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 
