舉例說明

正多邊形

一個有n個邊的多邊形，其施萊夫利符號為。例如，施萊夫利符號為的多邊形即為五邊形。

星形多邊形

星形多邊形(Star polygon)指的是正非凸多邊形，即邊長相等的凹多邊形或復雜多邊形。星形多邊形的施萊夫利符號若為

星芒形

當p和q不互質(zhì)時，此時的星形多邊形即稱為星芒形(star figure)。若p跟q的最大公因子為n，此一星芒形即是由n個

正多面體

正多面體的施萊夫利符號計做{p,q}，其中p代表每個面的頂點數(shù)，，而q代表每個頂點和幾個面相連

各種正多面體的施萊夫利符號如下:

正四面體 : {3,3}

立方體 : {4,3}

正八面體 : {3,4}

正十二面體 : {5,3}

正二十面體 : {3,5}

Schl?fli symbols may also be defined for regular tessellations of Euclidean or hyperbolic space in a similar way.

In addition to the 5 convex regular polyhedra, there are four other nonconvex ones formed with star polygon faces or vertex figures.

The Schl?fli symbols of the four Kepler-Poinsot solids are: (nonconvex regular polyhedra)

小星形十二面體 : {5/2,5}

大十二面體 : {5,5/2}

大星形十二面體 : {5/2,3}

大二十面體 ; {3,5/2}

Three regular tilings of the Euclidean plane:

Hexagonal tiling : {6,3}

Square tiling : {4,4}

Triangular tiling : {3,6}

Regular polychora (4-space)

The Schl?fli symbol of a regular polychoron is of the form {p,q,r}. It has regular polygonal faces, {p,q} cells, {q,r} regular polyhedral vertex figures, and regular polygonal edge figures.

There are six convex regular and 10 nonconvex polychora. There is also one regular tesselation of Euclidean 3-space: the cubic honeycomb.

The smallest convex polychoron is {3,3,3}, the pentachoron, and the largest is {5,3,3}, the 120-cell.

All 10 nonconvex regular polychora are combinations of , {5/2}, sides.

Higher dimensions

For higher dimensional polytopes, the Schl?fli symbol is defined recursively as {p1, p2, ..., pn ? 1} if the facets have Schl?fli symbol {p1,p2, ..., pn ? 2} and the vertex figures have Schl?fli symbol {p2,p3, ..., pn ? 1}.

There are only 3 regular polytopes in 5 dimensions and above: the simplex, {3,3,3,...,3}; the cross-polytope, {3,3, ... ,3,4}; and the measure polytope, {4,3,3,...,3}. There are no non-convex regular polytopes above 4 dimensions.

Dual polytopes

For dimension 2 or higher, every polytope has a dual.

If a polytope has Schl?fli symbol {p1,p2, ..., pn ? 1} then its dual has Schl?fli symbol {pn ? 1, ..., p2,p1}.

If the sequence is the same forwards and backwards, the polytope is self-dual. Every regular polytope in 2 dimensions (polygon) is self-dual.

Extended Schl?fli symbols for polyhedra and tilings

For polyhedra, one extended Schl?fli symbol is used in the 1954 paper by Coxeter enumerating the paper tiled uniform polyhedra.

Every regular polyhedron or tiling {p,q} has these five operations that create semiregular polyhedra. The short-hand notation is equivalent to the longer name. For instance, t{3,3} simply means truncated tetrahedron.

The vertical notation is used for dual-symmetric operations - those that generate the same polyhedron from {p,q} as {q,p}.

A second extended notation, also used by Coxeter applies to all dimensions, and are specified by a t followed by a list of indices corresponding to Wythoff construction mirrors. (They also correspond to ringed nodes in a Coxeter-Dynkins diagram.)

In each a Wythoff construction operational name is given first. Second some have alternate terminology (given in parentheses) apply only for a given dimension. Specifically omnitruncation and expansion, as well as dual relations apply differently in each dimension.

Operation Extended

Schl?fli

Symbols Coxeter-

Dynkins

Diagram Wythoff

symbol Face

(2) Face

(1) Face

(0)

Parent??t0{p,q} Image:Dynkins-100.png q | 2 p --

Rectified??t1{p,q} Image:Dynkins-010.png 2 | p q -- t

Birectified

(or dual)??t2{p,q} Image:Dynkins-001.png p | 2 q -- --

Truncated??t0,1{p,q} Image:Dynkins-110.png 2 q | p t --

Bitruncated

(or truncated dual)??t2,3{p,q} Image:Dynkins-011.png 2 p | q --

Cantellated

(or expanded)??t0,2{p,q} Image:Dynkins-101.png p q | 2

Cantitruncated

(or omnitruncated)??t0,1,2{p,q} Image:Dynkins-111.png 2 p q | t t

Snub??s{p,q} Image:Dynkins-sss.png | 2 p q

Image:Polyhedron truncation example3.png Image:Polyhedron kaleidoscopeTriAngles4.png

Generating triangles

Prisms

A p-gonal prism, with vertex figure p.4.4, can represented by either??or .

Extended for uniform polychora and 3-space honeycombs

Image:Polychoron tetrahedral domain.png

Example tetrahedron in cubic honeycomb cell.

There are 3 right dihedral angles (2 intersecting perpendicular mirrors):

Edges 1 to 2, 0 to 2, and 1 to 3.

Summary chart of truncation operationsEvery regular polytope can be seen as the images of a fundamental region in a small number of mirrors. In a 4-dimensional polytope (or 3-dimensional cubic honeycomb) the fundamental region is bounded by four mirrors. A mirror in 4-space is a three-dimensional hyperplane, but it is more convenient for our purposes to consider only its two-dimensional intersection with the three-dimensional surface of the hypersphere; thus the mirrors form an irregular tetrahedron.

Each of the sixteen regular polychora is generated by one of four symmetry groups, as follows:

group [3,3,3]: the 5-cell {3,3,3}, which is self-dual;

group [3,3,4]: 16-cell {3,3,4} and its dual tesseract {4,3,3};

group [3,4,3]: the 24-cell {3,4,3}, self-dual;

group [3,3,5]: 600-cell {3,3,5}, its dual 120-cell {5,3,3}, and their ten regular stellations.

(The groups are named in Coxeter notation.)

A set of up to 13 (nonregular) uniform polychora can be generated from each regular polychoron and its dual. Eight of the Andreini tessellations (uniform honeycombs in Euclidean 3-space) are analogously generated from the cubic honeycomb {4,3,4}.

For a given symmetry simplex, a generating point may be placed on any of the four vertices, 6 edges, 4 faces, or the interior volume. On each of these 15 elements there is a point whose images, reflected in the four mirrors, are the vertices of a uniform polychoron.

The extended Schl?fli symbols are made by a t followed by inclusion of one to four subscripts 0,1,2,3. If there's one subscript, the generating point is on a corner of the fundamental region, i.e. a point where three mirrors meet. These corners are notated as

0: vertex of the parent polychoron (center of the dual's cell)

1: center of the parent's edge (center of the dual's face)

2: center of the parent's face (center of the dual's edge)

3: center of the parent's cell (vertex of the dual)

(For the two self-dual polychora, "dual" means a similar polychoron in dual position.) Two or more subscripts mean that the generating point is between the corners indicated.

The following table defines all 15 forms. Each trunction form can have from one to four cell types, located in positions 0,1,2,3 as defined above. The cells are labeled by polyhedral truncation notation.

An n-gonal prism is represented as :

The green background is shown on forms that are equivalent from either the parent or dual.

The red background shows truncations of the parent, and blue as truncations of the dual.

Operation Extended

Schl?fli

symbols Coxeter-

Dynkins

Diagram Cell

(3) Cell

(2) Cell

(1) Cell

(0)

Parent t0{p,q,r} Image:Dynkins-1000.png??-- -- --

Rectified t1{p,q,r} Image:Dynkins-0100.png??-- --

Birectified

(or rectified dual) t2{p,q,r} Image:Dynkins-0010.png??-- --

Trirectifed

(or dual) t3{p,q,r} Image:Dynkins-0001.png -- -- --

Truncated t0,1{p,q,r} Image:Dynkins-1100.png??-- --

Bitruncated t1,2{p,q,r} Image:Dynkins-0110.png??-- --

Tritruncated

(or truncated dual) t2,3{p,q,r} Image:Dynkins-0011.png??-- --

Cantellated t0,2{p,q,r} Image:Dynkins-1010.png??--

Bicantellated

(or cantellated dual) t1,3{p,q,r} Image:Dynkins-0101.png? ?--

Runcinated

(or expanded) t0,3{p,q,r} Image:Dynkins-1001.png

Cantitruncated t0,1,2{p,q,r} Image:Dynkins-1110.png??--

Bicantitruncated

(or cantitruncated dual) t1,2,3{p,q,r} Image:Dynkins-0111.png? ?--

Runcitruncated t0,1,3{p,q,r} Image:Dynkins-1101.png

Runcicantellated

(or runcitruncated dual) t0,2,3{p,q,r} Image:Dynkins-1011.png

Runcicantitruncated

(or omnitruncated) t0,1,2,3{p,q,r} Image:Dynkins-1111.png

References

Coxeter, Longuet-Higgins, Miller, Uniform polyhedra, Phil. Trans. 1954, 246 A, 401-50. (Extended Schl?fli notation used)