The paper addresses the capabilities and limitations of extrafoveal processing during a categorical visual search. Previous research has established that a target could be identified from the very first or without any saccade, suggesting that extrafoveal perception is necessarily involved. However, the limits in complexity defining the processed information are still not clear. We performed four experiments with a gradual increase of stimuli complexity to determine the role of extrafoveal processing in searching for the categorically defined geometric shape. The (...) series of experiments demonstrated a significant role of extrafoveal processing while searching for simple two‐dimensional shapes and its gradual decrease in a condition with more complicated three‐dimensional shapes. The factors of objects’ spatial orientation and distractor homogeneity significantly influenced both reaction time and the number of saccades required to identify a categorically defined target. An analysis of the individual p‐value distributions revealed pronounced individual differences in using extrafoveal analysis and allowed examination of the performance of each particular participant. The condition with the forced prohibition of eye movements enabled us to investigate the efficacy of covert attention in the condition with complicated shapes. Our results indicate that both foveal and extrafoveal processing are simultaneously involved during a categorical search, and the specificity of their interaction is determined by the spatial orientation of objects, type of distractors, the prohibition to use overt attention, and individual characteristics of the participants. (shrink)

For more than a century, there has been some discussion about whether medieval Arabic al-jabr has its roots in Indian or Greek mathematics. Since the 1930s, the possibility of Babylonian ultimate roots has entered the debate. This article presents a new approach to the problem, pointing to a set of quasi-algebraic riddles that appear to have circulated among Near Eastern practical geometers since c. 2000 BCE, and which inspired first the so-called “algebra” of the Old Babylonian scribal school and later (...) the geometry of Elements II. The riddles also turn up in ancient Greek practical geometry and Jaina mathematics. Eventually they reached European mathematics via the Islamic world. However, no evidence supports a derivation of medieval Indian algebra or the original core of al-jabr from the riddles. (shrink)

Fractal geometry offers a new approach to describing the morphology of fern leaves. Traditional morphology is based on the Euclidean concept of shape as an area defined by a boundary. This approach has not proven successful with fern leaves because they are so elaborate. Fractal geometry treats forms as relationships between parts rather than as areas. In fern fronds there are often constant relationships between parts. Four fractal methodologies for describing these relations within leaves are explored in this paper. These (...) include recursive line branching algorithms, iterated function systems (IFS), modifications of IFS, and L-systems. The methods are evaluated by comparing their results with measurements and appearances of various ferns. Fractal methods offer objective, quantifiable and succinct descriptions of fern-leaves. We conclude that fractal geometry offers simple descriptions of some elaborate fern shapes and that it will probably have application in investigating different aspects of ferns and other organisms. (shrink)

The paper argues that Roger Bacon adhered to a unique form of geometrical atomism, according to which elemental matter can be analysed into cubic (when at rest) or pyramidal (when in motion) portions. Bacon addressed geometrical atomism from the perspective of the Aristotelian review, using his interpretation of Aristotelian principles to render the theory plausible. He was mostly concerned with solving the contradiction between the angular shapes of the portions and the shape of the elemental spheres. His motivation for (...) doing so, I argue, was his conviction in the applicability of geometry onto natural philosophy, and his goal was to render matter capable of geometrical analysis. In this way, he was convinced, geometrical properties can be considered efficient causes of matter’s spatial formation and motion. (shrink)

In this paper, based on video recordings of Orientation and Mobility (O&M) lessons for visually-disabled students, I will examine how occasioned maps (Psathas, 1979 ; Garfinkel, 2002 ), drawn in the student’s palm are interactionally traced, felt, and noticed in order to represent the shape of a crossing for all practical purposes. Touching will be examined from the perspective of the live production of "trails" on a specific region of the body, the palm of the hand. We will begin to (...) question how such hand-drawing map episodes occur during O&M courses, stressing how the coparticipants establish a participation framework that facilitates the making of the drawing, hand-map drawings are based on lines that are neither evanescent nor permanent. Their “persistence” is not an intrinsic feature but a systematic multimodal accomplishment. We will show how the drawn lines become depictions of the streets and contribute to producing two contrasting geometrical representations of the layout of a junction. (shrink)

How is knowledge of geometry developed and acquired? This central question in the philosophy of mathematics has received very different answers. Spelke and colleagues argue for a “core cognitivist”, nativist, view according to which geometric cognition is in an important way shaped by genetically determined abilities for shape recognition and orientation. Against the nativist position, Ferreirós and García-Pérez have argued for a “culturalist” account that takes geometric cognition to be fundamentally a culturally developed phenomenon. In this paper, I (...) argue that when understood as moderate versions supported by the state-of-the-art research, the nativist and culturalist views are in fact possible to reconcile. While Ferreirós and García-Pérez present the work of Spelke and colleagues as implying that geometric cognition is genetically determined, I argue that they fail to appreciate the role that Spelke and colleagues see for cultural factors. On this basis, I provide theoretical and terminological clarifications and show that moderate versions of the nativist and culturalist view are in fact consistent with each other. I then propose a unifying theoretical framework for future study that can integrate the two accounts in ontogeny by moving beyond the crude nature (nativism) vs. nurture (culturalism) dichotomy. (shrink)

This is a revised and updated edition of Graham Nerlich's classic book The Shape of Space. It develops a metaphysical account of space which treats it as a real and concrete entity. In particular, it shows that the shape of space plays a key explanatory role in space and spacetime theories. Arguing that geometrical explanation is very like causal explanation, Professor Nerlich prepares the ground for philosophical argument, and, using a number of novel examples, investigates how different spaces would affect (...) perception differently. This leads naturally to conventionalism as a non-realist metaphysics of space, an account which Professor Nerlich criticises, rejecting its Kantian and positivistic roots along with Reichenbach's famous claim that even the topology of space is conventional. He concludes that there is, in fact, no problem of underdetermination for this aspect of spacetime theories, and offers an extensive discussion of the relativity of motion. (shrink)

Permutative logic is a non-commutative conservative extension of linear logic suggested by some investigations on the topology of linear proofs. In order to syntactically reflect the fundamental topological structure of orientable surfaces with boundary, permutative sequents turn out to be shaped like q-permutations. Relaxation is the relation induced on q-permutations by the two structural rules divide and merge; a decision procedure for relaxation has been already provided by stressing some standard achievements in theory of permutations. In these pages, we provide (...) a parallel procedure in which the problem at issue is approached from the point of view afforded by geometry of 2-manifolds and solved by making specific surfaces interact. (shrink)

Although disoriented young children reorient themselves in relation to the shape of the surrounding surface layout, cognitive accounts of this ability vary. The present paper tests three theories of reorientation: a snapshot theory based on visual image-matching computations, an adaptive combination theory proposing that diverse environmental cues to orientation are weighted according to their experienced reliability, and a modular theory centering on encapsulated computations of the shape of the extended surface layout. Seven experiments test these theories by manipulating four properties (...) of objects placed within a cylindrical space: their size, motion, dimensionality, and distance from the space’s borders. Their findings support the modular theory and suggest that disoriented search behavior centers on two processes: a reorientation process based on the geometry of the 3D surface layout, and a beacon-guidance process based on the local features of objects and surface markings. Ó 2010 Elsevier Inc. All rights reserved. (shrink)

Scholars have long recognised the interest of the Stoics' thought on geometrical limits, both as a specific topic in their physics and within the context of the school's ontological taxonomy. Unfortunately, insufficient textual evidence remains for us to reconstruct their discussion fully. The sources we do have on Stoic geometrical themes are highly polemical, tending to reveal a disagreement as to whether limit is to be understood as a mere concept, as a body or as an incorporeal. In my view, (...) this disagreement held among the historical Stoics, rather than simply reflecting a doxographical divergence in transmission. This apparently Stoic disagreement has generated extensive debate, in which there is still no consensus as to a standard Stoic doctrine of limit. The evidence is thin, and little of it refers in detail to specific texts, especially from the school's founders. But in its overall features the evidence suggests that Posidonius and Cleomedes differed from their Stoic precursors on this topic. There are also grounds for believing that some degree of disagreement obtained between the early Stoics over the metaphysical status of shape. Assuming the Stoics did so disagree, the principal question in the scholarship on Stoic ontology is whether there were actually positions that might be called "standard" within Stoicism on the topic of limit. In attempting to answer this question, my discussion initially sets out to illuminate certain features of early Stoic thinking about limit, and then takes stock of the views offered by late Stoics, notably Posidonius and Cleomedes. Attention to Stoic arguments suggests that the school's founders developed two accounts of shape: on the one hand, as a thought-construct, and, on the other, as a body. In an attempt to resolve the crux bequeathed to them, the school's successors suggested that limits are incorporeal. While the authorship of this last notion cannot be securely identified on account of the absence of direct evidence, it may be traced back to Posidonius, and it went on to have subsequent influence on Stoic thinking, namely in Cleomedes' astronomy. (shrink)

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)

Focused attention is needed to perceive change (Rensink et al., 1997; Psychological Science, 8: 368-373) . But how much attentional processing is given to an item? And does this depend on the nature of the task? To answer these questions, "flicker" displays were created, where an original and a modified image continually alternated, with brief blanks between them. Each image was an array of simple figures, half being horizontal and the other half vertical. In half the trials, one of the (...) items changed orientation; in the remainder, all items remained the same. Observers were asked to detect whether a change was occurring in each trial. -/- Within-observer comparisons for different shapes of items yielded two classes of speed: relatively fast search (50 ms/item) for simple lines, and slower search (90 ms/item) for compound figures made of two or more lines. Evidently, more processing was given to compound figures, even though shape was irrelevant to the task. A similar set of experiments asked observers to detect changes in contrast polarity. Here, no differences were found for different shapes, indicating that geometric information is not processed as much when the task involves only nongeometric properties. (shrink)

The ability to perceive the shapes of things as enduring through changes in how they stimulate our sense organs is vital to our sense of stability in the world. But what sort of capacity is shape constancy, and how is it reflected in perceptual experience? This paper defends a pluralist account of shape constancy: There are multiple kinds of shape constancy centered on geometrical properties at various levels of abstraction, and properties at these various levels feature in the content (...) of perceptual experience, governing patterns of apparent shape similarity. I propose that the varieties of shape constancy are subserved by the syntactic complexity of perceptual shape representations. By assigning discrete constituents to various abstract shape parameters, these representations attune us to the preservation of certain abstract shape properties through changes in more determinate shape properties. Finally, I draw broader lessons concerning the nature and function of perceptual constancy. (shrink)

Current cognitive architectures are either working at the abstract, symbolic level, or the low, emergent level related to neural modeling. The best way to understand phenomena is to see, or imagine them, hence the need for a geometric model of mental processes. Geometric models should be based on an intermediate level of modeling that describe mental states in terms of features relevant from the first-person perspective but also linked to neural events. Concepts should be represented as geometrical objects (...) that have sufficiently rich structures to show their properties and their relations to other concepts. The best way to create such geometrical representations of concepts is through the approximate description of the physical states of neural networks. The evolution of brain states is then represented as a trajectory linking successful concepts, and topological constraints on the shape of such trajectory define grammar and logic. (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)

Even simple geometric shapes are seen as animate and goal-directed when they move in certain ways. Previous research has revealed a great deal about the cues that elicit such percepts, but much less about the consequences for other aspects of perception and cognition. Here we explored whether simple shapes that are perceived as animate and goal-directed are prioritized in memory. We investigated this by asking whether subjects better remember the locations of displays that are seen as animate (...) vs. inanimate, con- trolling for lower-level factors. We exploited the ‘Wolfpack effect’: moving darts (or discs with ‘eyes’) that stay oriented toward a particular target are seen to be actively pursuing that target, even when they actually move randomly. (In contrast, shapes that stay oriented perpendicular to a target are correctly perceived to be drifting randomly.) Subjects played a ‘matching game’ – clicking on pairs of panels to reveal animations with moving shapes. Across four experiments, the locations of Wolfpack animations (compared to control animations equated on lower-level visual factors) were better remembered, in terms of more efficient matching. Thus perceiving animacy influences subsequent visual memory, perhaps due to the adaptive significance of such stimuli. (shrink)

The present article addresses the logic of the sphere, or the Sphere Model of Consciousness developed by Patrizio Paoletti over three decades of research. M.E.D. Ed., 2002; Flussi, territori, luogo II. M.E.D. Ed., 2002; Fare il punto nave. M.E.D. Ed., 2005; In: Proceedings conference at Leslie and Susan Gonda Multidisciplinary Brain Research Center. Bar Ilan University. Faculty of Neuroscience, Israel, 2007; Osservazione—Quaderni di Pedagogia per il Terzo Millennio, Ed. 3P, 2011; Mediazione—Quaderni di Pedagogia per il Terzo Millennio, Ed. 3P, 2011). (...) The SMC model has been experimentally applied in the educational field and lies at the base of neuroscientific and psychoeducational research conducted by the Paoletti Foundation. In recent years, it has been studied by several researchers in the field of neuroscience. Following a logical-mathematical introduction regarding the properties of the spherical shape, we illustrate for the first time the structuring of the model and its neural foundations. We emphasize the central space of the sphere, defined by the geometric model and highlight its descriptive and heuristic properties in relation to the study of consciousness. Finally, we present educational applications of the model, particularly with respect to the center of the sphere, defined in the current context as the “place of pre-existence.”. (shrink)

The present article addresses the logic of the sphere, or the Sphere Model of Consciousness developed by Patrizio Paoletti over three decades of research. M.E.D. Ed., 2002; Flussi, territori, luogo II. M.E.D. Ed., 2002; Fare il punto nave. M.E.D. Ed., 2005; In: Proceedings conference at Leslie and Susan Gonda Multidisciplinary Brain Research Center. Bar Ilan University. Faculty of Neuroscience, Israel, 2007; Osservazione—Quaderni di Pedagogia per il Terzo Millennio, Ed. 3P, 2011; Mediazione—Quaderni di Pedagogia per il Terzo Millennio, Ed. 3P, 2011). (...) The SMC model has been experimentally applied in the educational field and lies at the base of neuroscientific and psychoeducational research conducted by the Paoletti Foundation. In recent years, it has been studied by several researchers in the field of neuroscience. Following a logical-mathematical introduction regarding the properties of the spherical shape, we illustrate for the first time the structuring of the model and its neural foundations. We emphasize the central space of the sphere, defined by the geometric model and highlight its descriptive and heuristic properties in relation to the study of consciousness. Finally, we present educational applications of the model, particularly with respect to the center of the sphere, defined in the current context as the “place of pre-existence.”. (shrink)

For Aristotle, the shape of a physical body is perceptible per se (DA II.6, 418a8-9). As I read his position, shape is thus a causal power, as a physical body can affect our sense organs simply in virtue of possessing it. But this invites a challenge. If shape is an intrinsically powerful property, and indeed an intrinsically perceptible one, then why are the objects of geometrical reasoning, as such, inert and imperceptible? I here address Aristotle’s answer to that problem, focusing (...) on the version of it that he presents in De caelo III.8. I argue that if we grant that Aristotle conceived of the shape of a sensible body as some kind of causal power, then the satisfactory resolution of that challenge pushes us to interpret him as having conceived of it as being, more specifically, an impure power—that is, as a property that is not only intrinsically powerful but also, in some way, intrinsically non-powerful as well. This is a notable result not only insofar as it illuminates Aristotle’s conception of shape but also insofar as it contributes to our knowledge of Aristotle’s theory of dunameis and his ontology more broadly. (shrink)

Charlotte Angas Scott (1858–1932) was an internationally renowned geometer, the first British woman to earn a doctorate in mathematics, and the chair of the Bryn Mawr mathematics department for forty years. There she helped shape the burgeoning mathematics community in the United States. Scott often motivated her research as providing a “geometric treatment” of results that had previously been derived algebraically. The adjective “geometric” likely entailed many things for Scott, from her careful illustration of diagrams to her choice (...) of references and citations. This article will focus on Scott’s striking and consistent use of geometric to describe a reality of dynamic points, lines, planes, and spaces that could be manipulated analogously to physical objects. By providing geometric interpretations of algebraic derivations, Scott committed to an early-nineteenth-century aesthetic vision of a “whole” analytical geometry that she adapted to modern research areas. (shrink)

This chapter contains sections titled: The Imaginary Status of Shape: The “Diachronic” Argument The Dominant Synchronic Argument An Alternative Interpretation Shape and Idealism.

This paper probes the foundations and limits of visual representation in the sciences through a close reading of the diagrams that accompanied definitions of the geometric point in the first century of printed editions of Euclid’s Elements. I begin with the modal form for such diagrams of Euclid’s “small and unsensible shape,” showing how it incorporates a broad spectrum of conventions and practices related to the point’s philosophical and practical roles in the surrounding Euclidean geometry. I then explore the (...) form’s several variations in order to consider the role of “mere representation” in geometric exegesis, and conclude by characterizing the curious relationship between things and their images and that relationship’s implications for understanding scientific knowledge and practice. (shrink)

Long tradition in philosophy and in empirical psychology has it that the perceptual recovery of enduring objective size and shape proceeds through initial spatial appearance experiences—like the sensed changing visual field size of a receding car, or the shifting shape appearance of a coin as it rotates in depth. The present paper carefully frames and then critically examines such proposals. It turns out that these are contingent, empirical matters, requiring close examination of relevant research in perception science in order to (...) decide. The paper concludes with extended discussions of the empirical study of the perception of slant, size, and shape. (shrink)

This book concerns Spinoza's theory of knowledge and closely related issues: Spinoza's conceptions of geometrical figure or shape, number, and observational sci.

Objective: Based upon notions to a mental vision of the Vitruvian Man, to determine if any obvious asymmetries exist within Leonardo da Vinci’s timeless schematic—which is famous for its highly symmetrical presentation. Methods: A qualitative analysis performed upon a Vitruvian Man print (taken from the namesake Wikipedia article) to: closely examine if the man’s head is positioned to noticeably tilt toward either direction—left or right—of a dissecting line superimposed for equally splitting (vertically) the circle in the schematic; and, to closely (...) examine the man’s eyes for any artistic asymmetry therein drawn. Results: The man’s head is determined tilting toward his right half/hemisphere of the circle. Also noticeable: the man’s right eye appears being of a much brighter look relative to a darkness observable about his left eye; and, an outline to an inverted equilateral triangle is identifiably shaded surrounding his left eye, whereas a more contrastingly circular shape appears as so shaded around his right eye. Discussion: Once a certain awareness is drawn to such artistic anomalies, they can become so clearly observable that one may wonder if Leonardo da Vinci did as much intentionally. And if so, then why? This paper details a hypothesis necessarily built upon the assumption that Leonardo intentionally embedded such cues within his Vitruvian Man whereby speculations toward a hemispheric brain theory can be established of such cryptic nature, given a then-pronouncedly authoritarian Catholic Church, whereupon he may have figured it possible to receive credit posthumously if ever such notions were proven valid. Incidentally, the concerning analysis as to why the Vitruvian Man’s head tilts in such a fashion aligns with Dr. Iain McGilchrist’s peer-reviewed findings to a hemispheric hypothesis. (shrink)

This essay explores the research practice of French geometer Michel Chasles, from his 1837 Aperçu historique up to the preparation of his courses on ‘higher geometry’ between 1846 and 1852. It argues that this scientific pursuit was jointly carried out on a historiographical and a mathematical terrain. Epistemic techniques such as the archival search for and comparison of manuscripts, the deconstructive historiography of past geometrical methods, and the epistemologically motivated periodization of the history of mathematics are shown to have played (...) a crucial role in the shaping of Chasles's own theories. In particular, we present Chasles's approach to the ‘material history’ of algebraic symbolism and argue that it motivated and informed his subsequent invention of a novel notational technology for the writing of geometrical proofs and propositions. In return, this technology allowed Chasles to carry out a programme for the modernization of geometry in keeping with epistemic requirements he had also delineated via a form of historical writing. (shrink)

An analysis is carried out involving reversible thermodynamic operations on arbitrarily shaped small cavities in perfectly conducting material. These operations consist of quasistatic deformations and displacements of cavity walls and objects within the cavity. This analysis necessarily involves the consideration of Casimir-like forces. Typically, even for the simplest of geometrical structures, such calculations become quite complex, as they need to take into account changes in singular quantities. Much of this complexity is reduced significantly here by working directly with the change (...) in electromagnetic fields as a result of the deformation and displacement changes. A key result of this work is the derivation that for such cavity structures, classical electromagnetic zero-point radiation is the appropriate spectrum at a temperature of absolute zero to ensure that the reversible deformation operations obey both isothermal and adiabatic conditions. In addition, a generalized Wien displacement law is obtained from the demand that the change in entropy of the radiation in these arbitrarily shaped structures must be a state function of temperature and frequency. (shrink)

In his 1622 work The Assayer, Galileo commented on the necessity of mathematics for understanding the natural world. "Philosophy is written in this very great book. . . . It is written in mathematical language and the characters are triangles, circles and other geometrical figures." More than 300 years later, debating math education at the 1958 International Congress of Mathematicians, French mathematician Jean Dieudonné interjected: "Down with Euclid! Death to triangles!".

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)

Some things look more complex than others. For example, a crenulate and richly organized leaf may seem more complex than a plain stone. What is the nature of this experience—and why do we have it in the first place? Here, we explore how object complexity serves as an efficiently extracted visual signal that the object merits further exploration. We algorithmically generated a library of geometric shapes and determined their complexity by computing the cumulative surprisal of their internal skeletons—essentially (...) quantifying the “amount of information” within each shape—and then used this approach to ask new questions about the perception of complexity. Experiments 1–3 asked what kind of mental process extracts visual complexity: a slow, deliberate, reflective process (as when we decide that an object is expensive or popular) or a fast, effortless, and automatic process (as when we see that an object is big or blue)? We placed simple and complex objects in visual search arrays and discovered that complex objects were easier to find among simple distractors than simple objects are among complex distractors—a classic search asymmetry indicating that complexity is prioritized in visual processing. Next, we explored the function of complexity: Why do we represent object complexity in the first place? Experiments 4–5 asked subjects to study serially presented objects in a self‐paced manner (for a later memory test); subjects dwelled longer on complex objects than simple objects—even when object shape was completely task‐irrelevant—suggesting a connection between visual complexity and exploratory engagement. Finally, Experiment 6 connected these implicit measures of complexity to explicit judgments. Collectively, these findings suggest that visual complexity is extracted efficiently and automatically, and even arouses a kind of “perceptual curiosity” about objects that encourages subsequent attentional engagement. (shrink)

Recent research indicates that interpersonal communication is noisy, and that people exhibit considerable insensitivity to problems in communication. Using a dyadic referential communication task, the goal of which is accurate information transfer, this study examined the extent to which interlocutors are sensitive to problems in communication and use other‐initiated repairs (OIRs) to address them. Participants were randomly assigned to dyads (N = 88 participants, or 44 dyads) and tried to communicate a series of recurring abstract geometric shapes to (...) a partner across a text–chat interface. Participants alternated between directing (describing shapes) and matching (interpreting shape descriptions) roles across 72 trials of the task. Replicating prior research, over repeated social interactions communication success improved and the shape descriptions became increasingly efficient. In addition, confidence in having successfully communicated the different shapes increased over trials. Importantly, matchers were less confident on trials in which communication was unsuccessful, communication success was lower on trials that contained an OIR compared to those that did not contain an OIR, and OIR trials were associated with lower Director Confidence. This pattern of results demonstrates that (a) interlocutors exhibit (a degree of) sensitivity to problems in communication, (b) they appropriately use OIRs to address problems in communication, and (c) OIRs signal problems in communication. (shrink)

Geometrical intuitions spontaneously drive visuo-spatial reasoning in human adults, children and animals. Is their emergence intrinsically linked to visual experience, or does it reflect a core property of cognition shared across sensory modalities? To address this question, we tested the sensitivity of blind-from-birth adults to geometrical-invariants using a haptic deviant-figure detection task. Blind participants spontaneously used many geometric concepts such as parallelism, right angles and geometrical shapes to detect intruders in haptic displays, but experienced difficulties with symmetry and (...) complex spatial transformations. Across items, their performance was highly correlated with that of sighted adults performing the same task in touch (blindfolded) and in vision, as well as with the performances of uneducated preschoolers and Amazonian adults. Our results support the existence of an amodal core-system of geometry that arises independently of visual experience. However, performance at selecting geometric intruders was generally higher in the visual compared to the haptic modality, suggesting that sensory-specific spatial experience may play a role in refining the properties of this core-system of geometry. (shrink)

The article explores digital literacy practices in children’s everyday lives at Norwegian preschools and some of the ways in which young children appropriate basic digital literacy skills through guided participation in situated activities. Building on an ethnomethodological perspective, the analyses are based on 70 hours of video recordings documenting the activities in which 45 children, aged 5-6, and eight preschool teachers participated. Through the detailed analysis of two categorization activities – identifying geometrical shapes and identifying feelings/thoughts –the use of (...) digital tools in the social organization of the activities are examined. The article finds that children’s digital literacy activities encompass visual, verbal, audio and embodied competencies that are made relevant, and thus accessible for learning, in interaction between the children and between the adults and children in the form of norms and guidelines for what constitutes correct categorizations (geometrical shapes and green and red feelings) in the situated activities, and that are appropriated and actualized by the children in interaction with their peers. The findings also show how the categorization practices in preschools deal with symbols and labels in ways that create and sustain socially organized ways of knowing, seeing and acting upon the world. Digital media are embedded in routines, procedures and socialites that are part of these categorization practices; they are part of how children are instructed to experience, interpret, understand and act in the world. Moreover, the different technologies created different conditions for the children’s participation. It was found that peer interaction was part of the digital literacy activities that involved such mobile technologies as smartphones and tablets, while when using non-mobile technologies, e.g. smartboards, the activities were structured more as ‘classic’ classroom activities, primarily guided by the teacher and the didactic material presented through the smartboard. (shrink)

The aim of the study was to assess the effects of visual aesthetic perception on event-related potentials . Eight subjects assigned an aesthetic judgment and a 10-step beauty estimation to the target stimuli, consisting of famous artistic pictures and geometric shapes. In a further task, the subjects performed a motor response to the previously judged pictures and geometric shapes. ERPs were recorded through 54 scalp electrodes during both tasks. The P3b amplitude was increased during the categorization (...) of the geometric shapes compared to the artistic figures and during the vision of the beautiful targets preceding the motor response. The categorization of the aesthetic qualities of geometrical shapes seems to induce a higher level of attention, while a higher arousal variation was elicited by the recognition of beauty, in any form that was presented. (shrink)

Dual process accounts of affective learning state that the learning of likes and dislikes reflects a learning mechanism that is distinct from the one reflected in expectancy learning, the learning of signal relationships, and has different empirical characteristics. Affective learning, for example, is said not to be affected by: (a) extinction training; (b) occasion setting; (c) cue competition; and (d) awareness of the CS-US contingencies. These predictions were tested in a series of experiments that employed simple Pavlovian conditioning procedures. Neutral (...) visual pictures of geometric shapes, or tactile conditional stimuli (CS) were paired with aversive electrotactile unconditional stimuli (US). Dependent measures were physiological (skin conductance, blink startle modulation) or verbal (US expectancy, on-line and off-line ratings of CS pleasantness). Different combinations of these dependent measures were employed across different experiments in an attempt to assess affective and expectancy learning simultaneously. Changes in CS pleasantness as indexed by ratings or blink startle modulation were readily observed. However, contrary to the predictions from dual-process accounts, results indicated that acquired CS unpleasantness is subject to extinction, occasion setting, cue competition, and not found in absence of CS-US contingency awareness. (shrink)

The problem of statistical inference based on a partial measurement (“coarse-graining”) requires the specification of an a priori distribution. We reformulate the ordinary theory such as to encompass systematically a wide range of a priori distributions (“relative coarse-graining”). This is done in a mathematical setting which admits an interpretation in both classical probability and quantum mechanics. The formalism is illustrated in a few simple examples, such as the die whose geometrical shape is known, the spin in thermal equilibrium with an (...) unknown reservoir, and the position measurement of a one-dimensional particle. It is shown that some of the limitations of the usual theory are a consequence of the fact that it is restricted to “equidistributed” (symmetric) a priori states. (shrink)

Forums, I extensively analysed Tartaglia’s corpus: science of weights, geometry, arithmetic, mathematics and physics–trajectories of the projectiles, fortifications, included its intelligibility science in the military architecture. The latter is exposed in Book VI of the Quesiti et invention diverse. In Quesiti there is La Gionta del sesto libro—a kind of appendix to the Book VI containing drawings of the geometric shape of the Italian fortifications. It is based on Euclidean geometry and other figures where a scale is displayed. The (...) interest—included intellectual history and cultural foundations of science—is: what is the role–played by La Gionta del sesto libro in the Quesiti? Is it independent booklet/speeches? If yes, when Tartaglia did write it? In this paper, I present an historical–philological–foundational hypothesis on the Tartaglia’s fortifications corpus, as exposed in the Book VI and La Gionta del sesto libro; with respect to dates of editions of Quesiti. This goal is important to make historically clear both the editorial role–played by Curzio Troiano Navò in the Tartaglia’s corpus before—after Tartaglia’s death and the particular interest and development of subject of fortifications by Tartaglia between 1537–1546 and 1554. (shrink)

The self-reference effect refers to better memory for self-relevant than for other-relevant information. Generally, the SRE is found in conditions in which links between the stimuli and the self are forged in the encoding phase. To investigate the possibility that such conditions are not prerequisites for the SRE, this research developed two conditions by using two recognition tasks involving abstract geometric shapes. One was the cue-in-encoding condition in which self- and other-cues were presented to construct links with AGSs (...) during the encoding phase, and the other was the cue-in-retrieval condition in which self- and other-cues were presented to construct links with AGSs during the retrieval phase. The SRE was found in both conditions. The findings reveal that self-cues merely presented during the retrieval phase are sufficient to induce the SRE. Links between the stimuli and the self constructed during the encoding phase may not be necessary prerequisites for the SRE. (shrink)

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)

Many elements of Derrida's “psychoanalytic” philosophy support the view that for most of his career he remained a philosopher-son: the disruptive attitude toward a systematic style, the tendency to exaggerate, the fervor—so like that of Heidegger, Freud, and especially Nietzsche—with which he repeatedly stages confrontations with his father-masters. However, Derrida never claimed, with the postmoderns, that every kind of reality is socially and historically constructed. His “mistake” is rather to have confused ideal objects, such as geometrical shapes and theorems, (...) with social objects, such as promises, contracts, and bills; for ideal objects do not depend on their embeddedness in a trace-structure as contracts and marriages undoubtedly do. The “Husserlian” theme of the centrality of writing becomes a powerful tool when applied to social reality: inscriptions are the ground of an enormous ontology that everywhere surrounds us. Derrida's achievement resides in his denunciation of the unconscious denigration of this ground and in his having foreseen the coming of our present “era of writing.”. (shrink)

Quick reference guide includes illustrated explanations of the most common terms used in general math classes. Discusses how students can use manipulatives and basic math tools to improve their understanding. With measurement conversion tables, guides to geometric shapes, and more.

The Stroop effect is a well-documented phenomenon, demonstrating both interference and facilitation effects. Many versions of the Stroop task were created, according to the purposes of its applications, varying in numerous aspects. While many versions are developed to investigate the mechanisms of the effect itself, the Stroop effect is also considered a general measure of attention, inhibitory control, and executive functions. In this paper, we implement “eStroop”: a new digital version based on verbal responses, measuring the main processes involved in (...) the traditional effect. eStroop features four categories of stimuli in four different colors: geometrical shapes, neutral words, congruent words, and incongruent words. The results of the administration to 307 University students confirm the Stroop effect and offer baseline data for future research and clinical testing. Direct comparisons with other recent versions of the task are discussed, offering insights into differences and similarities between different task variables. (shrink)