Description |
xiii, 321 pages : illustrations ; 22 cm |
Series |
Dover books on advanced mathematics |
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Dover books on advanced mathematics.
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Contents |
Machine derived contents note: I. Polygons And Polyhedra -- 1�Regular polygons -- 1�Polyhedra -- 1�The five Platonic Solids -- 1�Graphs and maps -- 1�"A voyage round the world" -- 1�Euler's Formula -- 1�Regular maps -- 1�Configurations -- 1�Historical remarks -- Ii. Regular And Quasi-Regular Solids -- 2�Regular polyhedra -- 2�Reciprocation -- 2�Quasi-regular polyhedra -- 2�Radii and angles -- 2�Descartes' Formula -- 2�Petrie polygons -- 2�The rhombic dodecahedron and triacontahedron -- 2�Zonohedra -- 2�Historical remarks -- Iii. Rotation Groups -- 3�Congruent transformations -- 3�Transformations in general -- 3�Groups -- 3�Symmetry opperations -- 3�The polyhedral groups -- 3�The five regular compounds -- 3�Coordinates for the vertices of the regular and quasi-regular solids -- 3�The complete enumeration of finite rotation groups -- 3�Historical remarks -- Iv. Tessellations And Honeycombs -- 4�The three regular tessellations -- 4�The quasi-regular and rhombic tessellations -- 4�Rotation groups in two dimensions -- 4�Coordinates for the vertices -- 4�Lines of symmetry -- 4�Space filled with cubes -- 4�Other honeycombs -- 4�Proportional numbers of elements -- 4�Historical remarks -- V. The Kaleidoscope -- 5�"Reflections in one or two planes, or lines, or points" -- 5�Reflections in three or four lines -- 5�The fundamental region and generating relations -- 5�Reflections in three concurrent planes -- 5�"Reflections in four, five, or six planes" -- 5�Representation by graphs -- 5�Wythoff's construction -- 5�Pappus's observation concerning reciprocal regular polyhedra -- 5�The Petrie polygon and central symmetry -- 5穢 Historical remarks -- Vi. Star-Polyhedra -- 6�Star-polygons -- 6�Stellating the Platonic solids -- 6�Faceting the Platonic solids -- 6�The general regular polyhedron -- 6�A digression on Riemann surfaces -- 6�Ismorphism -- 6�Are there only nine regular polyhedra? -- 6�Scwarz's triangles -- 6�Historical remarks -- Vii. Ordinary Polytopes In Higher Space -- 7�Dimensional analogy -- 7�"Pyramids, dipyramids, and prisms" -- 7�The general sphere -- 7�Polytopes and honeycombs -- 7�Regularity -- 7�The symmetry group of the general regular polytope -- 7�Scha;fli's criterion -- 7�The enumeration of possible regular figures -- 7�The characteristic simplex -- 7� Historical remarks -- Viii. Truncation -- 8�The simple truncations of the genral regular polytope -- 8�"Cesàro's construction for {3, 4, 3}" -- 8�Coherent indexing -- 8�"The snub {3, 4, 3}" -- 8�"Gosset's construction for {3, 3, 5}" -- 8�"Partial truncation, or alternation" -- 8�Cartesian coordinates -- 8�Metrical properties -- 8�Historical remarks -- Ix. PoincarÉ'S Proof Of Euler'S Formula -- 9�Euler's Formula as generalized by Schla;fli -- 9�Incidence matrices -- 9�The algebra of k-chains -- 9�Linear dependence and rank -- 9�The k-circuits -- 9�The bounding k-circuits -- 9�The condition for simple-connectivity -- 9�The analogous formula for a honeycomb -- 9�Polytopes which do not satisfy Euler's Formula -- X. "Forms, Vectors, And Coordinates" -- 10�Real quadratic forms -- 10�Forms with non-positive product terms -- 10�A criterion for semidefiniteness -- 10�Covariant and contravariant bases for a vector space -- 10�Affine coordinates and reciprocal lattices -- 10�The general reflection -- 10�Normal coordinates -- 10�The simplex determined by n + 1 dependent vectors -- 10�Historical remarks -- Xi. The Generalized Kaleidoscope -- 11�Discrete groups generated by reflectins -- 11�Proof |
Notes |
Unabridged and corrected republication of the second edition published by Macmillan in 1963 |
Bibliography |
Bibliography: pages 306-314 |
Subject |
Polytopes.
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LC no. |
73084364 |
ISBN |
0486614808 |
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