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Longer titles found: Coxeter–Dynkin diagram (view)

searching for Dynkin diagram 133 found (297 total)

Uniform 2 k1 polytope (410 words) [view diff] exact match in snippet view article find links to article

was named by their Coxeter symbol as 2k1 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence. It can be named

Uniform 1 k2 polytope (410 words) [view diff] exact match in snippet view article find links to article

family was named by their Coxeter symbol 1k2 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. It can be named

Uniform 10-polytope (822 words) [view diff] exact match in snippet view article find links to article

polytopes, generated by marking one or more nodes of the D10 Coxeter-Dynkin diagram. Of these, 511 (2×256−1) are repeated from the B10 family and 256 are

Great icosahedron (490 words) [view diff] exact match in snippet view article find links to article

(nonconvex regular polyhedra), with Schläfli symbol {3,5⁄2} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five

Great truncated icosidodecahedron (407 words) [view diff] exact match in snippet view article find links to article

vertices. It is given a Schläfli symbol t0,1,2{5⁄3,3}, and Coxeter-Dynkin diagram, . Cartesian coordinates for the vertices of a great truncated icosidodecahedron

Stellated truncated hexahedron (141 words) [view diff] exact match in snippet view article find links to article

is represented by Schläfli symbol t'{4,3} or t{4/3,3}, and Coxeter-Dynkin diagram, . It is sometimes called quasitruncated hexahedron because it is related

Runcinated 6-cubes (286 words) [view diff] exact match in snippet view article find links to article

6-cube Type Uniform 6-polytope Schläfli symbol t0,3{4,3,3,3,3} Coxeter-Dynkin diagram 4-faces Cells Faces Edges 7680 Vertices 1280 Vertex figure Coxeter group

One-dimensional symmetry group (1,839 words) [view diff] exact match in snippet view article find links to article

group [∞], or Coxeter-Dynkin diagram representing two reflections, and the translational symmetry as [∞]+, or Coxeter-Dynkin diagram as the composite of

Nonconvex great rhombicuboctahedron (236 words) [view diff] exact match in snippet view article find links to article

vertices. It is represented by the Schläfli symbol rr{4,3⁄2} and Coxeter-Dynkin diagram of . Its vertex figure is a crossed quadrilateral. This model shares

Great truncated cuboctahedron (171 words) [view diff] exact match in snippet view article find links to article

vertices. It is represented by the Schläfli symbol tr{4/3,3}, and Coxeter-Dynkin diagram . It is sometimes called the quasitruncated cuboctahedron because it

Uniform 6-polytope (1,491 words) [view diff] exact match in snippet view article find links to article

32+4−1=35 forms, derived by marking one or more nodes of the Coxeter-Dynkin diagram. All 35 are enumerated below. They are named by Norman Johnson from

Pentic 6-cubes (662 words) [view diff] exact match in snippet view article find links to article

Type uniform 6-polytope Schläfli symbol t0,4{3,34,1} h5{4,34} Coxeter-Dynkin diagram = 5-faces 4-faces Cells Faces Edges 1440 Vertices 192 Vertex figure

Uniform 9-polytope (894 words) [view diff] exact match in snippet view article find links to article

polytopes, generated by marking one or more nodes of the D9 Coxeter-Dynkin diagram. Of these, 255 (2×128−1) are repeated from the B9 family and 128 are

Stericated 5-simplexes (1,044 words) [view diff] exact match in snippet view article find links to article

omnitruncated 5-simplex facets with 3 facets around each ridge. It has Coxeter-Dynkin diagram of . The full snub 5-simplex or omnisnub 5-simplex, defined as an alternation

Great snub icosidodecahedron (872 words) [view diff] exact match in snippet view article find links to article

It can be represented by a Schläfli symbol sr{5⁄2,3}, and Coxeter-Dynkin diagram . This polyhedron is the snub member of a family that includes the great

Truncation (geometry) (1,045 words) [view diff] exact match in snippet view article

} {\displaystyle {\begin{Bmatrix}5\\3\end{Bmatrix}}} , and Coxeter-Dynkin diagram or has a uniform truncation, the truncated icosidodecahedron, represented

Cantellated 8-simplexes (457 words) [view diff] exact match in snippet view article find links to article

8-simplex Type uniform 8-polytope Schläfli symbol rr{3,3,3,3,3,3,3} Coxeter-Dynkin diagram 7-faces 6-faces 5-faces 4-faces Cells Faces Edges 1764 Vertices 252

Uniform 7-polytope (1,015 words) [view diff] exact match in snippet view article find links to article

polytopes, generated by marking one or more nodes of the D7 Coxeter-Dynkin diagram. Of these, 63 (2×32−1) are repeated from the B7 family and 32 are unique

Stericated 5-cubes (937 words) [view diff] exact match in snippet view article find links to article

5-cube Type Uniform 5-polytope Schläfli symbol 2r2r{4,3,3,3} Coxeter-Dynkin diagram 4-faces 242 Cells 800 Faces 1040 Edges 640 Vertices 160 Vertex figure

Great inverted snub icosidodecahedron (811 words) [view diff] exact match in snippet view article find links to article

indexed as U69. It is given a Schläfli symbol sr{5⁄3,3}, and Coxeter-Dynkin diagram . In the book Polyhedron Models by Magnus Wenninger, the polyhedron

Mitchell's group (189 words) [view diff] exact match in snippet view article find links to article

18, 24, 30, 42. Coxeter gives it group symbol [1 2 3]3 and Coxeter-Dynkin diagram . Mitchell's group is an index 2 subgroup of the automorphism group

Truncated 6-simplexes (428 words) [view diff] exact match in snippet view article find links to article

doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram. The truncated 6-simplex is one of 35 uniform 6-polytopes based on the

Cantellated 5-simplexes (455 words) [view diff] exact match in snippet view article find links to article

{\displaystyle r\left\{{\begin{array}{l}3,3,3\\3\end{array}}\right\}} Coxeter-Dynkin diagram or 4-faces 27 6 r{3,3,3} 6 rr{3,3,3} 15 {}x{3,3} Cells 135 30 {3,3}

Runcinated 5-simplexes (409 words) [view diff] exact match in snippet view article find links to article

5-simplex Type Uniform 5-polytope Schläfli symbol t0,3{3,3,3,3} Coxeter-Dynkin diagram 4-faces 47 6 t0,3{3,3,3} 20 {3}×{3} 15 { }×r{3,3} 6 r{3,3,3} Cells 255

Icosahedral bipyramid (134 words) [view diff] exact match in snippet view article find links to article

together at their bases. It is the dual of a dodecahedral prism, Coxeter-Dynkin diagram , so the bipyramid can be described as . Both have Coxeter notation

Tetrahedral bipyramid (198 words) [view diff] exact match in snippet view article find links to article

the dual of a tetrahedral prism, , so it can also be given a Coxeter-Dynkin diagram, , and both have Coxeter notation symmetry [2,3,3], order 48. Being

Runcinated 5-orthoplexes (431 words) [view diff] exact match in snippet view article find links to article

5-orthoplex Type Uniform 5-polytope Schläfli symbol t0,3{3,3,3,4} Coxeter-Dynkin diagram 4-faces 162 Cells 1200 Faces 2160 Edges 1440 Vertices 320 Vertex figure

Minuscule representation (297 words) [view diff] exact match in snippet view article find links to article

multiplicity of the zero weight is the number of short nodes of the Dynkin diagram. (The highest weight of that quasi-minuscule representation is the highest

Uniform 8-polytope (1,007 words) [view diff] exact match in snippet view article find links to article

Wythoffian uniform polytopes, from 3x64-1 permutations of the D8 Coxeter-Dynkin diagram with one or more rings. 127 (2x64-1) are repeated from the B8 family

7-demicubic honeycomb (395 words) [view diff] exact match in snippet view article find links to article

symbol h{4,3,3,3,3,3,4} h{4,3,3,3,3,31,1} ht0,7{4,3,3,3,3,3,4} Coxeter-Dynkin diagram = = Facets {3,3,3,3,3,4} h{4,3,3,3,3,3} Vertex figure Rectified 7-orthoplex

Quasiregular polyhedron (1,484 words) [view diff] exact match in snippet view article find links to article

Coxeter-Dynkin diagram The icosidodecahedron { 3 5 } {\displaystyle {\begin{Bmatrix}3\\5\end{Bmatrix}}} , vertex configuration (3.5)2, Coxeter-Dynkin diagram

7-simplex (655 words) [view diff] exact match in snippet view article find links to article

This polytope is a facet in the uniform tessellation 331 with Coxeter-Dynkin diagram: This polytope is one of 71 uniform 7-polytopes with A7 symmetry. Klitzing

Rectified 7-simplexes (513 words) [view diff] exact match in snippet view article find links to article

figure of the 251 honeycomb. It is called 05,1 for its branching Coxeter-Dynkin diagram, shown as . E. L. Elte identified it in 1912 as a semiregular polytope

3-4 duoprism (293 words) [view diff] exact match in snippet view article find links to article

diagrams Type Prismatic uniform polychoron Schläfli symbol {3}×{4} Coxeter-Dynkin diagram Cells 3 square prisms, 4 triangular prisms Faces 3+12 squares, 4 triangles

Pentellated 6-simplexes (1,136 words) [view diff] exact match in snippet view article find links to article

doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram. This configuration matrix represents the expanded 6-simplex, with 12

Rectified 8-simplexes (456 words) [view diff] exact match in snippet view article find links to article

labeling it as S1 8. It is also called 06,1 for its branching Coxeter-Dynkin diagram, shown as . The Cartesian coordinates of the vertices of the rectified

Truncated 5-simplexes (389 words) [view diff] exact match in snippet view article find links to article

5-simplex Type Uniform 5-polytope Schläfli symbol t{3,3,3,3} Coxeter-Dynkin diagram 4-faces 12 6 {3,3,3} 6 t{3,3,3} Cells 45 30 {3,3} 15 t{3,3} Faces 80

Rectified 6-simplexes (385 words) [view diff] exact match in snippet view article find links to article

labeling it as S1 6. It is also called 04,1 for its branching Coxeter-Dynkin diagram, shown as . Rectified heptapeton (Acronym: ril) (Jonathan Bowers) The

Uniform tiling (1,971 words) [view diff] exact match in snippet view article find links to article

Schläfli symbol for a right triangle domain: (p q 2) → {p, q}. The Coxeter-Dynkin diagram is a triangular graph with p, q, r labeled on the edges. If r = 2, then

Runcic 7-cubes (335 words) [view diff] exact match in snippet view article find links to article

Type uniform 7-polytope Schläfli symbol t0,2{3,34,1} h3{4,35} Coxeter-Dynkin diagram 5-faces 4-faces Cells Faces Edges 16800 Vertices 2240 Vertex figure

Cantellated 5-cubes (436 words) [view diff] exact match in snippet view article find links to article

{\displaystyle r\left\{{\begin{array}{l}4\\3,3,3\end{array}}\right\}} Coxeter-Dynkin diagram = 4-faces 122 10 80 32 Cells 680 40 320 160 160 Faces 1520 80 480 320

Runcinated 5-cubes (469 words) [view diff] exact match in snippet view article find links to article

5-cube Type Uniform 5-polytope Schläfli symbol t0,2,3{4,3,3,3} Coxeter-Dynkin diagram 4-faces 202 10 80 80 32 Cells 1240 40 240 320 320 160 160 Faces 2960

8-demicube (339 words) [view diff] exact match in snippet view article find links to article

is the vertex figure for the uniform tessellation, 251 with Coxeter-Dynkin diagram: H.S.M. Coxeter: Coxeter, Regular Polytopes, (3rd edition, 1973), Dover

Truncated great dodecahedron (187 words) [view diff] exact match in snippet view article find links to article

dodecahedron Truncated small stellated dodecahedron Dodecadodecahedron Truncated great dodecahedron Great dodecahedron Coxeter-Dynkin diagram Picture

Regular polytope (5,265 words) [view diff] exact match in snippet view article find links to article

polytopes are in bijection with Coxeter groups with linear Coxeter-Dynkin diagram (without branch point) and an increasing numbering of the nodes. Reversing

Great grand 120-cell (161 words) [view diff] exact match in snippet view article find links to article

Vertices 120 Vertex figure {5/2,3} Schläfli symbol {5,5/2,3} Coxeter-Dynkin diagram Symmetry group H4, [3,3,5] Dual Great icosahedral 120-cell Properties

9-demicube (332 words) [view diff] exact match in snippet view article find links to article

161 Schläfli symbol {3,36,1} = h{4,37} s{21,1,1,1,1,1,1,1} Coxeter-Dynkin diagram = 8-faces 274 18 {31,5,1} 256 {37} 7-faces 2448 144 {31,4,1} 2304 {36}

Small stellated 120-cell (286 words) [view diff] exact match in snippet view article find links to article

Vertices 120 Vertex figure {5,3} Schläfli symbol {5/2,5,3} Coxeter-Dynkin diagram Symmetry group H4, [3,3,5] Dual Icosahedral 120-cell Properties Regular

Great icosahedral 120-cell (177 words) [view diff] exact match in snippet view article find links to article

Vertices 120 Vertex figure {5/2,5} Schläfli symbol {3,5/2,5} Coxeter-Dynkin diagram Symmetry group H4, [3,3,5] Dual Great grand 120-cell Properties Regular

Truncated great icosahedron (322 words) [view diff] exact match in snippet view article find links to article

dodecahedron Truncated great stellated dodecahedron Great icosidodecahedron Truncated great icosahedron Great icosahedron Coxeter-Dynkin diagram Picture

Great 120-cell (213 words) [view diff] exact match in snippet view article find links to article

Edges 720 Vertices 120 Vertex figure {5/2,5} Schläfli symbol {5,5/2,5} Coxeter-Dynkin diagram Symmetry group H4, [3,3,5] Dual self-dual Properties Regular

Grand stellated 120-cell (217 words) [view diff] exact match in snippet view article find links to article

720 Vertices 120 Vertex figure {5,5/2} Schläfli symbol {5/2,5,5/2} Coxeter-Dynkin diagram Symmetry group H4, [3,3,5] Dual self-dual Properties Regular

AN (487 words) [view diff] exact match in snippet view article find links to article

aneuploid means "not euploid") An, in mathematics, a root system and its Dynkin diagram An, in mathematics, conventional notation for the alternating group

Grand 120-cell (225 words) [view diff] exact match in snippet view article find links to article

Vertices 120 Vertex figure {3,5/2} Schläfli symbol {5,3,5/2} Coxeter-Dynkin diagram Symmetry group H4, [3,3,5] Dual Great stellated 120-cell Properties

Runcinated 6-simplexes (529 words) [view diff] exact match in snippet view article find links to article

doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram. Prismatotruncated heptapeton (Acronym: patal) (Jonathan Bowers) The

Expansion (geometry) (521 words) [view diff] exact match in snippet view article

truncation for polygons, e{p} = e1{p} = t0,1{p} = t{p} and has Coxeter-Dynkin diagram . A regular {p,q} polyhedron (3-polytope) expands into a polyhedron

List of finite-dimensional Nichols algebras (1,993 words) [view diff] exact match in snippet view article find links to article

several Dynkin diagrams (for inequivalent types of Weyl chambers). Each Dynkin diagram has one vertex per irreducible V i {\displaystyle V_{i}} and edges depending

Icosahedral 120-cell (284 words) [view diff] exact match in snippet view article find links to article

120 Vertex figure {5,5/2} Schläfli symbol {3,5,5/2} Symmetry group H4, [3,3,5] Coxeter-Dynkin diagram Dual Small stellated 120-cell Properties Regular

9-simplex (659 words) [view diff] exact match in snippet view article find links to article

9-polytope Family simplex Schläfli symbol {3,3,3,3,3,3,3,3} Coxeter-Dynkin diagram 8-faces 10 8-simplex 7-faces 45 7-simplex 6-faces 120 6-simplex 5-faces

Steriruncic tesseractic honeycomb (192 words) [view diff] exact match in snippet view article find links to article

image) Type Uniform honeycomb Schläfli symbol h3,4{4,3,3,4} Coxeter-Dynkin diagram = 4-face type r{4,3,4} t{4,3,4} t0,1,3{4,3,4} {3,3}×{} Cell type t{4

Bitruncated 16-cell honeycomb (194 words) [view diff] exact match in snippet view article find links to article

honeycomb Schläfli symbol t1,2{3,3,4,3} h2,3{4,3,3,4} 2t{3,31,1,1} Coxeter-Dynkin diagram = = 4-face type Truncated 24-cell Bitruncated tesseract Cell type Cube

Great stellated 120-cell (224 words) [view diff] exact match in snippet view article find links to article

720 Vertices 120 Vertex figure {3,5} Schläfli symbol {5/2,3,5} Coxeter-Dynkin diagram Symmetry group H4, [3,3,5] Dual Grand 120-cell Properties Regular

9-cube (428 words) [view diff] exact match in snippet view article find links to article

Regular 9-polytope Family hypercube Schläfli symbol {4,37} Coxeter-Dynkin diagram 8-faces 18 {4,36} 7-faces 144 {4,35} 6-faces 672 {4,34} 5-faces 2016

8-demicubic honeycomb (504 words) [view diff] exact match in snippet view article find links to article

Coxeter group Schläfli symbol Coxeter-Dynkin diagram Vertex figure Symmetry Facets/verf B ~ 8 {\displaystyle {\tilde {B}}_{8}} = [31,1,3,3,3,3,3,4] = [1+

8-demicubic honeycomb (504 words) [view diff] exact match in snippet view article find links to article

Coxeter group Schläfli symbol Coxeter-Dynkin diagram Vertex figure Symmetry Facets/verf B ~ 8 {\displaystyle {\tilde {B}}_{8}} = [31,1,3,3,3,3,3,4] = [1+

10-simplex (768 words) [view diff] exact match in snippet view article find links to article

10-polytope Family simplex Schläfli symbol {3,3,3,3,3,3,3,3,3} Coxeter-Dynkin diagram 9-faces 11 9-simplex 8-faces 55 8-simplex 7-faces 165 7-simplex 6-faces

Runcinated tesseractic honeycomb (243 words) [view diff] exact match in snippet view article find links to article

Uniform 4-honeycomb Schläfli symbol t0,3{4,3,3,4} t0,3{4,3,31,1} Coxeter-Dynkin diagram 4-face type runcinated tesseract tesseract rectified tesseract cuboctahedral

10-cube (438 words) [view diff] exact match in snippet view article find links to article

Regular 10-polytope e Family hypercube Schläfli symbol {4,38} Coxeter-Dynkin diagram 9-faces 20 {4,37} 8-faces 180 {4,36} 7-faces 960 {4,35} 6-faces 3360

Small stellated dodecahedron (743 words) [view diff] exact match in snippet view article find links to article

dodecahedron Truncated small stellated dodecahedron Dodecadodecahedron Truncated great dodecahedron Great dodecahedron Coxeter-Dynkin diagram Picture

Steric tesseractic honeycomb (189 words) [view diff] exact match in snippet view article find links to article

(No image) Type Uniform honeycomb Schläfli symbol h4{4,3,3,4} Coxeter-Dynkin diagram = 4-face type {4,3,3} t0,3{4,3,3} {3,3,4} {3,3}×{} Cell type {4,3} {3

Runcicantellated tesseractic honeycomb (265 words) [view diff] exact match in snippet view article find links to article

image) Type Uniform 4-honeycomb Schläfli symbol t0,2,3{4,3,3,4} Coxeter-Dynkin diagram 4-face type bitruncated tesseract truncated octahedral prism tesseract

Stericantic tesseractic honeycomb (192 words) [view diff] exact match in snippet view article find links to article

image) Type Uniform honeycomb Schläfli symbol h2,4{4,3,3,4} Coxeter-Dynkin diagram = 4-face type rr{4,3,3} t0,1,3{3,3,4} t{3,3,4} {3,3}×{} Cell type rr{4

Truncated tesseractic honeycomb (244 words) [view diff] exact match in snippet view article find links to article

Uniform 4-honeycomb Schläfli symbol t{4,3,3,4} t{4,3,31,1} Coxeter-Dynkin diagram 4-face type truncated tesseract 16-cell Cell type Truncated cube Tetrahedron

5-cube (826 words) [view diff] exact match in snippet view article find links to article

Description Schläfli symbol Coxeter-Dynkin diagram Vertices Edges Coxeter notation Symmetry Order 5-cube {4,3,3,3} 32 80 [4,3,3,3] 3840 tesseractic prism

Runcicantitruncated tesseractic honeycomb (265 words) [view diff] exact match in snippet view article find links to article

Type Uniform 4-honeycomb Schläfli symbol t0,1,2,3{4,3,3,4} Coxeter-Dynkin diagram 4-face type Truncated 24-cell Truncated octahedral prism 4-8 duoprism

Great stellated dodecahedron (429 words) [view diff] exact match in snippet view article find links to article

dodecahedron Truncated great stellated dodecahedron Great icosidodecahedron Truncated great icosahedron Great icosahedron Coxeter-Dynkin diagram Picture

Steriruncicantic tesseractic honeycomb (193 words) [view diff] exact match in snippet view article find links to article

image) Type Uniform honeycomb Schläfli symbol h2,3,4{4,3,3,4} Coxeter-Dynkin diagram = 4-face type t0123{4,3,3} tr{4,3,3} 2t{4,3,3} t{3,3}×{} Cell type tr{4

Wythoff symbol (682 words) [view diff] exact match in snippet view article find links to article

Wythoff symbol is functionally similar to the more general Coxeter-Dynkin diagram, in which each node represents a mirror and the arcs between them –

Cantellated tesseractic honeycomb (290 words) [view diff] exact match in snippet view article find links to article

4-honeycomb Schläfli symbol t0,2{4,3,3,4} or rr{4,3,3,4} rr{4,3,31,1} Coxeter-Dynkin diagram 4-face type t0,2{4,3,3} t1{3,3,4} {3,4}×{} Cell type Octahedron Rhombicuboctahedron

Runcitruncated tesseractic honeycomb (265 words) [view diff] exact match in snippet view article find links to article

image) Type Uniform 4-honeycomb Schläfli symbol t0,1,3{4,3,3,4} Coxeter-Dynkin diagram 4-face type t0,1,3{4,3,3} t1{3,4,3} t1{3,4}×{} 4-8 duoprism Cell type

Eight-dimensional space (718 words) [view diff] exact match in snippet view article find links to article

Coxeter group. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram. The 8-demicube is a unique polytope from the D8 family, and 421, 241

Dodecadodecahedron (599 words) [view diff] exact match in snippet view article find links to article

dodecahedron Truncated small stellated dodecahedron Dodecadodecahedron Truncated great dodecahedron Great dodecahedron Coxeter-Dynkin diagram Picture

Cantic 8-cube (210 words) [view diff] exact match in snippet view article find links to article

uniform 8-polytope Schläfli symbol t0,1{3,35,1} h2{4,3,3,3,3,3,3} Coxeter-Dynkin diagram 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure ( )v{

Bitruncation (284 words) [view diff] exact match in snippet view article find links to article

Space 4-polytope or honeycomb Schläfli symbol Coxeter-Dynkin diagram Cell type Cell image Vertex figure S 3 {\displaystyle \mathbb {S} ^{3}} Bitruncated

8-simplex (860 words) [view diff] exact match in snippet view article find links to article

Regular 8-polytope Family simplex Schläfli symbol {3,3,3,3,3,3,3} Coxeter-Dynkin diagram 7-faces 9 7-simplex 6-faces 36 6-simplex 5-faces 84 5-simplex 4-faces

Steritruncated tesseractic honeycomb (263 words) [view diff] exact match in snippet view article find links to article

image) Type Uniform 4-honeycomb Schläfli symbol t0,1,4{4,3,3,4} Coxeter-Dynkin diagram 4-face type Runcinated tesseract Truncated tesseract Tesseract 4-8 duoprism

Cantitruncated tesseractic honeycomb (265 words) [view diff] exact match in snippet view article find links to article

Uniform 4-honeycomb Schläfli symbol tr{4,3,3,4} tr{4,3,31,1} Coxeter-Dynkin diagram 4-face type t0,1,2{4,3,3} t0,1{3,3,4} {3,4}×{} Cell type Truncated cuboctahedron

Seven-dimensional space (499 words) [view diff] exact match in snippet view article find links to article

Coxeter group. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram. The 7-demicube is a unique polytope from the D7 family, and 321, 231

Rhombic dodecahedral honeycomb (401 words) [view diff] exact match in snippet view article find links to article

Rhombic dodecahedral honeycomb Type convex uniform honeycomb dual Coxeter-Dynkin diagram = Cell type Rhombic dodecahedron V3.4.3.4 Face types Rhombus Space group

Great grand stellated 120-cell (365 words) [view diff] exact match in snippet view article find links to article

1200 Vertices 600 Vertex figure {3,3} Schläfli symbol {5/2,3,3} Coxeter-Dynkin diagram Symmetry group H4, [3,3,5] Dual Grand 600-cell Properties Regular

Tridecahedron (767 words) [view diff] exact match in snippet view article find links to article

can be represented by {11}×{} or t{2, 11}; can be used in a Coxeter-Dynkin diagram to represent it; its Wythoff symbol is 2 11 | 2; in Conway polyhedron

Tridecahedron (767 words) [view diff] exact match in snippet view article find links to article

can be represented by {11}×{} or t{2, 11}; can be used in a Coxeter-Dynkin diagram to represent it; its Wythoff symbol is 2 11 | 2; in Conway polyhedron

Truncated 5-cubes (450 words) [view diff] exact match in snippet view article find links to article

Truncated 5-cube Type uniform 5-polytope Schläfli symbol t{4,3,3,3} Coxeter-Dynkin diagram 4-faces 42 10 32 Cells 200 40 160 Faces 400 80 320 Edges 400 80 320

Rectified 600-cell (492 words) [view diff] exact match in snippet view article find links to article

4-polytope Uniform index 34 Schläfli symbol t1{3,3,5} or r{3,3,5} Coxeter-Dynkin diagram Cells 600 (3.3.3.3) 120 {3,5} Faces 1200+2400 {3} Edges 3600 Vertices

Duopyramid (650 words) [view diff] exact match in snippet view article find links to article

represented by a composite Schläfli symbol {p} + {q}, and Coxeter-Dynkin diagram . The regular 16-cell can be seen as a 4-4 duopyramid or 4-4 fusil,

A5 polytope (249 words) [view diff] exact match in snippet view article find links to article

# Coxeter plane graphs Coxeter-Dynkin diagram Schläfli symbol Name [6] [5] [4] [3] A5 A4 A3 A2 1 {3,3,3,3} 5-simplex (hix) 2 t1{3,3,3,3} or r{3,3,3,3}

Cantellated 7-orthoplexes (246 words) [view diff] exact match in snippet view article find links to article

7-orthoplex Type uniform 7-polytope Schläfli symbol rr{3,3,3,3,3,4} Coxeter-Dynkin diagram 6-faces 5-faces 4-faces Cells Faces Edges 7560 Vertices 840 Vertex figure

Truncated cuboctahedron (977 words) [view diff] exact match in snippet view article find links to article

of uniform patterns with vertex configuration (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra

A6 polytope (246 words) [view diff] exact match in snippet view article find links to article

# A6 [7] A5 [6] A4 [5] A3 [4] A2 [3] Coxeter-Dynkin diagram Schläfli symbol Name 1 t0{3,3,3,3,3} 6-simplex Heptapeton (hop) 2 t1{3,3,3,3,3} Rectified 6-simplex

Heptellated 8-simplexes (524 words) [view diff] exact match in snippet view article find links to article

8-dimensional space with three facets around each ridge. It has Coxeter-Dynkin diagram of . This polytope is one of 135 uniform 8-polytopes with A8 symmetry

A7 polytope (246 words) [view diff] exact match in snippet view article find links to article

# Coxeter-Dynkin diagram Schläfli symbol Johnson name Ak orthogonal projection graphs A7 [8] A6 [7] A5 [6] A4 [5] A3 [4] A2 [3] 1 t0{3,3,3,3,3,3} 7-simplex

B5 polytope (253 words) [view diff] exact match in snippet view article find links to article

Graph B4 / D5 [8] Graph B3 / A2 [6] Graph B2 [4] Graph A3 [4] Coxeter-Dynkin diagram and Schläfli symbol Johnson and Bowers names 1 h{4,3,3,3} 5-demicube

Trapezohedron (1,448 words) [view diff] exact match in snippet view article find links to article

degenerate, to be uniform. A dual-uniform star p/q-trapezohedron has Coxeter-Dynkin diagram . Wikimedia Commons has media related to Trapezohedra. Diminished trapezohedron

Goursat tetrahedron (644 words) [view diff] exact match in snippet view article find links to article

corresponding to the reflection order, being π/dihedral angle. A 4-node Coxeter-Dynkin diagram represents this tetrahedral graph with order-2 edges hidden. If many

Tetrahedron (8,822 words) [view diff] exact match in snippet view article find links to article

triangle or obtuse triangle faces. For a regular k-polytope, the Coxeter-Dynkin diagram of the characteristic k-orthoscheme is the k-polytope's diagram without

Truncated icosidodecahedron (850 words) [view diff] exact match in snippet view article find links to article

sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra

Conway group Co1 (925 words) [view diff] exact match in snippet view article find links to article

involution of type 2B in the monster group is of the form 21+24Co1. The Dynkin diagram of the even Lorentzian unimodular lattice II1,25 is isometric to the

Cantellated 7-simplexes (435 words) [view diff] exact match in snippet view article find links to article

r\left\{{\begin{array}{l}3,3,3,3,3\\3\end{array}}\right\}} Coxeter-Dynkin diagram or 6-faces 5-faces 4-faces Cells Faces Edges 1008 Vertices 168 Vertex

B6 polytope (252 words) [view diff] exact match in snippet view article find links to article

# Coxeter plane graphs Coxeter-Dynkin diagram Schläfli symbol Names B6 [12] B5 / D4 / A4 [10] B4 [8] B3 / A2 [6] B2 [4] A5 [6] A3 [4] 1 {3,3,3,3,4} 6-orthoplex

A8 polytope (234 words) [view diff] exact match in snippet view article find links to article

# Coxeter-Dynkin diagram Schläfli symbol Johnson name Ak orthogonal projection graphs A8 [9] A7 [8] A6 [7] A5 [6] A4 [5] A3 [4] A2 [3] 1 t0{3,3,3,3,3,3

Schläfli orthoscheme (1,073 words) [view diff] exact match in snippet view article find links to article

them all without repetition. For a regular k-polytope, the Coxeter-Dynkin diagram of the characteristic k-orthoscheme is the k-polytope's diagram without

Runcinated 8-simplexes (353 words) [view diff] exact match in snippet view article find links to article

Type uniform 8-polytope Schläfli symbol t1,4{3,3,3,3,3,3,3} Coxeter-Dynkin diagram 7-faces 6-faces 5-faces 4-faces Cells Faces Edges 11340 Vertices 1260

B7 polytope (255 words) [view diff] exact match in snippet view article find links to article

# Coxeter plane graphs Coxeter-Dynkin diagram Schläfli symbol Johnson and Bowers Name B7 [14] B6 [12] B5 [10] B4 [8] B3 [6] B2 [4] A5 [6] A3 [4] 1 {3,3

Triangular prismatic honeycomb (920 words) [view diff] exact match in snippet view article find links to article

honeycomb Type Convex honeycomb Schläfli symbol ht0,1,2,3{6,3,2,∞} Coxeter-Dynkin diagram Cells hexagonal antiprism octahedron tetrahedron Vertex figure Symmetry

Goddard–Thorn theorem (1,270 words) [view diff] exact match in snippet view article find links to article

bounds on the root multiplicities of the Kac–Moody Lie algebra whose Dynkin diagram is the Leech lattice, and Borcherds's construction of a generalized

16-cell (7,175 words) [view diff] exact match in snippet view article find links to article

characteristic 5-cell of the regular 16-cell is represented by the Coxeter-Dynkin diagram , which can be read as a list of the dihedral angles between its mirror

Goddard–Thorn theorem (1,270 words) [view diff] exact match in snippet view article find links to article

bounds on the root multiplicities of the Kac–Moody Lie algebra whose Dynkin diagram is the Leech lattice, and Borcherds's construction of a generalized

Octahedral symmetry (1,865 words) [view diff] exact match in snippet view article find links to article

octahedral symmetry, which represent the three mirrors of a Coxeter-Dynkin diagram. The product of the reflections produce 3 rotational generators. The

Pentellated 6-cubes (513 words) [view diff] exact match in snippet view article find links to article

6-cube Type Uniform 6-polytope Schläfli symbol t0,5{4,3,3,3,3} Coxeter-Dynkin diagram 5-faces 4-faces Cells Faces Edges 1920 Vertices 384 Vertex figure 5-cell

Order-5 dodecahedral honeycomb (663 words) [view diff] exact match in snippet view article find links to article

Uniform hyperbolic honeycomb Schläfli symbol {5,3,5} t0{5,3,5} Coxeter-Dynkin diagram Cells {5,3} (regular dodecahedron) Faces {5} (pentagon) Edge figure

Hexic 7-cubes (242 words) [view diff] exact match in snippet view article find links to article

Type uniform 7-polytope Schläfli symbol t0,5{3,34,1} h6{4,35} Coxeter-Dynkin diagram 5-faces 4-faces Cells Faces Edges 4704 Vertices 448 Vertex figure Coxeter

Regular 4-polytope (1,648 words) [view diff] exact match in snippet view article find links to article

visualizations can be found in the external links below. The Coxeter-Dynkin diagram graphs are also given below the Schläfli symbol. The Schläfli–Hess 4-polytopes

E8 lattice (3,559 words) [view diff] exact match in snippet view article find links to article

notation, Gosset's honeycomb is denoted by 521 and has the Coxeter-Dynkin diagram: This honeycomb is highly regular in the sense that its symmetry group

Uniform 4-polytope (4,310 words) [view diff] exact match in snippet view article find links to article

uniform duoprisms, products of two regular polygons. A duoprism's Coxeter-Dynkin diagram is . Its vertex figure is a disphenoid tetrahedron, . This family overlaps

Grand 600-cell (373 words) [view diff] exact match in snippet view article find links to article

Vertices 120 Vertex figure {3,5/2} Schläfli symbol {3,3,5/2} Coxeter-Dynkin diagram Symmetry group H4, [3,3,5] Dual Great grand stellated 120-cell Properties

Complex polytope (6,937 words) [view diff] exact match in snippet view article find links to article

by an empty Schläfli symbol {}, or Coxeter-Dynkin diagram . The dot or node of the Coxeter-Dynkin diagram itself represents a reflection generator while

B8 polytope (126 words) [view diff] exact match in snippet view article find links to article

# Element counts Coxeter-Dynkin diagram Schläfli symbol Name B8 [16] B7 [14] B6 [12] B5 [10] B4 [8] B3 [6] B2 [4] A7 [8] A5 [6] A3 [4] 1 t0{3,3,3,3,3,3

Hexicated 7-simplexes (1,272 words) [view diff] exact match in snippet view article find links to article

7-dimensional space with three facets around each ridge. It has Coxeter-Dynkin diagram of . Great petated hexadecaexon (Acronym: guph) (Jonathan Bowers) The

Point groups in three dimensions (5,081 words) [view diff] exact match in snippet view article find links to article

dimensions are also called Coxeter groups and can be given by a Coxeter-Dynkin diagram and represent a set of mirrors that intersect at one central point.

3-transposition group (3,289 words) [view diff] exact match in snippet view article find links to article

orthogonal group and S3 is the group of diagram automorphisms for the D4 Dynkin diagram. The missing cases with n small above either do not satisfy the condition

Hook length formula (5,141 words) [view diff] exact match in snippet view article find links to article

functions, Algebra I Analiz, 9.2 (1997), 73-146. Proctor, Robert (1999). "Dynkin diagram classification of λ-minuscule Bruhat lattices and of d-complete posets"

Schwarz triangle (10,972 words) [view diff] exact match in snippet view article find links to article

perpendicular mirrors that can be ignored in this diagram. The Coxeter-Dynkin diagram represents this triangular graph with order-2 edges hidden. A Coxeter

600-cell (29,746 words) [view diff] exact match in snippet view article find links to article

characteristic 5-cell of the regular 600-cell is represented by the Coxeter-Dynkin diagram , which can be read as a list of the dihedral angles between its mirror