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24
Computer Graphics and Geometric Ornamental Design
, 2002
"... and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made. ..."
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Cited by 14 (3 self)
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and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made.
On polygonal dual billiard in the hyperbolic plane
 REG. CHAOTIC DYNAMICS
, 2003
"... We study the polygonal dual billiard map in the hyperbolic plane. We show that for a class of convex polygons called large all orbits of the dual billiard map escape to infinity. We also analyse the dynamics of the dual billiard map when the dual billiard table is a regular polygon with all right an ..."
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Cited by 6 (2 self)
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We study the polygonal dual billiard map in the hyperbolic plane. We show that for a class of convex polygons called large all orbits of the dual billiard map escape to infinity. We also analyse the dynamics of the dual billiard map when the dual billiard table is a regular polygon with all right angles.
Great Problems of Mathematics: A Course Based on Original Sources
 American Mathematical Monthly
, 1992
"... how progress is repeatedly stifled by certain ways of thinking until some quantum leap ushers in a new era. In addition to allowing a firsthand look at the mathematical mindscape of the time, no other method would show so clearly the evolution of mathematical rigor and the conception of what constit ..."
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Cited by 5 (3 self)
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how progress is repeatedly stifled by certain ways of thinking until some quantum leap ushers in a new era. In addition to allowing a firsthand look at the mathematical mindscape of the time, no other method would show so clearly the evolution of mathematical rigor and the conception of what constitutes an acceptable proof. Thus most homework assignments focus on gaps and difficult points in the original texts. Since mathematics is not created in a social vacuum, we supplement the mathematical content with cultural, biographical, and mathematical history, as well as a variety of prose readings, ranging from Plato's dialogue Socrates and the Slave Boy to modern writings such as an excerpt on "Mathematics and the End of the World" from [8]. They form the basis of regular class discussions. Two good sources for such readings are [11, 18]. To encourage student involvement, the discussions are led by one or two students, and everybody is expected to contribute. As the finale
From Pythagoras To Einstein: The Hyperbolic Pythagorean Theorem
, 1998
"... A new form of the Hyperbolic Pythagorean Theorem, which has a striking intuitive appeal and offers a strong contrast to its standard form, is presented. It expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right angled triangle as the "Einstein sum" of the squares of ..."
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Cited by 5 (5 self)
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A new form of the Hyperbolic Pythagorean Theorem, which has a striking intuitive appeal and offers a strong contrast to its standard form, is presented. It expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right angled triangle as the "Einstein sum" of the squares of the hyperbolic lengths of the other two sides, Fig. 1, thus completing the long path from Pythagoras to Einstein.
A course on geometric group theory.
"... These notes are based on a series of lectures I gave at the Tokyo Institute of Technology from April to July 2005. They constituted a course entitled “An introduction to geometric group theory ” totalling about 20 hours. The audience consisted of fourth year students, graduate students as well as se ..."
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Cited by 3 (0 self)
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These notes are based on a series of lectures I gave at the Tokyo Institute of Technology from April to July 2005. They constituted a course entitled “An introduction to geometric group theory ” totalling about 20 hours. The audience consisted of fourth year students, graduate students as well as several staff members. I therefore tried to present a logically coherent introduction to the subject, tailored to the background of the students, as well as including a number of diversions into more sophisticated applications of these ideas. There are many statements left as exercises. I believe that those essential to the logical developments will be fairly routine. Those related to examples or diversions may be more challenging. The notes assume a basic knowledge of group theory, and metric and topological spaces. We describe some of the fundamental notions of geometric group theory, such as quasiisometries, and aim for a basic overview of hyperbolic groups. We describe group presentations from first principles. We give an outline description of fundamental groups and covering spaces, sufficient to allow us to illustrate various results with more explicit examples. We also give a crash course on hyperbolic geometry. Again the presentation is
Spacetime Geometry Translated into the Hegelian and Intuitionist Systems
"... Abstract: Kant noted the importance of spatial and temporal intuitions (synthetics) in geometric reasoning, but intuitions lend themselves to different interpretations and a more solid grounding may be sought in formality. In mathematics David Hilbert defended formality, while L. E. J. Brouwer cited ..."
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Abstract: Kant noted the importance of spatial and temporal intuitions (synthetics) in geometric reasoning, but intuitions lend themselves to different interpretations and a more solid grounding may be sought in formality. In mathematics David Hilbert defended formality, while L. E. J. Brouwer cited intuitions that remain unencompassed by formality. In this paper, the conflict between formality and intuition is again investigated, and it is found to impact on our interpretations of spacetime as translated into the language of geometry. It is argued that that language as a formal system works because of an auxiliary innateness that carries sentience, or feeling. Therefore, the formality is necessarily incomplete as sentience is beyond its reach. Specifically, it is argued that sentience is covertly connected to spacetime geometry when axioms of congruency are stipulated, essentially hiding in the formality what is sensecertain. Accordingly, geometry is constructed from primitive intuitions represented by onepointedness and routeinvariance. Geometry is recognized as a twosided language that permitted a Hegelian passage from Euclidean geometry to Riemannian geometry. The concepts of general relativity, quantum mechanics and entropyirreversibility are found to be the consequences of linguistic type reasoning, and perceived conflicts (e.g., the puzzle of quantum gravity) are conflicts only within formal linguistic systems. Therefore, the conflicts do not survive beyond the synthetics because what is felt relates to inexplicable feeling, and because the question of synthesis returns only to Hegel’s absolute Notion.
Books on Demand
, 2002
"... The picture on the cover is a representation of an smanifold illustrating some of the behavior of lines in an smanifold. This book has been peer reviewed and recommended for publication by: ..."
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The picture on the cover is a representation of an smanifold illustrating some of the behavior of lines in an smanifold. This book has been peer reviewed and recommended for publication by:
Conceptions of the Continuum
"... Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question ..."
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Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question the idea from current set theory that the continuum is somehow a uniquely determined concept. Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions. 1. What is the continuum? On the face of it, there are several distinct forms of the continuum as a mathematical concept: in geometry, as a straight line, in analysis as the real number system (characterized in one of several ways), and in set theory as the power set of the natural numbers and, alternatively, as the set of all infinite sequences of zeros and ones. Since it is common to refer to the continuum, in what sense are these all instances of the same concept? When one speaks of the continuum in current settheoretical