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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.
I.

POLYGONS AND POLYHEDRA  
1·1  Regular polygons  
1·2  Polyhedra  
1·3  The five Platonic Solids  
1·4  Graphs and maps  
1·5  "A voyage round the world"  
1·6  Euler's Formula  
1·7  Regular maps  
1·8  Configurations  
1·9  Historical remarks  
II.  REGULAR AND QUASIREGULAR SOLIDS  
2·1  Regular polyhedra  
2·2  Reciprocation  
2·3  Quasiregular polyhedra  
2·4  Radii and angles  
2·5  Descartes' Formula  
2·6  Petrie polygons  
2·7  The rhombic dodecahedron and triacontahedron  
2·8  Zonohedra  
2·9  Historical remarks  
III.  ROTATION GROUPS  
3·1  Congruent transformations  
3·2  Transformations in general  
3·3  Groups  
3·4  Symmetry opperations  
3·5  The polyhedral groups  
3·6  The five regular compounds  
3·7  Coordinates for the vertices of the regular and quasiregular solids  
3·8  The complete enumeration of finite rotation groups  
3·9  Historical remarks  
IV.  TESSELLATIONS AND HONEYCOMBS  
4·1  The three regular tessellations  
4·2  The quasiregular and rhombic tessellations  
4·3  Rotation groups in two dimensions  
4·4  Coordinates for the vertices  
4·5  Lines of symmetry  
4·6  Space filled with cubes  
4·7  Other honeycombs  
4·8  Proportional numbers of elements  
4·9  Historical remarks  
V.  THE KALEIDOSCOPE  
5·1  "Reflections in one or two planes, or lines, or points"  
5·2  Reflections in three or four lines  
5·3  The fundamental region and generating relations  
5·4  Reflections in three concurrent planes  
5·5  "Reflections in four, five, or six planes"  
5·6  Representation by graphs  
5·7  Wythoff's construction  
5·8  Pappus's observation concerning reciprocal regular polyhedra  
5·9  The Petrie polygon and central symmetry  
5·x  Historical remarks  
VI.  STARPOLYHEDRA  
6·1  Starpolygons  
6·2  Stellating the Platonic solids  
6·3  Faceting the Platonic solids  
6·4  The general regular polyhedron  
6·5  A digression on Riemann surfaces  
6·6  Ismorphism  
6·7  Are there only nine regular polyhedra?  
6·8  Scwarz's triangles  
6·9  Historical remarks  
VII.  ORDINARY POLYTOPES IN HIGHER SPACE  
7·1  Dimensional analogy  
7·2  "Pyramids, dipyramids, and prisms"  
7·3  The general sphere  
7·4  Polytopes and honeycombs  
7·5  Regularity  
7·6  The symmetry group of the general regular polytope  
7·7  Schäfli's criterion  
7·8  The enumeration of possible regular figures  
7·9  The characteristic simplex  
7·10  Historical remarks  
VIII.  TRUNCATION  
8·1  The simple truncations of the genral regular polytope  
8·2  "Cesàro's construction for 3, 4, 3"  
8·3  Coherent indexing  
8·4  "The snub 3, 4, 3"  
8·5  "Gosset's construction for 3, 3, 5"  
8·6  "Partial truncation, or alternation"  
8·7  Cartesian coordinates  
8·8  Metrical properties  
8·9  Historical remarks  
IX.  POINCARÉ'S PROOF OF EULER'S FORMULA  
9·1  Euler's Formula as generalized by Schläfli  
9·2  Incidence matrices  
9·3  The algebra of kchains  
9·4  Linear dependence and rank  
9·5  The kcircuits  
9·6  The bounding kcircuits  
9·7  The condition for simpleconnectivity  
9·8  The analogous formula for a honeycomb  
9·9  Polytopes which do not satisfy Euler's Formula  
X.  "FORMS, VECTORS, AND COORDINATES"  
10·1  Real quadratic forms  
10·2  Forms with nonpositive product terms  
10·3  A criterion for semidefiniteness  
10·4  Covariant and contravariant bases for a vector space  
10·5  Affine coordinates and reciprocal lattices  
10·6  The general reflection  
10·7  Normal coordinates  
10·8  The simplex determined by n + 1 dependent vectors  
10·9  Historical remarks  
XI.  THE GENERALIZED KALEIDOSCOPE  
11·1  Discrete groups generated by reflectins  
11·2  Proof that the fundamental region is a simplex  
11·3  Representation by graphs  
11·4  "Semidefinite forms, Euclidean simplexes, and infinite groups"  
11·5  "Definite forms, spherical simplexes, and finite groups"  
11·6  Wythoff's construction  
11·7  Regular figures and their truncations  
11·8  "Gosset's figures in six, seven, and eight dimensions"  
11·9  Weyl's formula for the order of the largest finite subgroup of an infinite discrete group generated by reflections  
11·x  Historical remarks  
XII.  THE GENERALIZED PETRIE POLYGON  
12·1  Orthogonal transformations  
12·2  Congruent transformations  
12·3  The product of n reflections  
12·4  "The Petrie polygon of p, q, . . . , w"  
12·5  The central inversion  
12·6  The number of reflections  
12·7  A necklace of tetrahedral beads  
12·8  A rational expression for h/g in four dimensions  
12·9  Historical remarks  
XIII.  SECTIONS AND PROJECTIONS  
13·1  The principal sections of the regular polytopes  
13·2  Orthogonal projection onto a hyperplane  
13·3  "Plane projections an,ßn,?n"  
13·4  New coordinates for an and ßn  
13·5  "The dodecagonal projection of 3, 4, 3"  
13·6  "The triacontagonal projection of 3, 3, 5"  
13·7  Eutactic stars  
13·8  Shadows of measure polytopes  
13·9  Historical remarks  
XIV.  STARPOLYTOPES  
14·1  The notion of a starpolytope  
14·2  "Stellating 5, 3, 3"  
14·3  Systematic faceting  
14·4  The general regular polytope in four dimensions  
14·5  A trigonometrical lemma  
14·6  Van Oss's criterion  
14·7  The Petrie polygon criterion  
14·8  Computation of density  
14·9  Complete enumeration of regular starpolytopes and honeycombs  
14·x  Historical remarks  
Epilogue  
Definitions of symbols  
Table I: Regular polytopes  
Table II: Regular honeycombs  
Table III: Schwarz's triangles  
Table IV: Fundamental regions for irreducible groups generated by reflections  
Table V: The distribution of vertices of fourdimensional polytopes in parallel solid sections  
Table VI: The derivation of fourdimensional starpolytopes and compounds by faceting the convex regular polytopes  
Table VII: Regular compunds in four dimensions  
Table VIII: The number of regular polytopes and honeycombs  
Bibliography  
Index  