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 QUASI-REGULAR SOLIDS |
|
2·1 |
Regular polyhedra |
|
2·2 |
Reciprocation |
|
2·3 |
Quasi-regular 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 quasi-regular 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 quasi-regular 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. |
STAR-POLYHEDRA |
|
6·1 |
Star-polygons |
|
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 k-chains |
|
9·4 |
Linear dependence and rank |
|
9·5 |
The k-circuits |
|
9·6 |
The bounding k-circuits |
|
9·7 |
The condition for simple-connectivity |
|
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 non-positive 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. |
STAR-POLYTOPES |
|
14·1 |
The notion of a star-polytope |
|
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 star-polytopes 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 four-dimensional polytopes in parallel solid sections |
|
|
Table VI: The derivation of four-dimensional star-polytopes 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 |
|