Coxeter and Escher
H.S.M Coxeter’s (1907 – 2003) career spanned 60 years at the University of Toronto where he published twelve books on geometry. He was most noted for his work on regular polytopes and higher-dimensional geometries. In a period when the tendency was to approach geometry more and more through algebra, Coxeter championed the traditional approach. Coxeter’s work on geometric figures inspired some of Escher’s work and some of the innovations of Buckminster Fuller.
Maurits Escher (1898-1972) was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints including 1953's Relativity and Circle Limit III. These feature impossible constructions, explorations of infinity, architecture, and tessellations.Escher explored the concept of representing infinity on a two-dimensional plane and correspondences with Coxeter inspired Escher's interest in hyperbolic tessellations. Escher's wood engravings Circle Limit I–IV demonstrate this concept.
Coxeter discusses the math behind Escher's circle limit
HSM Coxeter talks about the Poincare disk model of hyperbolic geometry. He also talks about complex inversions. Clip from the "fantastic world of Escher"
Recommended Reads
Beauty of Geometry by ISBN: 0486409198Publication Date: 1999-07-02Written by a distinguished mathematician, the dozen absorbing essays in this versatile volume offer both supplementary classroom material and pleasurable reading for the mathematically inclined. The essays promise to encourage readers in the further study of elementary geometry, not just for its own sake, but also for its broader applications, which receive a full and engaging treatment. Beginning with an analytic approach, the author reviews the functions of Schlafli and Lobatschefsky and discusses number theory in a dissertation on integral Cayley numbers. A detailed examination of group theory includes discussion of Wythoff's construction for uniform polytopes, as well as a chapter on regular skew polyhedra in three and four dimensions and their topological analogues. A profile of self-dual configurations and regular graphs introduces elements of graph theory, followed up with a chapter on twelve points in PG (5, 3) with 95040 self-transformations. Discussion of an upper bound for the number of equal nonoverlapping spheres that can touch another same-sized sphere develops aspects of communication theory, while relativity theory is explored in a chapter on reflected light signals. Additional topics include the classification of zonohedra by means of projective diagrams, arrangements of equal spheres in non-Euclidean spaces, and regular honeycombs in hyperbolic space. 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 can play in a wide range of mathematical applications.
Below is a sampling of books from the University of Toronto Library’s collection written by or about Coxeter, as well as the proceedings form International Congress on M.C. Escher held in Rome in 1985.
The Fifty-Nine Icosahedra by ISBN: 1899618325Publication Date: 1997-08-01For this new edition, the plans and illustrations of all 59 icosahedra have been redrawn and there is a new introduction by Professor Coxeter. For an understanding of the process of stellation, this book should be a useful addition to any mathematics library.
Mathematical Recreations and Essays by ISBN: 0486253570Publication Date: 2010-05-06This classic work offers scores of stimulating, mind-expanding games and puzzles: arithmetical and geometrical problems, chessboard recreations, magic squares, map-coloring problems, cryptography and cryptanalysis, much more. "A must to add to your mathematics library" #151; The Mathematics Teacher. Index. References for Further Study. Includes 150 black-and-white line illustrations.
The Coxeter Legacy by ISBN: 0821837222Publication Date: 2006-02-28Donald Coxeter infused enthusiasm, even passion, for mathematics in people of any age, any background, any profession, any walk of life. Enchanted by Euclidean geometry, he was interested in the beauty, the description, and the exploration of the world around us. His involvement in art and with artists earned him admiration and friends in the intellectual community all over the globe. Coxeter's devotion to polytopes and his interest in the theory of configurations live on in his students and followers. Coxeter groups arise in various subjects in applied mathematics, and they have a permanent place in some of the most demanding and fascinating branches of abstract mathematics, such as Lie algebras, algebraic groups, Chevalley groups, and Kac-Moody groups. This collection of articles by outstanding researchers and expositors is intended to capture the essence of the Coxeter legacy. It is a mixture of surveys, up-to-date information, history, storytelling, and personal memories; and it includes a rich variety of beautiful illustrations.- M. C. Escher byISBN: 0444700110Publication Date: 1986-12-01The work of Dutch artist Maurits Cornelis Escher (1898-1972) continues to attract wide interest. Mathematicians, physicists, crystallographers, chemists, biologists, psychologists, psychiatrists, art historians, and specialists in computer graphics and visual communications attended this congress. The papers presented here in this illustrated volume confirm that Escher's works are not only good examples of the visualization of scientific problems but also stimulate real scientific research.
MC Escher, Images of Mathematics...
Escher's work covered a variety of subjects throughout his life. His early love of portraits, Roman and Italian landscapes and of nature, eventually gave way to regular division of the plane. Over 150 colorful and recognizable works testify to Escher's ingenuity and vision. He managed to capture the notion of hyperbolic space on a fixed 2-dimensional plane as well as translating the principles of regular division onto a number of 3-dimensional objects such as spheres, columns and cubes. A number of his prints combine both 2 and 3-dimensional images with startling effect.. His art continues to amaze and wonder millions of people all over the world. In his work we recognize his keen observation of the world around us and the expressions of his own fantasies. . M.C. Escher shows us that reality is wondrous, comprehensible and fascinating....