Regular PolytopesPolytopes are geometrical figures bounded by portions of lines, planes, or hyperplanes. In plane (two dimensional) geometry, they are known as polygons and comprise such figures as triangles, squares, pentagons, etc. In solid (three dimensional) geometry they are known as polyhedra and include such figures as tetrahedra (a type of pyramid), cubes, icosahedra, and many more; the possibilities, in fact, are infinite! H. S. M. Coxeter's book is the foremost book available on regular polyhedra, incorporating not only the ancient Greek work on the subject, but also the vast amount of information that has been accumulated on them since, especially in the last hundred years. The author, professor of Mathematics, University of Toronto, has contributed much valuable work himself on polytopes and is a wellknown authority on them. Professor Coxeter begins with the fundamental concepts of plane and solid geometry and then moves on to multidimensionality. Among the many subjects covered are Euler's formula, rotation groups, starpolyhedra, truncation, forms, vectors, coordinates, kaleidoscopes, Petrie polygons, sections and projections, and starpolytopes. Each chapter ends with a historical summary showing when and how the information contained therein was discovered. Numerous figures and examples and the author's lucid explanations also help to make the text readily comprehensible. Although the study of polytopes does have some practical applications to mineralogy, architecture, linear programming, and other areas, most people enjoy contemplating these figures simply because their symmetrical shapes have an aesthetic appeal. But whatever the reasons, anyone with an elementary knowledge of geometry and trigonometry will find this one of the best source books available on this fascinating study. 
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Fantastic book.
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Definitely the bible of geometry written by the man who popularized geometry which was looked down on by the modern cabal of mathematical luminaries who felt that even a diagram makes a paper unclean. Now geometry is central to physics, cosmology and many other sciences. Regular Polytopes is a musthave reference.
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algebra alternate analogous bounding hyperplanes cells centre Chapter circumradius compound congruent transformation convex coordinates corresponding cos2 Coxeter cube cuboctahedron defined denote diagonals dihedral angle direct dodecahedron edges elements equal equation equatorial polygon equilateral Euler's Formula eutactic star faces finite groups follows four dimensions fourdimensional fundamental region geometry graph group of order halfturn hedron Hence honeycomb hyperplanes icosahedron infinite inscribed integers invariant inversion ir/p isomorphic lattice lines of symmetry Mathematical measure polytope midpoints ndimensional nodes obtain occur octahedron opposite vertices orthogonal pgon parallel pentagonal permutations perpendicular Petrie polygon plane Platonic solids poly polyhedra projection pyramids reciprocal reflections regular polyhedron regular polytope rhombic rhombs rotation group Schlafli symbol Schoute sections segments sides space sphere starpolytopes stella octangula stellated subgroup surface symmetry group Table tetrahedron theorem theory tion translation triangles truncation values vectors vertex figure zonohedra zonohedron
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Page xiv  And the city lieth foursquare, and the length is as large as the breadth: and he measured the city with the reed, twelve thousand furlongs. The length, and the breadth, and the height of it are equal.