distilling	the	bosons	likewise	interact	to	yield	the	six	pairs	of fermions	(three	lepton	pairs	and	three	quark	pairs).	Geometrically	speaking,	the fermions	function	as	"dimensional	bounding	elements,"	local	features	of	global 49 bosonic	dimensionality,	with	local	and	global	aspects	intimately	interwoven. Needless	to	say,	fermion	generation	requires	clarification,	but	I	will	not	elaborate further	on	it	here	(see	Cosmos).	What	I	will	elaborate	on	is	the	specific	course	of boson	generation	that	traces	the	evolution	of	nature's	fundamental	forces. While	Table	5	can	be	read	as	a	process	of	dimensional	generation,	what	we acknowledged	earlier	for	Table	1	also	applies	to	it:	if	Table	5	is	taken	by	itself without	reading	change	into	it,	the	Table	provides	a	merely	static	picture	of	particle relationships,	a	snapshot	that	misses	the	kinetic	facts	of	cosmogony.	Therefore,	to see	more	clearly	the	developmental	basis	of	the	correlations	suggested	in	Table	5, we	must	fill	in	the	concrete	details	of	how	the	four	gauge	bosons	evolve	in	relation to	each	other.	This	is	done	in	Table	6. 50 51 Table	6	presents	the	particulate	counterparts	of	the	cosmogonic	events studied	in	Tables	3	and	4.	The	particles	are	not	to	be	regarded	as	mere	physical objects	appearing	in	space.	Each	spinning	particle	(or,	by	the	principle	of complementarity,	each	wave)	is	in	fact	to	be	understood	as	a	dynamic	dimension unto	itself,	an	archetypal	action	that	combines	matter	and	psyche	in	an	intimate way.	Phenomenologically,	we	can	describe	the	particles	as	constituting	lifeworlds, but	it	may	be	even	better	to	view	them	as	"braneworlds."	For	we	are	employing	a certain	kind	of	string	theory,	and,	in	the	M-theoretic	elaboration	of	string	theory, "branes"	enter	the	picture	as	higher-dimensional	versions	of	strings.	Therefore,	if strings	are	taken	as	one-dimensional	vibrations,	branes	(as	in	"membranes")	can	be vibrations	of	higher	dimension.	According	to	Greene,	our	whole	universe	might	be conceived	as	a	"braneworld"	(2004,	386).	What	we	are	dealing	with	presently	are four	such	universes. With	Table	6,	we	have	a	novel	iteration	of	the	cosmogonic	spiral.	Here	the four	braneworlds	are	represented	in	the	four	windings	of	the	spiral.	In	section	4,	we saw	the	consequence	of	considering	cosmogony	as	a	process	of	symmetry	breaking: starting	from	symmetry,	an	effective	dimensional	rendering	of	nature's	evolving forces	cannot	be	realized.	In	the	rendition	signified	by	Table	6,	the	issue	is	resolved by	regarding	the	dimensional	generation	of	particles	(waves,	braneworlds)	as resulting	from	the	dialectical	interplay	of	symmetry	and	asymmetry	that	we	have called	synsymmetry:	the	enantiomorphic	fusion	that	creates	braneworld	symmetry in	one	winding	carries	with	it	the	implication	of	an	enantiomorphic	diffusion	giving rise	to	new	asymmetry	for	the	creation	to	takes	place	in	the	next	winding. We	may	compare	the	overall	spiral	pattern	of	cosmogony	brought	out	in Tables	3,	4	and	6	to	mainstream	cosmological	depictions.	While	most	cosmologists now	appear	to	agree	that	the	three-dimensional	universe	was	characterized	early	on by	phases	of	expansion,	there	is	less	agreement	on	the	fate	of	the	universe.	Will	it continue	to	expand,	or	will	it	reverse	itself	and	begin	to	contract?	Cosmologists	Paul Steinhardt	and	Neil	Turok	(2002)	have	addressed	this	question	by	offering	a	new version	of	the	oscillating	universe	idea,	one	that	has	attracted	much	attention among	astrophysicists.	Taking	their	cue	from	string/M-theory,	Steinhardt	and	Turok propose	a	cyclic	model	in	which	the	universe	undergoes	countless	rounds	of expansion	and	contraction. Our	topo-phenomenological	version	of	string	theory	suggests	an	evolving psychophysical	cosmos	that	is	not	merely	cyclical	but	spiralic.	In	agreement	with Steinhardt	and	Turok,	the	notion	of	synsymmetry	requires	that	the	cosmic expansion	we	are	currently	experiencing	in	this	three-dimensional	Kleinian braneworld	will	be	followed	by	a	contraction,	after	which	another	period	of expansion	shall	ensue.	But	the	subsequent	expansion	will	not	just	repeat	the previous	one.	It	will	involve	the	opening	up	of	a	whole	new	dimension,	including new	forms	of	matter	and	a	new	force	of	nature	beyond	what	appears	in	Table	6.	I shall	have	more	to	say	about	this	prospect	before	I	conclude.	What	I	presently	want to	emphasize	is	that	the	general	picture	of	cosmic	development	I	am	proposing	is neither	of	a	simply	open	universe	whose	given	dimensions	expand	indefinitely,	nor of	a	closed	universe	featuring	endless	cycles	of	expansion	and	contraction.	We	must 52 imagine	instead	an	evolving	cosmos	whose	contractions	are	the	"labor	pains"	that accompany	the	birthing	of	new	dimensional	organisms.	Each	cosmic	organism expands	in	its	turn,	only	to	experience	cosmic	contractions	that	pave	the	way	for	the next	round	of	creative	cosmic	growth. The	windings	of	Table	6	unfold	in	close	parallel	with	those	of	Tables	3b	and 4,	allowing	us	to	see	the	connections	among	topodimensional,	Taoist,	and particulate	descriptions	of	cosmogony.	In	the	first	winding,	we	have	the	graviton	or gravitational	wave,	G.	Given	its	equivalence	to	εD0(!/2),	we	may	take	it	to	comprise the	sub-lemniscatory	zero-dimensional	braneworld	or	vibrating	universe.	Rather than	undergoing	its	own	evolution,	G	functions	purely	as	a	carrier	wave	for	the generation	of	higher-dimensional	waves	that	initially	reside	as	seeds	within	its	zerodimensional	field. The	second	winding	of	the	cosmogonic	spiral	concerns	the	generation	of	the strong	nuclear	force:	g	≡	εD1(!/2).	The	strong	force	corresponds	to	the	lemniscatory braneworld	consisting	of	one	dimension.	In	stage	1	of	this	winding,	we	find	a primordial	matrix	wherein	g	is	manifested	in	nascent	form,	an	embryonic	particlewave	supported	by	the	overtone	and	undertone	enantiomorphs	of	the	G	carrier wave:	g/G	and	G/g,	respectively.	The	projection	of	the	one-dimensional	braneworld has	not	yet	occurred. In	advancing	to	the	second	stage	of	this	projective	cycle,	G	enantiomorphs fuse	to	bring	symmetry	to	g.	Thus	gaining	maturity,	the	strong-force	universe expands,	and,	in	the	process,	G	is	absorbed	into	the	emergent	structure	of	g.	This "annihilation"	of	G	at	the	same	time	"freezes	it	out."	We	can	see	how	this	works	from our	analysis	in	sections	7	and	8. Coincident	with	the	projective	potentiation	of	g	as	the	one-dimensional braneworld	prevailing	in	stage	2,	G	is	subjected	to	a	depotentiating	projection.	With its	original	vitality	presently	masked,	it	is	manifested	as	an	objectified	dimension embedded	within	the	larger,	ostensibly	objective	universe	of	g.	It	is	in	this	form	that G	is	"frozen	out"	in	stage	2	(indicated	in	Table	6	by	its	enclosure	within parentheses).	So	the	primordial	gravitational	field	that	had	served	as	carrier	wave for	the	inchoate	strong	force	in	stage	1	is	now	in	eclipse,	with	an	attenuated remnant	of	it	operating	within	the	field	of	the	mature	strong	force.	There	will	be further	occlusions	of	G	in	subsequent	windings	and	this	will	eventuate	in	the attenuated	form	of	gravitation	known	to	us	today-the	form	that	conventional analysis	cannot	effectively	reconcile	with	the	other	three	forces	of	nature	to	yield	a consistent	theory	of	quantum	gravity. The	events	attending	the	transition	to	cycle	2	come	as	we	might	expect	at	this point	in	our	investigation.	In	stage	3,	the	gears	shift	from	forward	to	backward, expansion	to	contraction,	and	the	stage-2	projection	of	a	presumably	objective strong-force	universe	is	withdrawn	via	retrograde	recognition	of	its	sub-objective source.	Then,	in	stage	4,	the	retrojection	goes	further	and	G	emerges	from	eclipse, converging	harmonically	with	g,	thus	sowing	the	seeds	for	the	next	winding. With	the	opening	of	the	third	cosmogonic	epoch,	we	have	the	generation	of the	weak	boson	pair:	W	and	Z	≡	εD2(!/2).	Topodimensionally,	this	entails	the evolution	of	the	two-dimensional	Moebial	braneworld.	In	the	first	stage	of	this 53 winding,	we	find	ourselves	back	in	the	primordial	matrix	of	the	gravitational	force, with	the	embryonic	weak	force	carried	by	the	new	enantiomorphs	of	G:	(W,	Z)/G and	G/(W,	Z).	Then,	in	stage	2,	these	enantiomorphs	fuse	and	symmetry	is	enhanced in	a	first	projective	expansion	of	the	weak-force	universe,	coupled	with	a depotentiating	projection	of	the	gravitational	force	that	freezes	it	out.	At	the	same time,	the	potential	for	further	weak-force	development	is	carried	by	the enantiomorphs	of	the	strong	force,	(W,	Z)/g	and	g/(W,	Z).	The	final	stage	of	cycle	1 completes	the	maturation	of	the	weak	force.	Here	strong-force	enantiomorphs	fuse, the	strong	force	freezes	out,	and	the	weak	force	gains	full	symmetry	with	the projection	of	an	ostensibly	objective	two-dimensional	braneworld.	Needless	to	say, all	this	is	counteracted	in	shifting	to	the	retrograde	contractions	of	cycle	2:	The projections	of	the	weak	force	are	withdrawn,	the	repressions	of	the	strong	and gravitational	forces	are	lifted,	and	the	three	forces	enter	into	harmony,	sowing	the seeds	for	the	cosmogonic	epoch	to	come. The	third	round	of	particle	generation	is	given	in	the	Kleinian	epoch	that brings	in	the	electromagnetic	force,	γ	≡	εD3(!/2).	By	now	the	reader	is	well equipped	to	track	the	stages	of	projection	shown	for	this	winding.	In	Table	6	we	see the	development	of	γ	from	its	embryonic	beginnings	in	the	archaic	gravitational matrix	through	three	stages	of	expansion	facilitated	by	enantiomorphic	fusions	of the	lower-dimensional	force	particles.	The	process	culminates	in	stage	4	with	the projection	of	our	visible	three-dimensional	universe-visible	by	means	of	the particle	permitting	us	to	view	it,	the	photon. Before	considering	the	second	cycle	of	the	electromagnetic	winding,	I	would like	to	discuss	a	feature	of	Table	6	that	bears	interesting	implications	for cosmogony.	Only	the	electromagnetic	winding	is	indexed	chronologically	by cosmogonic	eras:	the	Planck	era	(t	<	10–43	sec),	the	GUT	era	(10–43	to	10–36	sec),	the Electroweak	era	(10–36	to	10–12	sec),	and	our	present	era	(t	>	10–12	sec).	Why	are there	no	chronologically	dated	eras	given	for	the	lower-dimensional	windings	of	the spiral?	It	is	because	the	forms	of	time	associated	with	them	are	not	chronological.	In section	4,	I	indicated	that	each	topodimensional	spinor	is	related	to	a	different	order not	only	of	space	but	of	time	as	well,	though	I	did	not	elaborate	on	these	differences. It	does	seem	reasonable	to	hypothesize	that	different	lifeworlds	entail	different orders	of	time,	and	that	worlds	independent	of	our	three-dimensional	universe	are governed	by	forms	of	time	qualitatively	distinct	from	the	linear	time	so	familiar	to us.	While	I	have	explored	nonlinear	temporality	elsewhere	(Rosen	2006),	here	I	will only	observe	that	our	first	cosmogonic	winding	surely	defies	chronological	dating since	it	is	utterly	without	time,	expressing	as	it	does	the	timelessness	of	the	Tao	(on the	timeless	aspect	of	the	Tao,	see	Fagg	1985	and	Butler-Bowdon	2012). Now,	the	Kleinian	winding	of	the	cosmogonic	spiral	has	special	significance for	us.	It	is	our	own	winding.	The	dimensional	context	developed	in	it	is	the	threedimensional	world	that	frames	our	immediate	perceptions	and	serves	as	the concrete	ground	of	the	very	analysis	carried	out	in	this	paper.	The	photon	generated in	the	Kleinian	epoch	is	indeed	the	primary	means	by	which	we	make	observations, conduct	experiments,	perform	measurements,	and	generally	interact	with	our 54 environment.	So,	on	the	cosmogonic	map,	the	words	"you	are	here"	might	well	have been	marked	for	this	epoch. From	our	three-dimensional	vantage	point,	we	can	describe	the	retrograde stages	of	the	lower	dimensions	in	a	detached	fashion,	but	we	do	not	have	that	luxury when	it	comes	to	describing	the	second	cycle	of	our	own	dimension.	To	unify	the quantum	gravitational	field,	the	frozen-out,	depotentiated	braneworlds	must	be repotentiated	in	conjunction	with	the	electromagnetic	braneworld.	Since	the electromagnetic	world	is	the	analyst's	own	world,	and	since	this	is	a	psychophysical world	rather	than	merely	being	physically	"out	there,"	the	analyst	cannot	stand	aloof from	the	process,	treating	it	as	something	unfolding	objectively	before	her.	The analyst	is	here	on	the	cosmogonic	map,	and	he	must	participate	accordingly. To	be	sure,	in	entering	cycle	2,	we	have	already	taken	the	first	step	in	this direction	with	our	retrojective	reflections	on	the	psychophysical	nature	of	our universe.	Yet,	in	making	the	transition	to	stage	5,	our	perspective	remains	strictly three-dimensional,	as	it	was	in	stage	4.	What	we	need	to	do	in	the	subsequent	stages of	this	cycle	is	tangibly	engage	with	the	lower	dimensional	sub-objectivities	as	the repressions	of	them	are	lifted	and	their	original	potencies	are	restored.	But	just what	would	it	mean	for	the	scientist	to	forego	his	currently	objectifying	posture	in favor	of	one	that	is	deeply	participatory? 10.	TOWARD	A	PARTICIPATORY	COSMOGONY The	philosopher	of	science	Evelyn	Fox-Keller	calls	for	a	new	form	of	perception	in scientific	inquiry	that	she	names	"dynamic	objectivity"	(1985,	115).	The	old approach,	she	says,	involves	a	"static	objectivity"	in	which	"the	pursuit	of knowledge...begins	with	the	severance	of	subject	from	object"	(117).	In	contrast, dynamic	objectivity	aims	at	a	form	of	knowledge	that	grants	to	the	world around	us	its	independent	integrity	but	does	so	in	a	way	that	remains cognizant	of,	indeed	relies	on,	our	connectivity	with	that	world.	In	this, dynamic	objectivity	is	not	unlike	empathy,	a	form	of	knowledge	of	other persons	that	draws	explicitly	on	the	commonality	of	feelings	and	experience in	order	to	enrich	one's	understanding	of	another	in	his	or	her	own	right. (1985,	117) Dynamic	objectivity	employs	a	type	of	awareness	akin	to	the	retrograde	act of	withdrawing	projections.	This	is	evidenced	in	Fox-Keller's	citation	of	Piaget: "'Objectivity	consists	in...fully	realizing	the	countless	intrusions	of	the	self	in everyday	thought	and	the	countless	illusions	which	result....So	long	as	thought	has not	become	conscious	of	self,	it	is	prey	to	perpetual	confusions	between	objective and	subjective'"	(117).	According	to	Fox-Keller: Dynamic	objectivity	is	thus	a	pursuit	of	knowledge	that	makes	use	of subjective	experience	(Piaget	calls	it	consciousness	of	self)	in	the	interests	of a	more	effective	objectivity.	Premised	on	continuity	[of	self	and	other],	it recognizes	difference	between	self	and	other	as	an	opportunity	for	a	deeper 55 and	more	articulated	kinship.	The	struggle	to	disentangle	self	from	other	is itself	a	source	of	insight-potentially	into	the	nature	of	both	self	and other....To	this	end,	the	scientist	employs	a	form	of	attention	to	the	natural world	that	is	like	one's	ideal	attention	to	the	human	world:	it	is	a	form	of love.	The	capacity	for	such	attention,	like	the	capacity	for	love	and	empathy, requires	a	sense	of	self	secure	enough	to	tolerate	both	difference	and continuity.	(1985,	117–18) Writing	in	the	same	vein,	Fox-Keller	adduces	Ernest	Schachtel's	distinction between	"autocentric"	and	"allocentric"	perception.	Whereas	the	former	is "dominated	by	need	or	self-interest,"	the	latter	"is	perception	in	the	service	of	a	love 'which	wants	to	affirm	others	in	their	total	and	unique	being.'	It	is	an	affirmation	of objects	as	'part	of	the	same	world	of	which	man	is	a	part,'"	one	which	"permits	a fuller,	more	'global'	understanding	of	the	object	in	its	own	right"	(119).	Although Fox-Keller	pays	scant	attention	to	phenomenological	philosophy	as	such,	the	main thrust	of	her	presentation	is	much	in	keeping	with	phenomenology's	central	aim,	as expressed	in	its	well-known	slogan:	"To	the	things	themselves!"	And	it	seems	clear that	the	world	shared	by	the	"allocentric"	observer	and	the	objects	that	s/he observes	is	the	lifeworld	of	phenomenology. Fox-Keller	helps	us	gain	a	better	grasp	of	the	new	mode	of	scientific	inquiry by	offering	a	specific	example	of	one	of	its	premier	practitioners:	the	Nobel	prizewinning	biologist,	Barbara	McClintock.	In	stark	contrast	to	the	detached, dispassionate	attitude	of	the	Cartesian	scientist,	McClintock	speaks	of	obtaining	an intimate	feeling	for	the	plants	she	works	with:	"'I	don't	feel	I	really	know	the	story	if I	don't	watch	the	plant	all	the	way	along.	So	I	know	every	plant	in	the	field.	I	know them	intimately,	and	I	find	it	a	great	pleasure	to	know	them'"	(Fox-Keller	1985, 164).	In	another	place,	McClintock: describes	the	state	of	mind	accompanying	the	crucial	shift	in	orientation	that enabled	her	to	identify	chromosomes	she	had	earlier	not	been	able	to distinguish:	"I	found	that	the	more	I	worked	with	them,	the	bigger	and	bigger [the	chromosomes]	got,	and	when	I	was	really	working	with	them	I	wasn't outside,	I	was	down	there.	I	was	part	of	the	system....It	surprised	me	because I	actually	felt	as	if	I	was	right	down	there	and	these	were	my	friends....As	you look	at	these	things,	they	become	part	of	you.	And	you	forget	yourself." (McClintock	quoted	in	Fox-Keller	1985,	165) Fox-Keller	observes	that	McClintock's	vocabulary	"is	consistently	a	vocabulary	of affection,	of	kinship,	of	empathy,"	an	empathy	that	constitutes	"the	highest	form	of love:	love	that	allows	for	intimacy	without	the	annihilation	of	difference"	(164). Here	the	word	"love"	is	used	"neither	loosely	nor	sentimentally,	but	out	of	fidelity	to the	language	McClintock	herself	uses	to	describe	a	form	of	attention,	indeed	a	form of	thought"	(164). Fox-Keller	arrives	at	these	conclusions: 56 The	crucial	point	for	us	is	that	McClintock	can	risk	the	suspension	of boundaries	between	subject	and	object	without	jeopardy	to	science	precisely because,	to	her,	science	is	not	premised	on	that	division.	Indeed,	the	intimacy she	experiences	with	the	objects	she	studies...is	a	wellspring	of	her	powers	as a	scientist....In	this	world	of	difference,	division	is	relinquished	without generating	chaos.	Self	and	other,	mind	and	nature	survive	not	in	mutual alienation,	or	in	symbiotic	fusion,	but	in	structural	integrity.	(1985,	164–165) Finally,	after	recounting	the	goal	of	conventional	science,	Fox-Keller	observes that,	"To	McClintock,	science	has	a	different	goal:	not	prediction	per	se,	but understanding;	not	the	power	to	manipulate,	but	empowerment-the	kind	of	power that	results	from	an	understanding	of	the	world	around	us,	that	simultaneously reflects	and	affirms	our	connection	to	that	world"	(166). In	phenomenological	terms,	the	world	to	which	McClintock	is	connected	in feeling	and	embodied	empathy	is	the	lifeworld.	It	is	a	world	in	which	the	dialectic	of difference	and	identity	is	enacted	through	an	intimate	knowledge	of	other	that requires	and	is	inseparable	from	the	knowledge	of	self	(a	"consciousness	of	self"). McClintock's	"revolution	that	'will	reorganize...the	way	we	do	[scientific]	research'" (Fox-Keller	1985,	172)	depends	upon	descending	from	the	Cartesian	stratosphere and	immersing	ourselves	in	the	psychophysical	dimension	wherein	object	and subject,	symmetry	and	asymmetry,	continuity	and	discontinuity	mediate	one another	internally	in	an	encompassing	circular	flow.	We	have	not	forgotten	that	the source	of	this	phenomenological	circulation	lies	in	the	ancient	eddies	of	yin	and yang. Fox-Keller's	"dynamic	objectivity"	as	exemplified	by	McClintock	is	hardly	the only	instance	of	the	burgeoning	of	a	new	dialectical	science.	The	phenomenological initiative,	begun	early	in	the	twentieth	century,	has	been	advanced	by	thinkers	like Heelan	(1983)	and	Gendlin	(1991),	who	have	proposed	that	the	work	of	science	not proceed	from	"stratospheric"	perception,	but	from	the	intricacies	of	the	lifeworld	or lived	body.	A	dialectical	approach	to	science	also	is	advocated	by	biophysicist Koichiro	Matsuno	(1995).	Matsuno	has	called	for	a	"dialogical"	science	that	would supersede	the	old	"monologue"	carried	on	by	the	solitary	Cartesian	subject	looking down	upon	the	world	from	above.	In	Matsuno's	vision,	scientific	activity	would involve	a	community	of	subjects	concretely	engaged	with	each	other	in	dynamic	and generative	negotiations.	Whereas	the	Cartesian	subject	is	anonymous,	absent	from the	events	that	transpire,	the	participants	in	the	dialectical	community	would function	self-referentially	to	include	themselves	in	the	process	(Matsuno	exemplifies this	by	explicitly	including	himself	as	author	in	what	he	writes;	1995,	1998).	Other important	contributions	come	from	Plamen	Simeonov	(2012),	who	has	emphasized the	need	to	devise	first-person	methodologies	for	the	natural	sciences;	from	Arran Gare	(2013),	with	his	insistence	that	science	be	grounded	in	a	way	that	includes lived	subjectivity;	and	from	Louis	Kauffman's	(2015)	reflections	on	how mathematical	self-reference	is	related	to	topology	and	phenomenological philosophy.	Still	another	contribution	to	emergent	dialectical	science	is	offered	by the	Jungian	psychologist	Nathan	Schwartz-Salant	(2007).	Operating	selfreferentially,	Schwartz-Salant	employs	Merleau-Ponty	and	the	Klein	bottle	in 57 characterizing	the	deep	psychodynamics	of	human	relationships,	and	he	likens	the fields	operative	in	these	paradoxical	interactions	to	field	processes	in	fundamental physics	(see	also	Schwartz-Salant	2017). What	we	require	in	the	present	context	is	a	dialectical	cosmogony.	In	this approach,	the	analysis	of	cosmogony	is	situated	within	cosmogony	itself	("we	are part	of	the	world	we	are	trying	to	know,"	says	Gare;	2013,	25).	The	unquestioned objective	stance	analysts	have	tended	to	take	toward	cosmic	evolution	is	in	fact characteristic	of	the	fourth	and	final	stage	of	projection	in	the	Kleinian	epoch	of dimensional	generation.	This	is	the	stage	of	development	in	which	we	assume	that cosmogonic	events	are	"objectively	out	there,"	and	that	we	analysts	are	detached from	them,	with	our	lived	subjectivity	playing	no	role	in	what	we	see.	In	this	stage, the	common	sense	notion	of	an	external	universe	developing	on	its	own	is	so compelling	that	it	seems	absurd	for	us	to	think	otherwise.	But,	in	advancing	to	the stages	of	cycle	2,	the	point	comes	home	to	us	that	we	are	indeed	intimate participants	in	the	story	of	cosmic	creation.	Thus	entering	into	cosmogonic	process, the	classical	posture	of	analysis	gives	way	to	a	phenomenological	one	in	which	our own	process	of	development	plays	an	integral	role.	In	the	act	of	inwardly	grasping the	transformation	of	the	cosmos,	the	analyst	surpasses	the	projective	construction of	herself	as	an	isolated	onlooker	and	takes	part	in	the	drama	of	creating	a	world.	So, if	the	cosmos	is	self-evolving,	the	self	of	the	analyst	figures	essentially	in	the reflexive	enactment	of	this	process. But	let	me	try	to	be	clearer	about	what	the	involvement	of	the	analyst specifically	entails.	The	proposition	I	venture	to	suggest	is	that	a	fully	reflexive analysis	of	cosmogony	requires	that,	in	investigating	the	stages	of	cosmogonic retrojection,	the	analyst	must	gain	palpable	awareness	of	his	or	her	own	stages	of development.	Only	then	can	the	link	to	cosmic	development	be	realized	in	its existential	immediacy,	since,	only	then	would	the	analyst	realize	cosmic transformation	as	a	self-transformation,	not	just	a	transformation	of	what	is	other. Of	course,	the	analytical	self	in	question	cannot	merely	be	that	of	a	particular individual.	The	self	that	participates	in	the	archetypal	processes	of	cosmic	creation must	function	archetypally.	Yet	it	seems	we	need	to	begin	with	the	particular	person if	the	process	is	to	be	grounded	in	existential	reality.	Presumably,	in	the	course	of deeply	exploring	his	or	her	own	past,	the	analyst	would	cross	a	threshold	and	her personal	being	would	shade	into	the	transpersonal.	The	transpersonal	psychiatrist Stanislav	Grof	expressed	a	similar	idea	in	describing	the	transformation	of awareness	that	can	occur	in	the	act	of	re-experiencing	the	"perinatal"	stages	of development,	those	occurring	around	the	time	of	birth:	"All	we	can	say	is	that somewhere	in	the	process	of	confrontation	with	the	perinatal	level	of	the	psyche,	a strange	qualitative	Moebius-like	[!]	shift	seems	to	occur	in	which	deep	selfexploration	of	the	individual	unconscious	turns	into	a	process	of	experiential adventures	in	the	universe-at-large"	(1985,	36). In	the	projective	moment	of	cosmogony,	it	may	well	seem	a	flight	of	fancy	to link	the	stages	of	human	development	to	those	of	the	cosmos	as	a	whole.	The phenomenological	response	to	this	incredulity	extends	the	biological	dictum	that "ontogeny	recapitulates	phylogeny"	to	the	field	of	physics	and	says,	ontogeny recapitulates	cosmogony.	For,	if	it	is	true	that	we	participate	in	the	story	of	creation 58 in	a	full-fledged	way,	it	would	seem	that	our	own	history	would	be	inseparable	not only	from	that	of	the	broader	biological	world	but	from	nature	at	large.	Evidently then,	when	we	move	backward	through	the	stages	of	cycle	2	to	gain	"allocentric" awareness	of	the	braneworlds	belonging	to	nature's	archaic	past,	it	seems	we	must work	through	our	own	archaic	past	if	we	are	to	apprehend	those	worlds	in	the	most concrete,	immediate,	and	deeply	reflexive	way. Embryological	research	certainly	appears	to	support	the	idea	that	the	early development	of	the	human	individual	mirrors	the	development	of	the	species	as	a whole.	In	fact,	my	earlier	work	links	ontogeny	and	phylogeny	explicitly,	and	in	a detailed	way	(Rosen	2006).	Presently,	it	is	ontogeny	and	cosmogony	that	must	be linked.	Some	theorists	have	broadly	speculated	that	the	universe	functions	as	a	giant hologram	(Bohm	1980,	189).	Such	a	cosmos	should	possess	a	fractal	pattern	of	selfsimilarity,	with	the	structure	and	development	of	the	whole	being	mirrored recursively	on	every	scale	of	magnitude	down	to	the	smallest	part.	Then-if	probing the	early	history	of	an	individual	member	of	the	phylogenetic	order	opens	out	into phylogeny	as	a	whole-it	is	perhaps	not	unreasonable	to	hypothesize	a	deeper stratum	of	self-similarity	involving	the	history	of	the	cosmos	to	which	we	belong. Relevant	in	this	regard	is	the	vision	of	physicist	Lee	Smolin: Living	things	share	in	some	ways,	and	extend	in	other	ways,	the	basic properties	of	non-equilibrium	self-organized	systems	that	seem	to characterize	the	universe	on	every	scale,	from	the	cosmos	as	a	whole	to	the surface	of	planets....If	life,	order	and	structure	are	the	natural	state	of	the cosmos	itself,	then	our	existence,	indeed	our	spirit,	might	finally	be comprehended	as	created	naturally,	by	the	world,	rather	than	unnaturally and	in	opposition	to	it.	(1997,	160) In	a	similar	vein,	biophysical	theorist	Hector	Sabelli	asserts	that	"the	continuity	of evolution	requires	that	the	same	fundamental	forms	must	be	expressed	at	the physical,	biological,	and	psychological	levels	of	organization"	(2005,	431).	This	is consistent,	of	course,	with	the	psychophysical	nature	of	cosmogonic	process. In	The	Self-Evolving	Cosmos,	I	attempted	to	spell	out	more	specifically	the manner	in	which	the	analyst's	own	development	can	be	linked	to	that	of	the	cosmos at	large.	But	I	can	go	no	further	here	if	this	already	lengthy	paper	is	to	stop	short	of becoming	a	book! Let	me	conclude	by	returning	to	the	issue	raised	near	the	end	of	section	7. There	I	noted	that	the	guiding	principle	of	synsymmetry	implies	the	birth	of	a higher-dimensional	lifeworld	surpassing	the	Kleinian.	The	question	I	posed	was how	we	can	come	to	know	this	four-dimensional	reality	in	a	tangible	way.	What	I indicated	is	that	going	beyond	an	abstract	three-dimensional	analysis	of	the	fourth dimension	necessitates	the	dimensional	evolution	of	our	analytic	framework	itself. At	that	point,	it	was	already	implicit	that	the	backward	movement	through	the second	Kleinian	cycle	brings	this	evolution	about.	We	knew	that,	in	the	course	of cycle	2,	lower-dimensional	organisms	rise	from	obscurity,	and-operating	in synchrony	with	the	Kleinian	organism-sow	the	seeds	for	the	meta-Kleinian,	fourdimensional	framework.	But	we	knew	this	only	abstractly,	since	our	way	of	knowing 59 was	itself	restricted	to	the	three-dimensional	context.	So	the	obstacle	to	tangibly apprehending	the	fourth	dimension	is	our	inability	to	grasp	it	in	a	manner transcending	the	limits	of	our	three-dimensional	frame	of	analysis.	Clearly	the theoretical	anticipation	of	completing	the	second	Kleinian	cycle	is	no	substitute	for completing	it	in	actuality.	How	can	the	latter	be	achieved?	I	suggest	it	can	happen	by adopting	the	reflexive	phenomenological	posture	intimated	above.	Here	we participate	allocentrically	with	the	lower-dimensional	organisms	of	cycle	2, investigating	the	stages	of	development	as	stages	in	our	own	development, experiencing	the	transformations	occurring	as	self-transformations.	It	is	when	this process	is	brought	to	fruition	that	a	new	winding	of	the	cosmogonic	spiral	opens	up and	a	whole	new	world	is	introduced-a	world	we	will	have	come	to	know substantively	by	the	dimensional	expansion	of	our	capacity	for	knowing. While	my	own	epistemic	capacity	remains	distinctly	three-dimensional,	let me	offer	for	what	it	is	worth	a	provisional,	still	quite	abstract	impression	of	what may	be	in	store.	In	the	wider	turning	of	the	spiral,	a	new	and	more	dialectically intricate,	four-dimensional	braneworld	would	come	into	play	beyond	the	Kleinian world,	a	topological	action	pattern	laid	out	in	5	×	5	matrices.	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