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The Beauty of Geometry: Twelve Essays
These absorbing essays by a distinguished mathematician provide a compelling demonstration of the charms of mathematics. Stimulating and thought-provoking, this collection is sure to interest students, mathematicians, and any math buff with its lucid treatment of geometry and the crucial role geometry plays in a wide range of mathematical applications.
- GenresMathematicsNonfiction
288 pages, Paperback
First published July 2, 1999
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About the author
H.S.M. Coxeter
25 books13 followersHarold Scott MacDonald "Donald" Coxeter (1907-2003), CC, FRS, FRSC was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
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January 18, 2017
The Beauty of Geometry is a great book for those who love math and are interested in a more in-depth study of geometry! The book is a compilation of twelve geometric essays written by H.S.M Coxeter. Each one was published in a professional mathematics journal, ranging from "The Quaterly Journal of Mathematics" to "Proceedings of the London Mathematical Society" to "Acta Mathematica Acad. Sci. Hungaricae." Expect heavy mathematics, because these are peer-reviewed scientific articles.
His first essay reconciles the functions of Schlafli and Lobatschefsky concerning dividing polygons into right-angled triangles and dividing polyhedron into double-rectangular tetrahedra. The second essay regards integral Cayley numbers and their application to the eight square theorem. The third article deals with Wythoff's construction for uniform polytopes. The discussion goes into multiple dimensions and he uses both spherical and Euclidean simplexes. I had previously never seen the types of graphs that he used for the various polytopes (either a forest, tree, or isolated circuit) and they were all entertaining and challenging. In the fourth essay, he classifies zonohedra by means of projective diagrams. The fifth essay looks at regular polyhedra in three and four dimensions, and their topological analogues. Previously, I had not studied geometry past three dimensions. This is a recurring theme in the book. If you are interested in this, please start reading! I definitely found it challenging.
The sixth paper is about self-dual configurations and regular graphs, including the Pappus graph and Desargues graph. The seventh is a study of twelve point in PG(5,3) with 95040 self-transformations. The eighth chapter encompasses arrangements of equal spheres in non-Euclidean spaces. Coxeter calculates the packing density and covering densities of spheres in multiple arrangements, including two-dimensional, three-dimensional, and hyperbolic 4-space. The ninth essay is about an upper bound for the number of equal nonoverlapping spheres that can touch another of the same size. The tenth article is about regular honeycomb in hyperbolic space. The eleventh essay is on reflected light signals. This delves into relativity theory, with two unaccelerated observers emitting light signals. In the twelfth and final chapter, Coxeter breaks down the vast subject of geometry into branches: Euclidean geometry, ordered geometry, sphere packing, integral quaternions and integral octaves, projective geometry, conics in the real plane, conics and k-arcs in a finite plane, hyperbolic geometry, exterior-hyperbolic geometry, and relativity.
I am interested the relation that this material has with the study of physics. I have a bachelor of science in physics and am currently working toward becoming a high school physics teacher. I would like to bring outside reading material into my future classroom and I chose to read this book with the potential of using it as a supplemental text. The eleventh chapter correlates with physics the most, as it concerns relativity. Coxeter looks at reflected light signals with two unaccelerated observers emitting signals alternatively and recording the proper times of the sequence. Using this in a high school classroom would be helpful to establish complex mathematics as a reality, since a geometric perspective on relativity stresses the four dimensions. As a high school math student, I did not have a concept of how dimensions above three could be "meaningful." Relativity adds time to our typical three dimensions of space, resulting in what physicists refer to as a four-dimensional "space-time." The mathematics and physics involved to truly understand this chapter, however, are above the high school level and would, ultimately, not be a helpful addition to the classroom.
His first essay reconciles the functions of Schlafli and Lobatschefsky concerning dividing polygons into right-angled triangles and dividing polyhedron into double-rectangular tetrahedra. The second essay regards integral Cayley numbers and their application to the eight square theorem. The third article deals with Wythoff's construction for uniform polytopes. The discussion goes into multiple dimensions and he uses both spherical and Euclidean simplexes. I had previously never seen the types of graphs that he used for the various polytopes (either a forest, tree, or isolated circuit) and they were all entertaining and challenging. In the fourth essay, he classifies zonohedra by means of projective diagrams. The fifth essay looks at regular polyhedra in three and four dimensions, and their topological analogues. Previously, I had not studied geometry past three dimensions. This is a recurring theme in the book. If you are interested in this, please start reading! I definitely found it challenging.
The sixth paper is about self-dual configurations and regular graphs, including the Pappus graph and Desargues graph. The seventh is a study of twelve point in PG(5,3) with 95040 self-transformations. The eighth chapter encompasses arrangements of equal spheres in non-Euclidean spaces. Coxeter calculates the packing density and covering densities of spheres in multiple arrangements, including two-dimensional, three-dimensional, and hyperbolic 4-space. The ninth essay is about an upper bound for the number of equal nonoverlapping spheres that can touch another of the same size. The tenth article is about regular honeycomb in hyperbolic space. The eleventh essay is on reflected light signals. This delves into relativity theory, with two unaccelerated observers emitting light signals. In the twelfth and final chapter, Coxeter breaks down the vast subject of geometry into branches: Euclidean geometry, ordered geometry, sphere packing, integral quaternions and integral octaves, projective geometry, conics in the real plane, conics and k-arcs in a finite plane, hyperbolic geometry, exterior-hyperbolic geometry, and relativity.
I am interested the relation that this material has with the study of physics. I have a bachelor of science in physics and am currently working toward becoming a high school physics teacher. I would like to bring outside reading material into my future classroom and I chose to read this book with the potential of using it as a supplemental text. The eleventh chapter correlates with physics the most, as it concerns relativity. Coxeter looks at reflected light signals with two unaccelerated observers emitting signals alternatively and recording the proper times of the sequence. Using this in a high school classroom would be helpful to establish complex mathematics as a reality, since a geometric perspective on relativity stresses the four dimensions. As a high school math student, I did not have a concept of how dimensions above three could be "meaningful." Relativity adds time to our typical three dimensions of space, resulting in what physicists refer to as a four-dimensional "space-time." The mathematics and physics involved to truly understand this chapter, however, are above the high school level and would, ultimately, not be a helpful addition to the classroom.
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