Packing optimization problems have a wide spectrum of real-word applications. One of the applications of the problems is problem of placement of containers with spent nuclear fuel on the storage platform. The solution of the problem can be reduced to the solution of the problem of finding the optimal placement of a given set of congruent circles into a multiconnected domain taking into account technological restrictions. A mathematical model of the prob-lem is constructed and its peculiarities are considered. Our approach (...) is based on the mathematical modelling of rela-tions between geometric objects by means of phi-function technique. That allowed us to reduce the problem solving to nonlinear programming. Today, an important scientific problem is the problem of creating conditions for safe storage of spent nuclear fuel. In the process of creating any dry spent nuclear fuel storage, the following main stages can be identified: site selection, storage design, construction, operation and decommissioning. A full check for compliance of the repository and its elements with these standards usually begins at the design stage. At the stage of site selection, the inspection for compliance with safety standards is carried out only in terms of the impact of the repository as a whole on the environment. This approach cannot be considered fully appropriate, because, taking into account, for example, all the climatic features of the future storage site, it is possible to adjust the thermal storage regimes of spent nuclear fuel. Similarly, it can be considered necessary to analyze and select the shape of the storage site in order to accommo-date the maximum possible number of spent fuel containers. Such a choice, obviously, should be made taking into ac-count the norms of nuclear, radiation and thermal safety, as well as in compliance with technological limitations. The problem of finding the optimal placement of containers taking into account the given technological limitations can be formulated in the form of the problem of optimization of geometric design. Therefore, the purpose of the study is to build a mathematical model of the problem and study its characteristics to develop effective methods of solution. The proposed approach is based on mathematical modeling of relations between geometric objects using the method of phi-functions. This allowed to reduce the solution of the problem to the problem of nonlinear programming. (shrink)

© 2018, Springer International Publishing AG, part of Springer Nature. Aristotelian diagrams are used extensively in contemporary research in artificial intelligence. The present paper investigates the geometric and cognitive differences between two types of Aristotelian diagrams for the Boolean algebra B4. Within the class of 3D visualizations, the main geometric distinction is that between the cube-based diagrams and the tetrahedron-based diagrams. Geometric properties such as collinearity, central symmetry and distance are examined from a cognitive perspective, focusing on diagram design principles (...) such as congruence/isomorphism and apprehension. The cognitive effectiveness of the different visualizations is compared for the representation of implication versus opposition relations, and for subdiagram embeddings. (shrink)

This book gives an analysis of Hertz's posthumously published Principles of Mechanics in its philosophical, physical and mathematical context. In a period of heated debates about the true foundation of physical sciences, Hertz's book was conceived and highly regarded as an original and rigorous foundation for a mechanistic research program. Insisting that a law-like account of nature would require hypothetical unobservables, Hertz viewed physical theories as images of the world rather than the true design behind the phenomena. This paved the (...) way for the modern conception of a model. Rejecting the concept of force as a coherent basic notion of physics he built his mechanics on hidden masses and rigid connections, and formulated it as a new differential geometric language.Recently many philosophers have studied Hertz's image theory and historians of physics have discussed his forceless mechanics. The present book shows how these aspects, as well as the hitherto overlooked mathematical aspects, form an integrated whole which is closely connected to the mechanistic world view of the time and which is a natural continuation of Hertz's earlier research on electromagnetism. Therefore it is also a case study of the strong interactions between philosophy, physics and mathematics. Moreover, the book presents an analysis of the genesis of many of the central elements of Hertz's mechanics based on his manuscripts and drafts. Hertz's research program was cut short by the advent of relativity theory but its image theory influenced many philosophers as well as some physicists and mathematicians and its geometric form had a lasting influence on advanced expositions of mechanics. (shrink)

The design of the bronchial tree has largely been proposed as a model of optimal design from a physical-functional perspective. However, the distributive function of the airway may be more related to a geometrical than a physical problem. The bronchial tree must distribute a three dimensional volume of inspired air on a two dimensional alveolar surface, included in a limited volume. It is thus valid to ask whether an optimal bronchial tree from a physical perspective is also optimum from (...) a geometrical point of view. In this paper we generate a simple geometric model for the branching pattern of the bronchial tree, deducing relationships that permit estimation of the departures from the geometrical optimum of each bifurcation. We also, for comparative purposes, estimate the departures from the physical optimum. From the geometrical assumptions: i) a symmetrical dichotomic fractal design, ii) with minimum volume and iii) maximum dispersion of the terminal points; and several simulations we suggest that the optimality is characterized by a bifurcation angle Θ ∼ 60° and a length reduction scale γ = (1/2)1/3 = 0.7937. We propose distances from the physical and geometrical optimality defined as Euclidean distances from the expected optima. We show how the advanced relationships and the distances can be used to estimate departures from the optimality in bronchographs of four species. We found lower physical and geometrical departures in the distal zone than those of the proximal zones, as well as lower physical than geometrical departures from optimality. (shrink)

The Topos of Music is the upgraded and vastly deepened English extension of the seminal German Geometrie der Töne. It reflects the dramatic progress of mathematical music theory and its operationalization by information technology since the publication of Geometrie der Töne in 1990. The conceptual basis has been vastly generalized to topos-theoretic foundations, including a corresponding thoroughly geometric musical logic. The theoretical models and results now include topologies for rhythm, melody, and harmony, as well as a classification theory of musical (...) objects that comprises the topos-theoretic concept framework. Classification also implies techniques of algebraic moduli theory. The classical models of modulation and counterpoint have been extended to exotic scales and counterpoint interval dichotomies. The probably most exciting new field of research deals with musical performance and its implementation on advanced object-oriented software environments. This subject not only uses extensively the existing mathematical music theory, it also opens the language to differential equations and tools of differential geometry, such as Lie derivatives. Mathematical performance theory is the key to inverse performance theory, an advanced new research field which deals with the calculation of varieties of parameters which give rise to a determined performance. This field uses techniques of algebraic geometry and statistics, approaches which have already produced significant results in the understanding of highest-ranked human performances. The book's formal language and models are currently being used by leading researchers in Europe and Northern America and have become a foundation of music software design. This is also testified by the book's nineteen collaborators and the included CD-ROM containing software and music examples. (shrink)

© 2017 by the authors. Aristotelian diagrams visualize the logical relations among a finite set of objects. These diagrams originated in philosophy, but recently, they have also been used extensively in artificial intelligence, in order to study various knowledge representation formalisms. In this paper, we develop the idea that Aristotelian diagrams can be fruitfully studied as geometrical entities. In particular, we focus on four polyhedral Aristotelian diagrams for the Boolean algebra B4, viz. the rhombic dodecahedron, the tetrakis hexahedron, the (...) tetraicosahedron and the nested tetrahedron. After an in-depth investigation of the geometrical properties and interrelationships of these polyhedral diagrams, we analyze the correlation between logical and geometrical distance in each of these diagrams. The outcome of this analysis is that the Aristotelian rhombic dodecahedron and tetrakis hexahedron exhibit the strongest degree of correlation between logical and geometrical distance; the tetraicosahedron performs worse; and the nested tetrahedron has the lowest degree of correlation. Finally, these results are used to shed new light on the relative strengths and weaknesses of these polyhedral Aristotelian diagrams, by appealing to the congruence principle from cognitive research on diagram design. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Rhetoric 34.4 (2001) 292-307 [Access article in PDF] The Rhetoric of the Geometrical Method Spinoza's Double Strategy Christopher P. Long A double strategy may be apprehended in the first definitions, axioms and propositions of Spinoza's Ethics: the one is rhetorical, the other, systematic. Insofar as these opening passages constitute a geometrical argument that leads ultimately to the strict monism that lies at the heart of (...) Spinoza's philosophy, they function systematically as grounding principles. However, insofar as they are designed to uncover errors at the heart of the Cartesian system, they function rhetorically as a critique of Descartes's equivocal theory of substance. The geometrical mode of presentation lends an aura of necessity to the systematic strategy and tends to eclipse the rhetorical strategy that functions as a critique of Cartesian dualism. 1 To this extent, the systematic strategy itself functions rhetorically. On the other hand, to the extent that any seventeenth-century attempt to establish a monistic system would inevitably have had to involve a critique of Cartesian dualism, the rhetorical strategy itself functions systematically. Thus, although two distinct strategies are simultaneously discernible in the opening passages of the Ethics, the two work together, each arguing on a different front, to establish Spinoza's monism. By focusing on the rhetorical side of this double strategy, the underlying nature of Spinoza's critique of Descartes can be clarified and certain gaps in the systematic, geometrical argument for monism can be explained. Descartes's theory of substance The rhetorical strategy Spinoza employs at the outset of the Ethics involves a brilliant use of equivocation and ambiguity designed to draw a reader dedicated to Cartesian dualism into the heart of an argument designed to [End Page 292] undermine this dualism itself. The impetus behind this rhetorical critique is surely Spinoza's dissatisfaction with Descartes's theory of substance. 1. Descartes's equivocation Descartes's theory rests upon a fundamental equivocation: on the one hand, the term "substance" is meant strictly to designate that which is absolutely self-sufficient, namely, God; on the other hand, Descartes allows for a weaker understanding of substance, which pertains to those things (specifically res cogitans and res extensa) that require only the concurrence of God in order to exist (Principles I, 51-52; AT VIIIA24-25). 2 Thus, for Descartes, although God is, strictly speaking, the only substance, there remains a sense in which both res cogitans and res extensa may be understood as substances as well.Ultimately, this equivocal understanding of substance may be traced to a tension internal to Descartes between the Aristotelian notion of substance as underlying subject and the seventeenth-century mutation of that notion, which came to be defined in terms of causal self-sufficiency. 3 Jonathan Bennett traces the source of this mutation back to the view held by Aristotle that the existence of some property depends upon its being instantiated--that is, there can be no courage unless there exists someone courageous. He continues his interesting argument thus: This suggests that there is a relation of dependence running from properties to the things that have them and not conversely. The dependence would be logical: the thesis must be that "existent but uninstantiated property" is conceptually defective, not merely contrary to physics. Thus, one mark of a genuine thing, or substance, is its not being logically dependent for its existence on anything else. (Bennett 1984, 56) The development of this line of thinking leads to the conception of substance we find both in Descartes's Principles and in Spinoza's Ethics, namely, that a substance must not causally depend on anything outside of itself for its existence. 4 However, with Descartes, what emerges out of this tradition is a hybrid understanding of substance in which the term "substance" is not reserved simply for that which is utterly self-sufficient (God), but extends to those things that depend only upon the concurrence of God for their existence (rescogitans and resextensa). This bifurcated understanding of substance seems to stem from the distinction at work in [End Page 293] Aristotle... (shrink)

Context: Technology has not only changed the way we teach mathematical concepts but also the nature of knowledge, and thus what is possible to learn. While geometric transformations are recognized to be foundational to the formation of students’ geometric conceptions, little research has focused on how these notions can be introduced in elementary schooling. Problem: This project addressed the need for development of students’ reasoning about and with geometric transformations in elementary school. We investigated the nature of students’ understandings of (...) translations, rotations, scaling, and stretching in two dimensions in the context of use of the software application Graphs ’n Glyphs. More specifically, we explored the question “What is the nature of elementary students’ reasoning of geometric transformations when instruction exploits the technological tool Graphs ’n Glyphs?” Method: Using a design research perspective, we present the conduct of a teaching experiment with one pair of fourth-graders, studying translation and rotation. The project investigated how and to what extent activity using Graphs ’n Glyphs can elicit students’ reasoning about geometric transformations, and explored the constraints and affordances of Graphs ’n Glyphs for thinking-in-change about geometric transformations. Results: The students proved adept using the software with carefully designed tasks to explore the behavior of two-dimensional shapes, distinguish among transformations, and develop predictions. In relation to varied conditions of transformations, they formed generalizations about the way a shape behaves, including the use of referent points in predicting outcomes of translations, and recognizing the role of the center of rotation. Implications: The generalizations that the students developed are foundational for developing an understanding of the properties of transformations in the later years of schooling. Dynamic technological approaches to geometry may prove as valuable to elementary students’ understanding of geometry as it is for older students. Constructivist content: This study contributes to ongoing constructivism/constructionism conversations by concentrating on the transformation of ideas when engaging learners in activity through particular contexts and tools. Key Words: Geometry, transformations, constructionist technologies. (shrink)

In recent years, with the development of society and the rapid development of the animation industry, people are paying more and more attention to and requirements for animation production. As an indispensable part of animation production, picture composition plays a major role in animation production. It can give full play to the application of color matching and light and shadow design and enhance the depth and space of the animation screen. Tone space conversion refers to the conversion or representation of (...) color data in one color space into corresponding data in another color space. Its purpose is to distinguish and process color components such as hue and saturation in an image. This article first introduces the domestic and foreign research status of digital image preprocessing and analyzes the basic principles of several color space conversions in detail. Then, several color space conversion algorithms are studied, and the performance of the algorithms is compared and analyzed. The paper focuses on the hardware implementation and optimization of the algorithm for converting RGB color space into HSI color space to meet the real-time requirements. This article focuses on the mutual conversion between the RGB tone space and the HSI tone space and describes in detail how each color component in the HSI tone space is converted from the three RGB color components from a geometric perspective, and then the conversion is derived, and several general conversion methods of RGB to HSI tone space are introduced; two conversion methods of geometric derivation method and standard modulus algorithm are implemented in the software, and the comparison verification is carried out, and the comparison is made from the perspective of hardware implementation. The pros and cons of the two methods are discussed. Finally, the paper summarizes the shortcomings in the design and proposes further research directions in the future. (shrink)

Some ancient Greek coins from the island state of Aegina depict peculiar geometric designs. Hitherto they have been interpreted as anticipations of some Euclidean propositions. But this paper proposes geometrical constructions which establish connections to pre-Euclidean treatments of incommensurability. The earlier Aeginetan coin design from about 500 bc onwards appears as an attempt not only to deal with incommensurability but also to conceal it. It might be related to Plato’s dialogue Timaeus. The newer design from 404 bc onwards (...) reveals incommensurability, namely in the context of ‘doubling the square’. It thereby covers the same topic but a different geometry as passages in Plato’s dialogue Meno (385 bc). This coin design incorporates important elements of ancient Greek geometrical analysis of the fifth century bc like the gnomon, Hippocrates’ squaring of the lunule (ca. 430 bc), and a geometrical version of monetary equivalence. Through this venue, the design’s conceptual lineage might be traced as far back as Heraclitus’ cosmology of about 500 bc. (shrink)

For product packaging, the visual elements in it can further enhance the appeal of the package to customers. This article briefly introduces visual elements and packaging design and made an example analysis with the gift packaging design of Squirrel Design Studio. In the case study, the packaging design of the studio’s mirror, storage bag, and puzzle was rated by hierarchical analysis and questionnaires, and the packaging design was analyzed based on the rating results. A convolutional neural network (CNN) was also (...) used to evaluate packages in batches. The results showed that the CNN could make a batch evaluation of gift packaging design accurately; the three gift packaging designs were based on the studio’s logo, making the ratings similar; in addition, the packaging design patterns were composed of different geometric shapes to show the studio’s innovative design theme, and the squirrel silhouette and text description were used to strengthen the impression of the studio among customers. (shrink)

The study aims to contribute to understand the Vitruvian design system through the analysis of the temples described in the text leading to the three key concepts related: proportio, commensus and commodulatio. At first, it is proposed a conceptual discussion about the origin of the compositional elements, and further it is discussed the concept of proportio as an operation of sizing and scaling these elements individually. Analyzed the definition and types of elements, it is proposed to continue the discussion on (...) the relationship that must be built as a way of associating elements in a consistent way, leading to the notion of symmetria, understood as an ideal state in which the elements maintain dimensional and/or geometrical relationships between them, creating a relational harmony. From the operational point of view, it is sought to demonstrate that the Vitruvian symmetria is fetched from two types of relations: a modular and a dimensional. (shrink)

For VR systems, one of its core parts is to present people with a real and immersive 3D simulation environment. This paper uses real-time computer graphics technology, three-dimensional modeling technology, and binocular stereo vision technology to study the multivisual animation character objects in virtual reality technology; designs a binocular stereo vision animation system; designs and produces a three-dimensional model; and develops a virtual multivisual animation scene application. The main research content and work performed in the text include the (...) research of the basic graphics rendering pipeline process and the analysis and research of each stage of the rendering pipeline. It mainly analyzes the 3D graphics algorithm used in the three-dimensional geometric transformation of computer graphics and studies the basic texture technology, basic lighting model, and other image output processes used in the fragment processing stage. Combined with the development needs of the subject, the principles of 3D animation rendering production software and 3D graphics modeling are studied, and the solid 3D model displayed in the virtual reality scene is designed and produced. This article also reflects the application of virtual reality in multivisual animation character design from the side, so it has realistic value and application prospects. (shrink)

A geometric constraint solver for finding legal configurations for an under-constrained set of geometric components is proposed. While making drawings interactively, the user usually specifies few geometric constraints explicitly because some constraints are not clear to him- or her-self, or it is not practical to specify all constraints at any early design stage. Theoretically, the full geometric constraints are necessary to define a unique layout of every geometric components, but it is naturally not given throughout the process. Therefore, in such (...) an under-constrained situation, the lack of constraints must be supplemented properly to determine the final layout of components remaining undefined. For this purpose, we propose a geometric constraint solver that works in two phases: the former is to satisfy all the explicit constraints imposed by the user, and the latter is to choose appropriate value instances for every geometric component whose layout is not yet determined. We implemented a constraint-based interactive system for designing line drawings which is composed of the solver and the interactive module to let the user input geometric modi.cation commands incrementally, and proved its effectiveness by some experiments. (shrink)

Graphic pattern (e.g. geometric design) and number-based code (e.g. digital sequencing) can store and transmit complex information more efficiently than referential modes of representation. The analysis of the two genres and their relation to one another has not advanced significantly beyond a general classification based on motion-centred geometries of symmetry. This article examines an intriguing example of patchwork coverlets from the maritime societies of Oceania, where information referencing a complex genealogical system is lodged in geometric designs. By drawing attention (...) to the interplay of graphic pattern and number-based code and its role in the knowledge economies of maritime societies, the article offers new insight into possible ways of designing a digital informational surface that captures the behaviour of an operational system, allowing both for differentiation and integration. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:A Hypothesis Concerning the Character of Islamic ArtAsli GocerWhy Islamic art has the distinctive features it has continues to generate clashing explanations. The Islamic visual treasury has no figural images, for instance, and three-dimensional sculpture or large scale oil painting, but instead contains miniatures, vegetal ornaments, arabesque surface patterns, and complex geometrical designs. To account for the phenomena the following radically opposing theories have been offered: the (...) influence of Judaism, the Quranic prohibition against making images, 1 Islamic resentment of the glory of Byzantine icons, 2 logocentricism of classical Islam, 3 the spiritual dimension of Islam, 4 tacit Islamic assertion of “otherness,” 5 and contingency of history. 6 The merits of these theories have been roundly discussed in the literature, and so I will not repeat the task here. I will insist that none of these theories offer a coherent explanation for all arts, but my presumption is that such an explanation is possible. In nonvisual arts the Muslim prefers simple tunes over that of the symphony, for instance, whirling dervishes over the ballet, and lyric poetry over dramatic literature. In my view what explains the common characteristic of all these arts is a certain kind of Islamic attitude informed by a certain philosophical outlook. Although it would be hasty to suggest the existence of a monolithic [End Page 683] Islamic art, it would be equally rash to deny some striking commonalities among its diverse examples. 7The following observation by a Muslim scholar is notable in this regard: “Whether in the great courtyard of the Delhi Mosque or the Qarawiyyin in Fez, one feels oneself within the same artistic and spiritual universe despite all the local variations in material, structural techniques, and the like.” 8 Some cite geometry as what mediates that feeling to the observer despite the cultural divergences in style. 9 Even those who have given up the search for the universal or the spiritual in Islamic art cannot help but see in it the classical ideal of beauty. 10 It is this classical heritage that in fact holds one of the important keys to the puzzle at hand. In particular I shall claim that Islamic art is the living image of what Plato has in mind for art in a theocratic state. Clearly, to refer to Plato and art in the same breath invites trouble. As is well enough known, giving a general interpretation of Plato’s views on the nature of art is a very complex endeavor. 11 Even in the face of grave difficulties in textual exegesis, however, there is sufficient evidence that Plato has some settled views on art. Those pertinent to our discussion here involve two claims, that beauty and goodness are divine and that all human endeavor (including artistic creation) must imitate the divine. 12 Although Platonic strain in Islam has been noted in other areas, its persistent legacy has been overlooked in art. 13 I shall argue that the Platonic influence on Islamic thought can be traced not only to political philosophy 14 and geometry, 15 as it has been commonly accepted, but also aesthetics. 16 A philosophical explanation of the Islamic attitude toward art may therefore be derivable from the Platonic axiom that god is the paradigm of beauty, which is the cosmological principle of [End Page 684] order and harmony; and as all created things, art, too, must reflect the divine paradigm.Before defending this ahistorical hypothesis on the character of Islamic art so reviled in recent literature, a few caveats are in order. “Islamic art” contains redundant terms. Strictly speaking all art in Islam is “Islamic,” because for the Muslim art is a crucial point of contact with the divine. That art should direct the mind to the divine is also Plato’s view. 17 Second, the categories that have traditionally been used to describe Islamic art such as “aniconic” and “nonrepresentational” are unhelpful for the problem at hand. 18 When it comes to the issue of iconography in Islamic arts, there is no consensus, for Islamicists are notably divided on the semiotic dimension of the Islamic visual language. The question of whether the geometric surface pattern is essentially iconography, for instance, is yet... (shrink)

Discover how mathematical sequences abound in our natural world in this definitive exploration of the geography of the cosmos You need not be a philosopher or a botanist, and certainly not a mathematician, to enjoy the bounty of the world around us. But is there some sort of order, a pattern, to the things that we see in the sky, on the ground, at the beach? In A Beginner's Guide to Constructing the Universe, Michael Schneider, an education writer and computer (...) consultant, combines science, philosophy, art, and common sense to reaffirm what the ancients observed: that a consistent language of geometric design underpins every level of the universe, from atoms to galaxies, cucumbers to cathedrals. Schneider also discusses numerical and geometric symbolism through the ages, and concepts such as periodic renewal and resonance. This book is an education in the world and everything we can't see within it. Contains numerous b&w photos and illustrations. (shrink)

Hadrianoupolis lies on the principal western route from the Central Anatolian Plain through the mountains to Bartın and the Black Sea, 3 km west of the modern town of Eskipazar, near Karabük, in Roman Paphlagonia. It was a small but important site which controlled this major route and dominated a rich agricultural, especially vinicultural, enclave.In 2003 the local Archaeological Museum of Ereğli began a small-scale salvage excavation of the newly discovered main church of Hadrianoupolis, known as “Early Byzantine Church B”, (...) situated in the centre of the ancient city. Only the floor and foundation levels are preserved. The church was erected probably in the early 6th century AD and may have still been in use as late as the 7th century. The most important discoveries at Church B were the floor mosaics which show personifications of four Biblical rivers: Euphrates, Tigris, Phison and Geon. Animals, floral and geometric designs, and an extensive inscription are also represented.In 2006 archaeological excavations were begun in Hadrianoupolis by a team from the Dokuz Eylül University, İzmir, under the direction of Dr Ergün Laflı. As a result of our 2005 surveys, it has now been confirmed that Hadrianoupolis lay indeed on the site of modern Eskipazar, with finds dating from the 1st century B.C. to the 8th century A.D. It also was determined that the core of the ancient city extended as far as the modern village of Budaklar and the surrounding districts of Hacı Ahmetler, Çaylı and Eleler, along the Eskipazar-Mengen highway for 8 km east-west and 3 km north-south. The chora of Hadrianoupolis is much more extensive in size.The field surveys in 2005 identified the remains of at least 24 buildings at the site. Among them are two bath buildings of the Late Roman period, two Early Byzantine churches, a fortified structure of the Byzantine period, a theatre , a vaulted building, a domed building and some domestic buildings with mosaic floors. In 2006 trenches were opened to investigate two of the best preserved of these buildings: Bath A and Early Byzantine Church A. In 2007 “Bath A”, “Bath B”, a Late Roman villa, an absidal Early Byzantine building, as well as two Roman monumental rock-cut graves, were excavated.The 2006–2007 campaigns have established that Hadrianoupolis was a fortified regional centre during the late Roman and early Byzantine period , when it can easily be defined as a “polis” with civic buildings and a fairly large urban population, as well as an extensive agrarian rural population. Most of the visible surface remains belong to this period. (shrink)

The organization of the mobile atomic exhibition, Mostra Atomica, designed by the United States Information Service to travel through Italy in 1954–55, had to meet technical, scientific, artistic, and political challenges. The head of the group in charge of the exhibition was architect Peter G. Harnden whose pedigree in the intelligence and training in architecture were an ideal match for leading the unit dedicated to exhibitions. The political sensitivity of the Mostra Atomica also required the intervention of the Italian Ministry (...) of the Interior to guarantee safe mobility and secure shows. In every major town, American and British diplomats attended the local opening ceremony, while the very symbol of science diplomacy was Enrico Fermi, whose recorded message praised international cooperation. All in all, the USIS campaign promoting peaceful applications of nuclear physics was successful in reaching and involving Italian society. Visual and spatial aesthetics were particularly relevant: the geometrical design of the exposition rooms conveyed a strong sense of modernity that contrasted with the artistic heritage of Italian cities. The present article is based on archival files, newspaper reports, and photographs that document who was responsible for planning, setting up, and reporting this Cold War propaganda event. (shrink)

On 6 January 1795, the twenty-year-old Schelling—still a student at the Tübinger Stift—wrote to his friend and former roommate, Hegel: “Now I am working on an Ethics à la Spinoza. It is designed to establish the highest principles of all philosophy, in which theoretical and practical reason are united”. A month later, he announced in another letter to Hegel: “I have become a Spinozist! Don’t be astonished. You will soon hear how”. At this period in his philosophical development, Schelling had (...) been deeply under the spell of Fichte’s new philosophy and the Wissenschaftslehre. The text Schelling was writing at the time was the early Vom Ich als Prinzip der Philosophie, though his characterization of this text would much better fit the somewhat later work which is the focus of the current paper: Schelling’s 1801 Darstellung meines System der Philosophie (hereafter: Presentation). The Presentation is a text written more geometrico, following the style of Spinoza’s Ethics. While Spinoza’s influence and inspiration is stated explicitly and unmistakably in Schelling’s preface, the content of this composition might seem quite foreign to Spinoza’s philosophy, so much so, in fact, that Michael Vater—the astute translator and editor of the recent English translation of the text—has contended that “despite the formal similarities between Spinoza’s geometrical method and Schelling’s numbered mathematical-geometrical constructions, Schelling’s direct debts to Spinoza are few”. The Presentation is an extremely dense and difficult text, and while I agree that at first glance Schelling’s engagement with the concept of reason (Vernunft) and the identity formula ‘A=A’ seems to have little if anything to do with Spinoza (especially since Spinoza’s key terminology of ‘God’, ‘causa sui’, ‘substance’, ‘attribute’, and ‘mode’ is barely mentioned in the Presentation), I suspect that at a deeper level Schelling is attempting to transform Spinoza’s system by replacing God, Spinoza’s ultimate reality, with reason. Though this might at first seem bizarre, I believe it can be profitably motivated and explained upon further reflection. It is this transformation of Spinoza’s God into (the early) Schelling’s reason that is the primary subject of this study. I develop this paper in the following order. In the first part I provide a very brief overview of Schelling’s lifelong engagement with Spinoza’s philosophy, which will prepare us for my study of the 1801 Presentation. In the second part, I consider the formal structure and rhetoric of the Presentation against the background of Spinoza’s Ethics, and show how Schelling regularly imitates Spinoza’s tiniest rhetorical gestures. In the third and final part I turn to the opening of the Presentation, and argue that Schelling attempts there to distance himself from Fichte by developing a conception of reason as the absolute, or the identity of the subject and object, just as the thinking substance and the extended substance are identified in Spinoza’s God. (shrink)

Whether designing animals, insects, or plants, Nature draws upon symmetry and periodicity to play a fundamental role in defining the body plan. When implemented with the proper chemical mechanisms, these principles guide our bodies from single-celled embryos to bilaterally symmetric creatures with intricate periodic structures, such as the spine and rib cage. The properties of symmetry and periodicity also appear to be fundamental to visual perception. We will show that this is no coincidence, but is a consequence of the fact (...) that these properties are generated by the same underlying phenomenon, standing wave patterns formed from harmonic resonances. The symmetry in life forms arises from chemical harmonic resonances, whereas the symmetry in visual perception arises from harmonic resonances in the visual system. We will show that harmonic resonances have very interesting properties for the representation of geometrical form that make them eminently suitable for encoding geometric form for body plans as well as for visual perception. In particular, we will show that by being simultaneously invariant to perspective transformation and robust to deformation, such resonant representations have fundamental advantages over more traditional techniques. (shrink)

There are several models of the use of geometric and feature cues in reorientation. The adaptive combination approach posits that people integrate cues with weights that depend on cue salience and learning, or, when discrepancies are large, they choose between cues based on these variables. In a new paradigm designed to evaluate integration and choice, disoriented participants attempted to return to a heading direction, in a trapezoidal enclosure in which feature and geometric cues both unambiguously specified a heading, but later (...) the feature was moved. With discrepancies greater than 90 degrees, participants choose geometry. With smaller discrepancies, integration appeared in three of five situations; otherwise, participants used geometry alone. Variation depended on direction of feature movement and whether the nearest corner was acute or obtuse. The results have implications for contrasting adaptive combination and modularity theory, and for future research, offering a new paradigm for reorientation research, and for testing cue integration more broadly. (shrink)

The current literature that attempts to bridge between geometric morphometrics (GMM) and finite element analyses (FEA) of CT-derived data from bones of living animals and fossils appears to lack a sound biotheoretical foundation. To supply the missing rigor, the present article demonstrates a new rhetoric of quantitative inference across the GMM–FEA bridge—a rhetoric bridging form to function when both have been quantified so stringently. The suggested approach is founded on diverse standard textbook examples of the relation between forms and the (...) way strains in them are produced by stresses imposed upon them. One potentially cogent approach to the explanatory purposes driving studies of this class arises from a close scrutiny of the way in which computations in both domains, shape and strain, can be couched as minimizations of a scalar quantity. For GMM, this is ordinary Procrustes shape distance; in FEA, it is the potential energy that is stored in the deformed configuration of the solid form. A hybrid statistical method is introduced requiring that all forms be subjected to the same detailed loading designs (the same “probes”) in a manner careful to accommodate the variations of those same forms before they were stressed. The proper role of GMM is argued to be the construction of regressions for strain energy density on the largest-scale relative warps in order that biological explanations may proceed in terms of the residuals from those regressions: the local residual features of strain energy density. The method, evidently a hierarchical one, might be intuitively apprehended as a geometrical approach to a formal allometric analysis of strain. The essay closes with an exhortation. (shrink)

Gjirokastra’s buildings occupy a special place in the housing typology of Albanian popular dwellings in the feudal period. The “popular tower" is linked with its defensive character, therefore in many cases, it takes the name of a castle or defensive tower. This paper takes into consideration a typical example of the historical fortified dwelling in a well-known city of Albania, Gjirokastra. The methodology used in order to improve the way of thinking, the way of implementing, and the way of designing (...) future dwellings is very related to the historical attitude of the dwelling. Based on the historical documentation, of the fortified dwelling a real analysis is performed according to the logic of modulations. The purpose of this paper is to create a methodology for future generations of architects in order to better engage in the design process, respecting the valuable architectural history. The modern dwellings of Gjirokastra historical-urban context must adopt the historical architectural elements, the historical manners of construction, the typical construction materials, the operational mode, and people’s lifestyle, in order to preserve the city's historical values via a modern touch. KEYWORDS: Architectural elements, historical preservation values, geometric ratios, modular. (shrink)

The geometric shapes and natural forms, captured with exceptional precision in Ernst Haeckel's prints, still influence artists and designers to this day. This volume highlights the research and findings of this natural scientist. Powerful modern microscopes have confirmed the accuracy of Haeckel's prints, which even in their day, became world famous. Haeckel's portfolio, first published between 1899 and 1904 in separate installments, is described in the opening essays. The plates illustrate Haeckel's fundamental monistic notion of the "unity of all living (...) things" and the wide variety of forms are executed with utmost delicacy. Incipient microscopic organisms are juxtaposed with highly developed plants and animals. The pages, ordered according to geometric and "constructive" aspects, document the oness of the world in its most diversified forms. This collection of plates was not only well-received by scientists, but by artists and architects as well. Rene Binet, a pioneer of glass and iron constructions, Emile Galle, a renowned Art Nouveau designer, and the photographer Karl Blossfeld all make explicit reference to Haeckel in their work. (shrink)

This paper aims to address the difficulties of high school students in bridging their computational understanding with their visualization skills in understanding the notion of the limits in their calculus class. This research used a pre-experimental one-group pretest-posttest design research on 62 grade 10 students enrolled in the Science, Technology, and Engineering Program (STEP) in one of the public high schools in Zamboanga del Sur, Philippines. A series of remedial sessions were given to help them understand the function values, one-sided (...) limits, limits of a function, and its discontinuities with the aid of the GeoGebra. Results revealed that there is a significant improvement in their mathematics performance before and after the remediation, t(61) = -19.99, p<.05. The real-time feedback provided by the drawing interface of GeoGebra has provided the students the insights into how objects behave in geometrical space, bridging their understanding in locating the values shown by the computational algebraic processes. Additionally, the significant difference in the achievement reveals a large effect size (Cohen’s =2.54) which could be accounted for by the remediation using interactive activities in the GeoGebra. (shrink)

As it emerged from philosophical analyses and cognitive research, most concepts exhibit typicality effects, and resist to the efforts of defining them in terms of necessary and sufficient conditions. This holds also in the case of many medical concepts. This is a problem for the design of computer science ontologies, since knowledge representation formalisms commonly adopted in this field do not allow for the representation of concepts in terms of typical traits. However, the need of representing concepts in terms of (...) typical traits concerns almost every domain of real world knowledge, including medical domains. In particular, in this article we take into account the domain of mental disorders, starting from the DSM-5 descriptions of some specific mental disorders. On this respect, we favor a hybrid approach to the representation of psychiatric concepts, in which ontology oriented formalisms are combined to a geometric representation of knowledge based on conceptual spaces. (shrink)

Based on the design of many modern chemical instruments, information about a specimen is retrieved after the specimen undergoes agitation, manipulation and disturbance of its internal state. But can we retain the traditional ideal that instruments should reveal properties that are definable independently of all modes of detection? In this paper I argue that the capacity of chemical instruments to convert experimental phenomena to information places constraints on the way in which the specimen is characterized. During research, the specimen is (...) defined by those properties which permit its detection. Based on modern instrumentation, this constraint necessitates a conception of the specimen as a reactive system of dynamical properties. The dream of a purely transparent detection process violates the design of chemical instruments. This mutual dependence of instrument and specimen is illustrated by empirical studies of the geometrical configuration of DNA. (shrink)

“Walk on the Sun” is an interactive experience of image as music. As explorers move across images that are data projected onto the floor, their movements are visually tracked and used to select pixels in the images which they immediately hear as musical pitches played by various instruments. The sonification design maps color to one of 9 instruments, brightness to one of 50 pitches, and location in the image to panning position, creating 57,600 differentiable musical events. This high resolution and (...) interactive auditory presentation of pixel data enables the blind to explore images of the Sun from the STEREO space mission, nebula and galactic images from Hubble, as well as art masterpieces. Specifically, the blind can hear when hot spots cross the center of the Sun or the solar winds and corona are changing by sonifying virtual geometric structures, such as lines and circles, to create chords of music reflecting the changing content of the selected pixels within that structure as images are played as movies. Originally funded by a NASA/STSCI Ideas grant, the exhibit has toured to more than 12 cities in the US, visiting blind and science centers in the process and receiving enthusiastic response throughout. Plans for additional work furthering NASA wide image sonification standards are in process. (shrink)

This textbook is designed for an Introduction to Proofs course organized around the themes of number and space. Concepts are illustrated using both geometric and number examples, while frequent analogies and applications help build intuition and context in the humanities, arts, and sciences. Sophisticated mathematical ideas are introduced early and then revisited several times in a spiral structure, allowing students to progressively develop rigorous thinking. Throughout, the presentation is enlivened with whimsical illustrations, apt quotations, and glimpses of mathematical history and (...) culture. Early chapters integrate an introduction to sets, logic, and beginning proof techniques with a first exposure to more advanced mathematical structures. The middle chapters focus on equivalence relations, functions, and induction. Carefully chosen examples elucidate familiar topics, such as natural and rational numbers and angle measurements, as well as new mathematics, such as modular arithmetic and beginning graph theory. The book concludes with a thorough exploration of the cardinalities of finite and infinite sets and, in two optional chapters, brings all the topics together by constructing the real numbers and other complete metric spaces. Designed to foster the mental flexibility and rigorous thinking needed for advanced mathematics, Introduction to Mathematics suits either a lecture-based or flipped classroom. A year of mathematics, statistics, or computer science at the university level is assumed, but the main prerequisite is the willingness to engage in a new challenge. (shrink)

In this essay I revisit The Prince and the Discourses and argue that across the design of these two texts on the theme of conspiracy Machiavelli constructs an ambush on Medici princes. I reconsider Mary Dietz's (1986), and Langton's and Dietz's (1987) suggestion that Machiavelli's The Prince was a deceptive political act through an exploration of the link Dietz and Sheldon Wolin (2004) draw between Machiavelli's method and Renaissance artistry. I suggest that Machiavelli applied a one-point linear perspective – a (...) scientific and visual method of pictorial representation and geometrical modelling that emerged for the first time in the Renaissance – to the political field. I test this hypothesis on the theme of conspiracy in Machiavelli’s work by arguing that The Prince ultimately presents one vantage point – that of the prince – while the Discourses offers another – that of the conspirators. I argue that a blind spot is created by these two texts when they are jointly considered: conspirators seduced, recruited, and trained by the Discourses eliminate a prince caught off guard for having followed the advice on conspiracy in The Prince. (shrink)

That every first-order theory has a coherent conservative extension is regarded by some as obvious, even trivial, and by others as not at all obvious, but instead remarkable and valuable; the result is in any case neither sufficiently well-known nor easily found in the literature. Various approaches to the result are presented and discussed in detail, including one inspired by a problem in the proof theory of intermediate logics that led us to the proof of the present paper. It can (...) be seen as a modification of Skolem’s argument from 1920 for his “Normal Form” theorem. “Geometric” being the infinitary version of “coherent”, it is further shown that every infinitary first-order theory, suitably restricted, has a geometric conservative extension, hence the title. The results are applied to simplify methods used in reasoning in and about modal and intermediate logics. We include also a new algorithm to generate special coherent implications from an axiom, designed to preserve the structure of formulae with relatively little use of normal forms. (shrink)

In his Harmonics, Ptolemy constructs a complex set of theoretically ‘correct’ forms of musical scale, represented as sequences of ratios, on the basis of mathematical principles and reasoning. But he insists that their credentials will not have been established until they have been submitted to the judgement of the ear. They cannot be audibly instantiated with the necessary accuracy without the help of specially designed instruments, which Ptolemy describes in detail, discussing the uses to which each can be put and (...) cataloguing its limitations. The best known of these instruments is the monochord, but there are several more complex devices. This paper discusses one such instrument which is known from no other source, ancient or modern, whose design was prompted by the geometrical construction known as the helikôn. It has several remarkable peculiarities. I examine its design, its purposes, and the merits and shortcomings which Ptolemy attributes to it. An appendix describes an instrument I have built to Ptolemy’s specifications, in an attempt to find out how satisfactorily such a bizarre contraption will work; and it explains how various practical problems can be resolved. Keywords: Ptolemy; Greek harmonics; Scientific instruments; Helikôn; Experiment; Mathematics. (shrink)

A computational theory of consciousness should include a quantitative measure of consciousness, or MoC, that (i) would reveal to what extent a given system is conscious, (ii) would make it possible to compare not only different systems, but also the same system at different times, and (iii) would be graded, because so is consciousness. However, unless its design is properly constrained, such an MoC gives rise to what we call the boundary problem: an MoC that labels a system as conscious (...) will do so for some – perhaps most – of its subsystems, as well as for irrelevantly extended systems (e.g., the original system augmented with physical appendages that contribute nothing to the properties supposedly supporting consciousness), and for aggregates of individually conscious systems (e.g., groups of people). This problem suggests that the properties that are being measured are epiphenomenal to consciousness, or else it implies a bizarre proliferation of minds. We propose that a solution to the boundary problem can be found by identifying properties that are intrinsic or systemic: properties that clearly differentiate between systems whose existence is a matter of fact, as opposed to those whose existence is a matter of interpretation (in the eye of the beholder). We argue that if a putative MoC can be shown to be systemic, this ipso facto resolves any associated boundary issues. As test cases, we analyze two recent theories of consciousness in light of our definitions: the Integrated Information Theory and the Geometric Theory of consciousness. (shrink)

In 1922, three young men founded the avant-garde magazine 7 Arts to promote the arts and in particular the synthesis of all the arts as only architecture and cinema can achieve. Pierre Bourgeois, a poet, his brother Victor, an architect, and the painter Pierre-Louis Flouquet were soon joined by the composer Georges Monier and Karel Maes, painter, engraver and furniture designer. "The five" succeeded in harnessing the vital forces of the Belgian avant-garde and in placing their magazine at the very (...) heart of the European avant-garde. Throughout the six years of publication the pages of the magazine provided a platform for the principal protagonists of the Belgian and international artistic scene: De Stijl, the Bauhaus, the Purists, the Constructivists, the Futurists and many other champions of the geometric abstraction to which they gave the name Plastique pure or Pure Plastic. Their ambition was to have the arts penetrate every dimension of modern urban life and to transform it in the process. In 2020, CIVA is devoting a retrospective exhibition to 7 Arts, the impressive adventure that this work presents. (shrink)

Open peer commentary on the article “Elementary Students’ Construction of Geometric Transformation Reasoning in a Dynamic Animation Environment” by Alan Maloney. Upshot: This commentary assumes a constructionist perspective to discuss the choice of methods, conclusions and design goals that Panorkou and Maloney make in their study of students’ activities with the Graph ’n Glyphs microworld.

With In Focus Sacred Geometry, learn the fascinating history behind this ancient tradition as well as how to decipher the geometrical symbols, formulas, and patterns based on mathematical patterns. People have searched for the meaning behind mathematical patterns for thousands of years. At its core, sacred geometry seeks to find the universal patterns that are found and applied to the objects surrounding us, such as the designs found in temples, churches, mosques, monuments, art, architecture, and nature. Learn the (...) fundamental principles behind: Interpreting the sacred symbolism and principles behind shapes Calculating numbers used in sacred geometry Applying the golden ratio and Fibonacci’s Sequence to objects Translating symbolic and geometrical letters and alphabets This accessible and beautifully designed guide to sacred geometry includes a frameable poster of the main sacred geometrical shapes and their unique, innate arrangements. The In Focus series applies a modern approach to teaching the classic body, mind, and spirit subjects. Authored by experts in their respective fields, these beginner's guides feature smartly designed visual material that clearly illustrates key topics within each subject. As a bonus, each book includes reference cards or a poster, held in an envelope inside the back cover, that give you a quick, go-to guide containing the most important information on the subject. (shrink)

Some of the controversy surrounding the Duhemian claim that in science falsification is as inconclusive as verification is reconsidered. The D-Thesis, a particular version of this claim first discussed by Adolf Grünbaum, is formulated in a more precise and perspicuous fashion as a conjunction of two subtheses. Grünbaum's attempt to refute one of the subtheses by means of a geometrical counterexample and some subsequent discussions of this example are examined critically. An argument designed to prove the other subthesis is (...) analyzed and shown to be unsuccessful. It is concluded that the D-Thesis is as yet neither proven nor refuted. (shrink)

It is a matter of tacit consensus that rationalist adeptness in urban planning traces its foundations to the philosophy of the Renaissance thinker and mathematician Ren Descartes. This study suggests, in turn, that the planned urban environment of the Renaissance may have also led Descartes, and his intellectual peers, to tenets that became the foundations of modern philosophy and science. The geometric street pattern of the late middle ages and the Renaissance, the planned townscapes, street views and the formal garden (...) design, appeared as parables for the perfection of the universe and the supremacy of critical reason. It is within this urban metaphor that Descartes's philosophical narrative betrays perceptual and conceptual impact from the contrast between convoluted medieval townscapes and the emerging harmonious street patterns where defined vistas and predictable clarity of street views were paramount. The geometrically delineated street views of the Renaissance new town became the spark that lit the philosopher's sagacity in reflecting upon the concept of "clear and distinct ideas." Past suggestions that Descartes was led to his philosophical breakthroughs through his discovery of co-ordinate geometry reinforce further the stance that Renaissance planning predisposed rationalist thought. (shrink)

Wines with geographical indication can be classified and represented by such features as designations of origin, producers, vintage years, alcoholic strength, and grape varieties; these features allow us to define wines in terms of a set of necessary and/or sufficient conditions. However, wines can also be identified by other characteristics, involving their look, smell, and taste; in this case, it is hard to define wines in terms of necessary and/or sufficient conditions, as wine concepts exhibit typicality effects. This is a (...) setback for the design of computer science ontologies aiming to represent wine concepts, since knowledge representation formalisms commonly adopted in this field do not allow for the representation of concepts in terms of typical traits. To solve this problem, we propose to adopt a hybrid approach in which ontology-oriented formalisms are combined with a geometric representation of knowledge based on conceptual spaces. As in conceptual spaces, concepts are identified in terms of a number of quality dimensions. In order to determine those relevant for wine representation, we use the terminology developed by the Italian Association of Sommeliers to describe wines. This will allow us to understand typicality effects about wines, determine prototypes and better exemplars, and measure the degree of similarity between different wines. (shrink)