## 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 well-known authority on them. Professor Coxeter begins with the fundamental concepts of plane and solid geometry and then moves on to multi-dimensionality. Among the many subjects covered are Euler's formula, rotation groups, star-polyhedra, truncation, forms, vectors, coordinates, kaleidoscopes, Petrie polygons, sections and projections, and star-polytopes. 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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### Contents

FIG 2 | |

ROTATION GROUPS | |

FIG 3 | |

FIG 7 | |

FIG 7 | |

TRUNCATION | |

FIG 8 | |

FIG 8 | |

POINCARÉS PROOF OF EULERS FORMULA | |

FIG 10 | |

THE GENERALIZED KALEIDOSCOPE | |

TESSELLATIONS AND HONEYCOMBS | |

FIG 4 | |

FIG 4 | |

THE KALEIDOSCOPE | |

FIG 5 | |

FIG 5 | |

STARPOLYHEDRA | |

FIG 6 | |

FIG 6 | |

ORDINARY POLYTOPES IN HIGHER SPACE | |

FIG 11 | |

THE GENERALIZED PETRIE POLYGON | |

FIG 12 | |

SECTIONS AND PROJECTIONS | |

STARPOLYTOPES EPILOGUE | |

BIBLIOGRAPHY | |

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### Common terms and phrases

aform alternate analogous belong bounding hyperplanes cells centre characteristic simplex compound congruent transformation consists convex coordinates corresponding Coxeter cube defined denote diagonals dihedral angle dimensional direct dodecahedron edges elements equal equation equatorial polygon equilateral Euler’s Formula eutactic star faces finite groups follows four dimensions fourdimensional fundamental region geometry Gosset graph group of order halfturn Hence hyperplanes icosahedron infinite inscribed integers invariant inversion isomorphic lattice lines of symmetry Mathematical measure polytope midpoints ndimensional nodes number of dimensions obtain occur octahedron opposite vertices orthogonal orthoscheme parallel pentagonal permutations perpendicular Petrie polygon plane Platonic solids polyhedra projection pyramids quasiregular reciprocal regular polyhedron regular polytope rhombic rhombs rightangled rotation group Schläfli symbol Schoute sections segments sides space sphere starpolytopes stella octangula stellated subgroup surface symmetry group symmetry operation tetrahedron theorem translation triacontahedron triangles truncation values vectors vertex figure zonohedra zonohedron