Geometrical Heuristics for the Political Cosmology of Huainanzi 淮南子

Zoning is designed to delineate land uses in order to prevent urban development from incurring negative externalities. Morphologically speaking, zoning should result in spatial patterns that are closer to Euclidean (or regular) geometry than fractal (or irregular, self-similar) geometry. Previous computer simulations showed that zoning did not affect the fractal geometry of the urban spatial patterns of the Taipei metropolitan area. In this paper, we explore into the reasons why zoning fails in most Asian cities in general, and in Taipei in particular, through theoretical expositions. We argue that the preference for mixed uses is the main reason why zoning fails to achieve its goals. Furthermore, even a limited extent of mixed uses would result in spatial patterns characterised by fractal geometry rather than Euclidean geometry. We argue for a hybrid land regulation through the zoning and permit systems that take advantage of both the artificial and natural aspects of cities.

We review our scaling results for the diffusion-limited reactions A + A → 0 and A+B→0 on Euclidean and fractal geometries. These scaling results embody the anomalies that are observed in these reactions in low dimensions; we collect these observations under a single phenomenological umbrella. Although we are not able to fix all the exponents in our scaling expressions from first principles, we establish bounds that bracket the observed numerical results.

Fractals, engineering surfaces and tribology

الهندسة الكسرية | الهندسة الخوارزمية | التصميم الديناميكي | الهندسة الإقليدية | التكرارات الهندسية | الأتمتة | الاستدامة | Fractal Geometry | Algorithms | Dynamic Design | Euclidean Geometry | Geometrical Patterns | Automation | Sustainability

ABSTRACT: In this study we present some considerations about the End of Course Work undergraduate Full Degree in Mathematics / UFMT, drafted in 2011, and by taking title "A story about Cauchy and Euler's theorem on polyhedra" that gave birth to our research project Master of Education, begun in 2012, on the approaches of Euler's theorem on polyhedra in mathematics textbooks. At work in 2011 presented some considerations about the history of Euler's theorem for polyhedra which focus the demonstration presented by Cauchy (1789-1857), who tries to generalize it, relying on assumptions not observable in Euclidean geometry. Therefore, we seek the accessible literature on the history of mathematics; relate some aspects of the demonstration Cauchy with historical events on the development of mathematics in the nineteenth century, which allowed the acceptance of such a demonstration by mathematicians of his time. Keywords: History of Mathematics. Euler's Theorem on Polyhedra. Demonstration of Cauchy.

I have found the geometry of the mandala exists deep in the psyche, from its earliest anthropomorphic form to the Systema Munditotius mandala of C. G. Jung. In this article, I define the difference between Euclidean geometry and the descriptive geometry of Jacob Steiner and the geometry of Descartes’ modern mathematics. I analyze these geometries with the philosophical principles of Immanuel Kant, an important influence on Jung’s psychology.

The objective of the current study was to evaluate the spatial complexity of the cerebral pial surface through two-dimensional fractal analysis of the external linear contour of cerebral hemispheres and to investigate the correlation between the parameters determined using Euclidean and fractal geometries. Magnetic resonance brain images were obtained from 100 individuals (44 males and 56 females, aged 18–86 years). Five magnetic resonance images were selected from the MRI dataset of each brain, comprising four tomographic sections in the coronal plane and one section in the axial plane. Fractal dimension values of the linear contour of the pial surface of cerebral hemispheres were measured using the two-dimensional box counting method. Morphometric parameters based on Euclidean geometry were also determined (perimeter, area and their derivative values). In this study, the obtained fractal dimension values were shown to be sensitive to the tortuosity of the linear contour of cerebral hemispheres, which depends on the number of gyri and sulci and the complexity of their shape. Therefore, the fractal dimension can be considered as an objective quantitative parameter characterizing the spatial complexity of the pial surface of cerebral hemispheres. The present study revealed that the Euclidean geometry-based morphometric parameters most strongly associated with the fractal dimension of the cerebral linear contour were the perimeter and the parameters calculated from perimeter values, including the perimeter-to-area ratio, shape factor, and two-dimensional gyrification index. Fractal dimension values did not exhibit strong correlations with age. The data obtained in this study can be utilized for anatomical and anthropological studies. Furthermore, they hold practical applications in clinical contexts for diagnostic purposes, such as the diagnosis of congenital cerebral malformations and postnatal cerebral maldevelopment.

Morphometry is an integral part of most modern morphological studies and the classic morphological morphometric methods and techniques are often borrowed for research in other fields of medicine. The majority of morphometric techniques are derived from Euclidean geometry. In the past decades, the principles, parameters and methods of fractal geometry are increasingly used in morphological studies. The basic parameter of fractal geometry is fractal dimension. Fractal dimension allows you to quantify the degree of filling of space with a certain geometric object and to characterize the complexity of its spatial configuration. There are many anatomical structures with complex irregular shapes that cannot be unambiguously and comprehensively characterized by methods and techniques of traditional geometry and traditional morphometry: irregular linear structures, irregular surfaces of various structures and pathological foci, structures with complex branched, tree-like, reticulated, cellular or porous structure, etc. Fractal dimension is a useful and informative morphometric parameter that can complement existing quantitative parameters to quantify objective characteristics of various anatomical structures and pathological foci. Fractal analysis can qualitatively complement existing morphometric methods and techniques and allow a comprehensive assessment of the spatial configuration complexity degree of irregular anatomical structures. The review describes the basic principles of Euclidean and fractal geometry and their application in morphology and medicine, importance and application of sizes and their derivatives, topological, metric and fractal dimensions, regular and irregular figures in morphology, and practical application of fractal dimension and fractal analysis in the morphological studies and clinical practice.

Architectural forms are man-made and thus very much based in Euclidean geometry. However, in his book The Fractal Geometry of Nature, Mandelbrot makes the following comparison: The fractal new geometric art shows surprising kinship to Grand Master paintings or Beaux Arts architecture. An obvious reason is that classical visual arts, like fractals, involve very many scales of length and favor self similarity. Modern mathematics, music, painting, and architecture may seem to be related to one another. But this is a superficial impression, notably in the context of architecture. A Mies van der Rohe building is a scale bound throwback to Euclid, while a high period Beaux Arts building is rich in fractal aspects.

Although Lakatos’Proofs and Refutations (henceforth PR) is one of the classics of twentieth century philosophy of mathematics, few attempts have been made to build on its ideas.1 Rather, quite often PR has become an object of respectful reference and detailed exegetical efforts. Of course, Lakatos is credited for having taught philosophers that philosophy of mathematics has to take into account the history and practice of mathematics. This orientation towards the history and practice of the discipline certainly represents progress beyond the ahistorical traditions of logicism, formalism, platonism, and intuitionism. It should help to get rid of the most pernicious vice with which the philosophy of mathematics is plagued; to wit, elementarism, which considers elementary arithmetics of natural numbers or Euclidean geometry as typical for all of mathematics. I think, however, that there is more to learn from Lakatos than just a general orientation towards the history and practice of real-life mathematics. As I want to show in this paper, these insights concern the role of inventing and varying mathematical concepts. More precisely, I want to use Lakatos’ ideas of “concept-formation” and “concept-stretching” to sketch an evolutionary theory of mathematical knowledge, which takes axiomatic variation of concepts as the fundamental driving force of the ongoing evolution of mathematics (PR, 83ff.).2

The New 3 Rs:Repetition, Recollection, and Recognition Lissa Paul (bio) This is not really a critical essay—more like an item from a gossip column. But as I am writing and circulating it myself, it also has the quality of a leaked document. Or in dull academic terms it might be simply a description of a work-in-progress—though I like to think about what I'm doing as more interesting, perhaps even slightly scandalous or amoral. I'm working on a poetics of children's literature, one that moves out of the realm of criticism as interpretation. These new poetics are shaped by the nonlinear patterns of the "chaos" theory of physics, and by revisionist approaches to the Aristotelian concepts of mythos (plot), mimesis (imitation), and anagnorisis (recognition): the first major shifts in Western perception since Euclidian geometry and Platonic philosophy first shaped it. That may seem a rather extravagant claim, but please don't stop reading yet. My work-in-progress is more interesting—and easier—than it sounds though some background is necessary before I get to the gossipy bits. My obsessions with chaos theory and revisionist Aristotelian poetics developed out of a paper I did for the Narrative Theory in Children's Literature Section of the MLA in New Orleans in December 1988. In "Intimations of Imitations: Mimesis, Fractal Geometry and Children's Literature" (an expanded version is published in the May 1989 volume of Signal), I spoke about correspondences between two radically different models of perceiving the world: mimesis and fractal geometry. Mimesis is ancient (over twenty-five hundred years old), while fractal geometry is recent (the term coined only in 1975); mimesis originates in Aristotelian poetics, while fractal geometry originates in mathematics and is a description of irregular shapes. What struck me was that both theories articulate much the same vision: repetitions of self-similar structures produce unexpectedly varied ways of seeing the world. Mimesis, traditionally a static form that presumes to describe exact correspondences between the world and words, turns out, in the new poetics, to be notoriously diverse (in my paper, I talked about mimesis in the context of the cow poems in What is the Truth? by Ted Hughes—very different cow poems in the book are all recognizable representations of cows). Fractal geometry, by comparing relationships between scales, is a mathematical description of the shapes of irregular, natural forms likes snowflakes, tree bark, lightning, and clouds: shapes that are almost but never quite exactly alike (computer graphics make it possible to design and produce these images). Chaos theory (a misnomer really) is the name given to the study of the kind of order we've previously dismissed as disorderly or chaotic. In the end, mimesis, fractal geometry and chaos theory argue not for a world that is explainable, but for one that is infinitely varied. Rather than insisting on ultimate truths, they offer a language for the discussion of relationships that make variation visible. Because I was discussing the complex connections between mimesis, fractal geometry and poetry in my ten-minute oral paper at the MLA, I used a mnemonic—repetition, recollection and recognition—as a catch phrase to describe the pattern I had understood. I knew (from my readings on orality and literacy and from my knowledge of oral formulaic poetry) that it was the sort of line people could remember on one hearing. It was also convenient short-hand for the ways that both mimesis and fractal geometry represent the world as repetitions of self-similar patterns. My guilty secret (this is the scandalous bit) is that when I was working on the paper, I didn't fully understand the appropriateness of my triple catch phrase, repetition-recollection-recognition, as co-ordinates for a new poetics. Critics are supposed to know exactly what they mean. I didn't. But I did know that in "Intimations of Imitations," I was bringing together several critical lines that had formed my knowledge of the poetics of oral literature. I understood—more or less—that the works of Eric Havelock, Walter Ong, Peter Brooks and [End Page 55] Christopher Prendergast spoke to repetition and recollection, but I had not yet...

Preface to the Second Edition. PART 1: THE WORLD OF MATHEMATICS AND THE MATHEMATICS OF THE WORLD. Chapter 1. The Origin and Prehistory of Mathematics. Chapter 2. Mathematical Cultures I. Chapter 3. Mathematical Cultures II. Chapter 4. Women Mathematicians. PART 2: NUMBERS. Chapter 5. Counting. Chapter 6. Calculation. Chapter 7. Ancient Number Theory. Chapter 8. Numbers and Number Theory in Modren Mathematics. PART 3: COLOR PLATES. PART 4: SPACE. Chapter 9. Measurement. Chapter 10. Euclidean Geometry. Chapter 11. Post-Euclidean Geometry. Chapter 12. Modern Geometries. PART 5: ALGEBRA. Chapter 13. Prolems Leading to Algebra. Chapter 14. Equations and Methods. Chapter 15. Modern Algebra. PART 6: ANALYSIS. Chapter 16. The Calculus. Chapter 17. Real and Complex Aanlysis. PART 7: MATHEMATICAL INFERENCES. Chapter 18. Probability and Statistics. Chapter 19. Logic and Set Theory. Literature. Subject Index. Name Index.

The Main Trends in the Foundations of Geometry in the 19th Century

In contemporary philosophy, “visual thinking in mathematics” refers to studies of the kinds and roles of visual representations in mathematics. Visual representations include both external representations (i.e., diagrams) and mental visualization. Currently, three main areas and questions are being investigated. The first concerns the roles of diagrams, or the diagram-based reasoning, found in Euclid’s Elements. Second is the epistemic role of diagrams: the question of whether reasoning based on diagrams can be rigorous. This debate includes the question of whether beliefs based on visual input can be justified, and whether visual perception may lead to mathematical knowledge. The third observes that diagrams abound in (contemporary) mathematical practice, and so tries to understand the role they play, going beyond the traditional debates on the legitimacy of using diagrams in mathematical proofs. Looking at the history of mathematics, one will find that it is only recently that diagrammatic proofs have become discredited. For about 2,000 years, Euclid’s Elements was conceived as the paradigm of (mathematical) rigorous reasoning, and so until the 18th century, Euclidean geometry served as the foundation of many areas of mathematics. One includes the early history of analysis, where the study of curves draws on results from (Euclidean) geometry. During the 18th and 19th centuries, however, diagrams gradually disappear from mathematical texts, and around 1900 one finds the famous statements of Pasch and Hilbert claiming that proofs must not rely on figures. The development of formal logic during the 20th century further contributed to a general acceptance of a view that the only value of figures, or diagrams, is heuristic, and that they have no place in mathematical rigorous proofs. A proof, according to this view, consists of a discrete sequence of sentences and is a symbolic object. In the latter half of the 20th century, philosophers, sensitive to the practice of mathematics, started to object to this view, leading to the emergence of the study of visual thinking in mathematics.

Although several efforts produced by new mathematical education approaches for improving education systems and teaching, yet the results are not sufficient to adsorb the totality of innovations proposed, both in the contents and management. In this sense constructive debates and new ideas were welcomed and appreciated upon new aspects of science education, side new learning and Cognitive Modelling, for our interests. A parallel effort was produced by scientist-epistemologist-historians concerning the history of science and its foundations in science education. Historical foundations represent the most important intellectual part of the science, even if sometimes they were avoided or limited to specialist disciplines such as history of mathematics, history of physics, only. Nevertheless some results, such as the operative concept of mass by Mach, rather the coherence and validity of an algebraic–geometric group in a Euclidean geometry and in non-Euclidean geometry was firstly appointed by epistemological point of view by (e.g.,) Poincaré, etc... Thus, what kind of concrete relationship between science education (mathematics and physics) and history of science (idem) one can discuss correlated with foundations of science? and above all, how this relationship can be appointed? The history and epistemology of science help to understand evolution/involution of mathematical and physical sciences in the interpretation-modelling of a phenomenon and its interpretation-didactic-modelling, and how the interpretation can change for a different use of mathematical: e.g., mathematics à la Cauchy, non-standard analysis, constructive mathematics in physics. Based on previous studies, a discussion concerning mathematics education and history of science is presented. In our paper we will focus on learning modelling to discuss its efficacy and power both from educational point of view and the need of mathematics and physics teachers education. Some case–studies on the relationship between physics and mathematics in the history are presented, as well. Particularly we focus on a possible learning modelling activity within physics phenomenology to create a resonance among the above poles and mathematical modelling cycle to argue its efficacy, power and related with historical foundations of physical, mathematical sciences. Key words: modelling, mathematics, physics, history of foundations, epistemology of science.