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)

Quantum geometry on a discrete set means a directed graph with a weight associated to each arrow defining the quantum metric. However, these ‘lattice spacing’ weights do not have to be independent of the direction of the arrow. We use this greater freedom to give a quantum geometric interpretation of discrete Markov processes with transition probabilities as arrow weights, namely taking the diffusion form ∂+f = f for the graph Laplacian Δθ, potential functions q, p built from the probabilities, and (...) finite difference ∂+ in the time direction. Motivated by this new point of view, we introduce a ‘discrete Schrödinger process’ as ∂+ψ = ıψ for the Laplacian associated to a bimodule connection such that the discrete evolution is unitary. We solve this explicitly for the 2-state graph, finding a 1-parameter family of such connections and an induced ‘generalised Markov process’ for f = |ψ|2 in which there is an additional source current built from ψ. We also mention our recent work on the quantum geometry of logic in ‘digital’ form over the field F2 = {0, 1}, including de Morgan duality and its possible generalisations. (shrink)

Cet article explore la signification de l’espace comme un « lieu pratiqué » selon la notion reprise à Michel de Certeau, en examinant la construction d’un gymnase et ses effets sur les relations sociales et les réseaux disciplinaires. Tout comme le laboratoire ou le théâtre, le gymnase a été spécifiquement pensé pour permettre certaines actions et en témoigner, en reflétant des conceptions de l’entraînement et de l’éducation corporelle. Ses divers agencements de l’espace y favorisent une incorporation de la race, du (...) lieu, du genre et de l’identité. Le War Memorial Gymnasium de l’Université de Colombie britannique dans l’Ouest du Canada, construit en 1851, est utilisé pour illustrer la manière dont, sous des formes complexes, les géométries du pouvoir – ces multiples configurations de l’espace et des relations sociales à l’intérieur et autour du gymnase – produisent dans ses murs une culture didactique et sportive aux enjeux véritablement stratégiques et tactiques, qui active et articule les division de genre. (shrink)

In mathematics textbooks and special mathematical treatises, themes and theses of Arthur Schopenhauer's elementary geometry appear again and again. Since Schopenhauer's geometry or philosophy of geometry was considered exemplary in the 19th and early 20th centuries in its relation to figures and thus to the intuition, the two-hundred-year reception history sketched in this paper also follows the evaluation of intuition-related geometries, which depends on the mathematical paradigms.

Richard Tieszen [Tieszen, R. (2005). Philosophy and Phenomenological Research, LXX(1), 153–173.] has argued that the group-theoretical approach to modern geometry can be seen as a realization of Edmund Husserl’s view of eidetic intuition. In support of Tieszen’s claim, the present article discusses Husserl’s approach to geometry in 1886–1902. Husserl’s first detailed discussion of the concept of group and invariants under transformations takes place in his notes on Hilbert’s Memoir Ueber die Grundlagen der Geometrie that Hilbert wrote during the winter 1901–1902. (...) Husserl’s interest in the Memoir is a continuation of his long-standing concern about analytic geometry and in particular Riemann and Helmholtz’s approach to geometry. Husserl favored a non-metrical approach to geometry; thus the topological nature of Hilbert’s Memoir must have been intriguing to him. The task of phenomenology is to describe the givenness of this logos, hence Husserl needed to develop the notion of eidetic intuition. (shrink)

Picture geometry is often regarded as an area of technical knowledge that accompanies or provides useful information for basic research on visual culture and almost never as a methodological one. Despite the historical and conceptual connections between mathe­matics and the visual, even a basic geometric competence is by no means a common of image and visual culture researchers. At the same time, the overwhelming majority of this kind of work belong to the field of technical knowledge, the history of mathemat­ics, (...) or positivism based theories; in other words, they do not address the most important issues for humanities research. Rare non-technical works in this area do not find due recognition and dissemination among a wider community of specialists. The article offers the main result of the research, the subject of which was pictorial depth. Pictorial depth is the version of the specific experience of distance with which the image is traditionally associated. The problem of distance is a classical area of philosophical reflection on the image. The article presents an attempt to geometrically express one of its versions, namely pictorial and visual depth. At the same time, the important dimension of inacces­sibility and heterogeneity is preserved, that is, the experience of depth is not reduced to visual data or the literal geometry of a flat picture. By constructing a geometrically object that can be considered depth, it is possible to extract intuitively non-obvious properties that would not be available through direct analysis of experience and other methods of naked reflection. The article presents only some of the findings obtained during the study. (shrink)

Finsler geometry on the tangent bundle appears to be applicable to relativistic field theory, particularly, unified field theories. The physical motivation for Finsler structure is conveniently developed by the use of “gauge” transformations on the tangent space. In this context a remarkable correspondence of metrics, connections, and curvatures to, respectively, gauge potentials, fields, and energy-momentum emerges. Specific relativistic electromagnetic metrics such as Randers, Beil, and Weyl can be compared.

Geometry, etymologically the “science of measuring the Earth”, is a mathematical formalization of space. Just as formal concepts of number may be rooted in an evolutionary ancient system for perceiving numerical quantity, the fathers of geometry may have been inspired by their perception of space. Is the spatial content of formal Euclidean geometry universally present in the way humans perceive space, or is Euclidean geometry a mental construction, specific to those who have received appropriate instruction? The spatial content of the (...) formal theories of geometry may depart from spatial perception for two reasons: first, because in geometry, only some of the features of spatial figures are theoretically relevant; and second, because some geometric concepts go beyond any possible perceptual experience. Focusing in turn on these two aspects of geometry, we will present several lines of research on US adults and children from the age of three years, and participants from an Amazonian culture, the Mundurucu. Almost all the aspects of geometry tested proved to be shared between these two cultures. Nevertheless, some aspects involve a process of mental construction where explicit instruction seem to play a role in the US, but that can still take place in the absence of instruction in geometry. (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)

Les architectures molles, sculptées, transparentes, immatérielles prétendent se libérer des contraintes géométriques, comme si la géométrie ne revendiquait que la droite et la forme orthogonale ou le cercle! Certains architectes s'abandonnent aux " hasards " informatiques et construisent des édifices à la géométrie chahutée par un logiciel. Des urbanistes opposent encore le plan radioconcentrique au plan en damier en ce qui concerne l'expansion des villes et, refusant d'imaginer d'autres morphologies, laissent faire la promotion immobilière, les opportunités foncières et le chacun (...) pour soi. La géométrie, dans notre culture marquée par la philosophie grecque, est constitutive et de l'architecture et de l'urbanisme. Elle est mise en débat par le jeu extraordinairement varié des formes et de leurs agencements, aussi bien que par des régulations qui donnent une mesure au monde et suscitent des questionnements quant à ce qui est à la mesure de l'existence. Depuis le simple pas jusqu'aux théories les plus sophistiquées, la géométrie - qui n'a jamais cessé de se complexifier depuis Pythagore ou Euclide jusqu'aux géométries algébrique, infinitésimale et variable - se rappelle à nous. C'est ce rappel qu'il nous faut entendre, comme une invitation à penser aussi bien notre corps que le paysage, aussi bien la maison que la ville et la cité. Cet ouvrage collectif veut questionner géométriquement et philosophiquement l'urbain contemporain et les architectures qu'il provoque. En d'autres termes, il espère saisir à partir de la confrontation entre mathématiciens, géomètres, historiens, architectes, urbanistes, paysagistes et philosophes l'expérience existentielle de l'espace-temps des lieux. (shrink)

The Aristotelian square of oppositions is a well-known diagram in logic and linguistics. In recent years, several extensions of the square have been discovered. However, these extensions have failed to become as widely known as the square. In this paper we argue that there is indeed a fundamental difference between the square and its extensions, viz., a difference in informativity. To do this, we distinguish between concrete Aristotelian diagrams and, on a more abstract level, the Aristotelian geometry. We then introduce (...) two new logical geometries, and develop a formal, well-motivated account of their informativity. This enables us to show that the square is strictly more informative than many of the more complex diagrams. (shrink)

The paper presents the diagrammatic calculus CL, which combines features of tree, Euler-type, Venn-type diagrams and squares of opposition. In its basic form, `CL' (= Cubus Logicus) organizes terms in the form of a square or cube. By applying the arrows of the square of opposition to CL, judgments and inferences can be displayed. Thus CL offers on the one hand an intuitive method to display ontologies and on the other hand a diagrammatic tool to check inferences. The paper focuses (...) mainly on the adaptation of the square of opposition in CL and offers an algebraic notation, which corresponds to the diagrammatic representation. (shrink)

§1. Introduction. With Hrushovski's proof of the function field Mordell-Lang conjecture [16] the relevance of geometric stability theory to diophantine geometry first came to light. A gulf between logicians and number theorists allowed for contradictory reactions. It has been asserted that Hrushovski's proof was simply an algebraic argument masked in the language of model theory. Another camp held that this theorem was merely a clever one-off. Still others regarded the argument as magical and asked whether such sorcery could unlock the (...) secrets of a wide coterie of number theoretic problems.In the intervening years each of these prejudices has been revealed as false though such attitudes are still common. The methods pioneered in [16] have been extended and applied to a number of other problems. At their best, these methods have been integrated into the general methods for solving diophantine problems. Moreover, the newer work suggests limits to the application of model theory to diophantine geometry. For example, all such known applications are connected with commutative algebraic groups. This need not be an intrinsic restriction, but its removal requires serious advances in the model theory of fields. (shrink)

In 2003, Peratt demonstrated that rock art images worldwide bear a remarkable similarity to high-energy plasma discharge formations. In later papers, Peratt located the plasma discharge column in which all of these would have occurred at the Earth’s South Pole. This article accepts the relation between the rock art images and the plasma formations, but concludes that the geometry of the reconstruction is incompatible with the global occurrence of the rock art images. As a corollary, the finer details of the (...) reconstructed column must also be called into question. In particular, the reconstruction of the top cusp, the two upper plasmoids, and the filamentary sheath in a single column at the South Pole cannot be reliably deduced from the data as presented by Peratt. All evidence points to a worldwide distribution of the phenomena. (shrink)

La connaissances noyau de la géométrie euclidienne est liée au raisonnement déductif et non à la reconnaissance de motifs perceptuels.

We investigate the isomorphism types of combinatorial geometries arising from Hrushovski's flat strongly minimal structures and answer some questions from Hrushovski's original paper.

Does geometry constitues a core set of intuitions present in all humans, regarless of their language or schooling ? We used two non verbal tests to probe the conceptual primitives of geometry in the Munduruku, an isolated Amazonian indigene group. Our results provide evidence for geometrical intuitions in the absence of schooling, experience with graphic symbols or maps, or a rich language of geometrical terms.

A novel cognitive theory of semantics that proposes that the meanings of words can be described in terms of geometric structures.

Between 1873 and 1876, Felix Klein published a series of papers that he later placed under the rubric anschauliche Geometrie in the second volume of his collected works (1922). The present study attempts not only to follow the course of this work, but also to place it in a larger historical context. Methodologically, Klein’s approach had roots in Poncelet’s principle of continuity, though the more immediate influences on him came from his teachers, Plücker and Clebsch. In the 1860s, Clebsch reworked (...) some of the central ideas in Riemann’s theory of Abelian functions to obtain complicated results for systems of algebraic curves, most published earlier by Hesse and Steiner. These findings played a major role in enumerative geometry, whereas Plücker’s work had a strongly qualitative character that imbued Klein’s early studies. A leitmotif in these works can be seen in the interplay between real curves and surfaces as reflected by their transformational properties. During the early 1870s, Klein and Zeuthen began to explore the possibility of deriving all possible forms for real cubic surfaces as well as quartic curves. They did so using continuity methods reminiscent of Poncelet’s earlier approach. Both authors also relied on visual arguments, which Klein would later advance under the banner of intuitive geometry (anschauliche Geometrie). (shrink)

The city has featured as a central image in utopian thought. In planning the foundation of the new and ideal city there is a close interconnection between ideas about urban form and the vision of the moral good. The spatial structure of the ideal city in these visions is a framing device that embodies and articulates not only political philosophy but is itself an articulation of moral and cosmological systems. This paper analyses three different utopian moments in three different historical (...) epochs – Tommaso Campanella’s City of the Sun (1602), the Choson dynasty foundation of the city of Seoul (1395) and the modernist utopian urbanism of the controversial Le Corbusier (1887–1965). In this analysis attention is drawn to the cosmological and moral visions articulated in these three ‘images of the city’ (Lynch). The opposition between rationalistic/mechanistic and religious/traditional urban design can prove to be an oversimplification that obscures the complex interrelations between the moral geometry and the natural alignments of the ideal urban form. (shrink)

The controversy between Euclidean and non-Euclidean geometry arose new philosophical and scientific insights which were relevant to the later development of natural science. Here we want to consider Poincaré and Helmholtz’s positions as two of the most important and original ones who contributed to the subsequent development of the epistemology of natural sciences. Based in these conceptions, we will show that the role of philosophy is still important for some aspects of science.

The dispute that raged between Thomas Hobbes and John Wallis from 1655 until Hobbes's death in 1679 was one of the most intense of the ‘battles of the books’ in seventeenth-century intellectual life. The dispute was principally centered on geometric questions, but it also involved questions of religion and politics. This paper investigates the origins of the dispute and argues that Wallis’s primary motivation was not so much to refute Hobbes’s geometry as to demolish his reputation as an authority in (...) political, philosophical, and religious matters. It also highlights the very different conceptions of geometrical methodology employed by the two disputants. In the end, I argue that, although Wallis was successful in showing the inadequacies of Hobbes’s geometric endeavours, he failed in his quest to discredit the Hobbesian philosophy in toto. (shrink)

A timely collection of advanced, original material in the area of statistical methodology motivated by geometric problems, dedicated to the influential work of Kanti V. Mardia This volume celebrates Kanti V. Mardia's long and influential career in statistics. A common theme unifying much of Mardia’s work is the importance of geometry in statistics, and to highlight the areas emphasized in his research this book brings together 16 contributions from high-profile researchers in the field. Geometry Driven Statistics covers a wide range (...) of application areas including directional data, shape analysis, spatial data, climate science, fingerprints, image analysis, computer vision and bioinformatics. The book will appeal to statisticians and others with an interest in data motivated by geometric considerations. Summarizing the state of the art, examining some new developments and presenting a vision for the future, Geometry Driven Statistics will enable the reader to broaden knowledge of important research areas in statistics and gain a new appreciation of the work and influence of Kanti V. Mardia. (shrink)

This article explores the changing relationships between geometric and arithmetic ideas in medieval Europe mathematics, as reflected via the propositions of Book II of Euclid’s Elements. Of particular interest is the way in which some medieval treatises organically incorporated into the body of arithmetic results that were formulated in Book II and originally conceived in a purely geometric context. Eventually, in the Campanus version of the Elements these results were reincorporated into the arithmetic books of the Euclidean treatise. Thus, while (...) most of the Latin versions of the Elements had duly preserved the purely geometric spirit of Euclid’s original, the specific text that played the most prominent role in the initial passage of the Elements from manuscript to print—i.e., Campanus’ version—followed a different approach. On the one hand, Book II itself continued to appear there as a purely geometric text. On the other hand, the first ten results of Book II could now be seen also as possiblytranslatable into arithmetic, and in many cases even as inseparably associated with their arithmetic representation. (shrink)

As part of his program to unify linear algebra and geometry using the language of Clifford algebra, David Hestenes has constructed a (well-known) isomorphism between the conformal group and the orthogonal group of a space two dimensions higher, thus obtaining homogeneous coordinates for conformal geometry.(1) In this paper we show that this construction is the Clifford algebra analogue of a hyperbolic model of Euclidean geometry that has actually been known since Bolyai, Lobachevsky, and Gauss, and we explore its wider invariant (...) theoretic implications. In particular, we show that the Euclidean distance function has a very simple representation in this model, as demonstrated by J. J. Seidel.(18). (shrink)

L’emploi par Poincaré de la notion de convention au sujet des hypothèses géométriques signale un déplacement par rapport aux problématiques traditionnelles. La découverte des géométries non euclidiennes montre qu’il n’y a pas de cadre spatial unique ; plusieurs systèmes sont possibles. On affirme ainsi l’existence d’un aspect essentiel de la connaissance qui ne dérive pas des faits et ne relève ni de l’inné ni de l’intuition. L’introduction de la notion de convention, dont il s’agit de prendre la mesure, ouvre la (...) voie à une prise en compte des facteurs décisionnels en science. Nous nous proposons, dans cet article, d’analyser l’argumentation de Poincaré en reconstruisant sa position face aux doctrines de Kant et de Mill.By understanding geometrical hypotheses as conventions, Poincaré significantly alters the traditional philosophical discussion of space. The discovery of non-Euclidean geometries proves that there is not a single spatial framework ; several systems are possible. Thus, there is an essential feature of knowledge that derives neither from empirical facts nor from intuition or innate understanding. The introduction of the notion of convention, whose significance is at issue here, provides a way of understanding the element of free choice in science. This article aims at analyzing Poincaré’s reasoning, by contrasting his viewpoint with that of Kant and Mill. (shrink)

Geometry and Chronometry in Philosophical Perspective was first published in 1968. Minnesota Archive Editions uses digital technology to make long-unavailable books once again accessible, and are published unaltered from the original University of Minnesota Press editions. In this volume Professor Grünbaum substantially extends and comments upon his essay "Geometry, Chronometry, and Empiricism," which was first published in Volume III of the Minnesota Studies in the Philosophy of Science. Commenting on the essay when it first appeared J. J. C. Smart wrote (...) in Mind (England): "In my opinion Adolf Grünbaum's paper ... supersedes nearly all that has been written on the logical status of physical geometry and chronometry." The full text of the essay is given here with the author's extension of it and his discussion of some of the critical comment it has evoked, particularly, a critique published by Hilary Putnam. Adolph Grünbaum is Andrew Mellon Professor of Philosophy at the University of Pittsburgh and the current president of the Philosophy of Science Association. (shrink)

This book explores and compares the reflections on space and quantity found in the works of five philosophers: Spinoza, Leibniz, Bergson, Whitehead, and Deleuze. What unites these philosophers is a series of metaphysical concerns rooted in 17th-century rationalism and embraced in 20th-century philosophies of process and difference. At the heart of these concerns is the need for a comprehensive metaphysical account of the diversity and individuality of things. This demand leads to a shared critique of Cartesian and Newtonian conceptions of (...) space. The most problematic aspect of those notions of space is homogeneity. In essence, uniform space fails to explain the differences between locations, thus violating the principle of sufficient reason. Cartesian and Newtonian theories of space thereby fail to meet the metaphysical requirement for explaining diversity and individuality. The traditional concept of quantity faces similar issues. Motivated by these problems, these five philosophers develop an alternative conception of space and quantity. By examining these theories, the book sheds new light on an unexplored relation between rationalism and 20th-century Continental philosophy. A Geometry of Sufficient Reason will appeal to scholars and graduate students working in Continental philosophy, history of philosophy, metaphysics, and the history and philosophy of science. (shrink)

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a (...) “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “ n -opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m ” ( m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid). (shrink)

This paper examines a social contractarian model in which an actor cooperates by mimicry; that is, cooperates just in case there is majority cooperation in his orher vicinity. A computer simulation is developed to study the relation between initial and final proportions of such cooperators, as wel l as to chart the population dynamics themselves. The model turns out to be non-linear; item bodies a quintessentially chaotic threshold. The simulation also yields other unforeseen results, revealing a "geometry of delection" that (...) unites delecting cells into robust molecular formations which persist with in overall cooperative domains, or which under certain conditions undermine cooperativeness entirely. The model thus sheds so me light on the structura l dimension of mimicry that underlies social communication, conflict and its resolution. (shrink)

The analysis of the golden Mithras’ bas-relief in the Museum of the Baths of Diocletian in Rome confirms the Platonic Chiasma. The scene admits two diagonals starting from each corner. One passes through the sun and the other through the moon. The sun god is also shown with an object in his left hand, which may be a soul or a sacred heart. This would confirm that the slab shows the opposition between metempsychosis (lunar) and resurrection (solar). The analysis of (...) the Barberini fresco is of great interest. It shows that the Platonic Chiasma falls exactly within the celestial gates of Capricorn and Cancer. The solar line is superimposed on a line already present on the fresco. Finally, sacred geometry helps to explain some strange-looking reliefs from the Danube region. (shrink)

Thomas C. Vinci argues that Kant's Deductions demonstrate Kant's idealist doctrines and have the structure of an inference to the best explanation for correlated domains. With the Deduction of the Categories the correlated domains are intellectual conditions and non-geometrical laws of the empirical world. With the Deduction of the Concepts of Space, the correlated domains are the geometry of pure objects of intuition and the geometry of empirical objects.

Even though the discovery of the regular polyhedra is attributed to the Pythagoreans, there is some fascinating evidence that they may have been known in prehistoric Scotland. In the Ashmolean Museum at Oxford University there are five rounded stones with regularly spaced bumps. The high points of each bump mark the vertices of each of the regular polyhedra. The stone balls also appear to demonstrate the duals of three of the regular polyhedra. For example, if the six faces of the (...) cube become points, they become the six vertices of the octahedron. If the eight faces of the octahedron become points, they become the eight vertices of a cube. Remarkably, groves in the Ashmolean stones indicate each of the duals, including the fact that the tetrahedron is its own dual. Instead of assuming that these ancient Scots were experts in solid geometry, we have hypothesized that they must have discovered all of this by sphere stacking, which will be demonstrate later in this article. (shrink)

Fix an algebraically closed field of characteristic zero and let G be its geometry of transcendence degree one extensions. Let X be a set of points of G. We show that X extends to a projective subgeometry of G exactly if the partial derivatives of the polynomials inducing dependence on its elements satisfy certain separability conditions. This analysis produces a concrete representation of the coordinatizing fields of maximal projective subgeometries of G.

Elementary geometry can be axiomatized constructively by taking as primitive the concepts of the apartness of a point from a line and the convergence of two lines, instead of incidence and parallelism as in the classical axiomatizations. I first give the axioms of a general plane geometry of apartness and convergence. Constructive projective geometry is obtained by adding the principle that any two distinct lines converge, and affine geometry by adding a parallel line construction, etc. Constructive axiomatization allows solutions to (...) geometric problems to be effected as computer programs. I present a formalization of the axiomatization in type theory. This formalization works directly as a computer implementation of geometry. (shrink)

The concept of geometry may evoke a world of pure platonic shapes, such as spheres and cubes, but a deeper understanding of visual experience demands insight into the perceptual organization of naturalistic form. Japanese gardens excel as designed environments where the complex fractal geometry of nature has been simplified to a structural core that retains the essential properties of the natural landscape, thereby presenting an ideal opportunity for investigating the geometry and perceptual significance of such naturalistic characteristics. Here, fronto-parallel perspective, (...) asymmetrical structuring of the ground plane, spatial arrangement of garden elements, tuning of textural qualities and choice of naturalistic form, are presented as a set of physical features that facilitate a systematic analysis of Japanese garden design per se, as well as the geometry of the particular naturalistic features that it aims to enhance. Comparison with Western landscape design before and after contact between Western and Eastern hemispheres illustrates the degree of naturalness achieved in the Japanese garden, and suggests how classical Western landscape design generally differs in this regard. It further reveals how modern Western gardens culminate in a different naturalistic geometry, thus also a distinctly different vision of the natural landscape, even if these designs were greatly influenced by the gardens of China and Japan. (shrink)

This book reconstructs, both from the historical and theoretical points of view, Leibniz's geometrical studies, focusing in particular on the research Leibniz ...

I use recent work on Kant and diagrammatic reasoning to develop a reconsideration of central aspects of Kant’s philosophy of geometry and its relation to spatial intuition. In particular, I reconsider in this light the relations between geometrical concepts and their schemata, and the relationship between pure and empirical intuition. I argue that diagrammatic interpretations of Kant’s theory of geometrical intuition can, at best, capture only part of what Kant’s conception involves and that, for example, they cannot explain why Kant (...) takes geometrical constructions in the style of Euclid to provide us with an a priori framework for physical space. I attempt, along the way, to shed new light on the relationship between Kant’s theory of space and the debate between Newton and Leibniz to which he was reacting, and also on the role of geometry and spatial intuition in the transcendental deduction of the categories. (shrink)

This paper examines the discrepancy between the attitudes of many historians of mathematics to sixteenth-century geometry and those of museum curators and others interested in practical mathematics and in instruments. It argues for the need to treat past mathematical practice, not in relation to timeless criteria of mathematical worth, but according to the agenda of the period. Three examples of geometrical activity are used to illustrate this, and two particular contexts are presented in which mathematical practice localised in the sixteenth (...) century takes on a special historical significance. (shrink)