User:Tomruen/Uniform honeycombs
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This is a list of uniform tessellations in dimensions 2-8, constructed by vertex figures that are uniform polytopes with circumradii equal to 1.0.
Lattice refers to the vertex arrangement of a uniform tessellation/honeycomb. There are also dual lattices, which place vertices at the tessellation facet centers.
The number of vertices in the vertex figure is equal to the kissing number of the tessellation.
Circumradius 1 vertex figures
[edit]BSA | Dim | Uniform polytope (vertex figure) | Vertices | Fam | Uniform tessellation | Tessellation, packing and vertex figure | Dual tessellation and facet |
---|---|---|---|---|---|---|---|
2D | ![]() ![]() ![]() t0,1{3} expanded 2-simplex (hexagon) |
6 | ![]() ![]() ![]() honeycomb A2 lattice |
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2D | ![]() ![]() ![]() {6} hexagon |
6 | H~2 | ![]() ![]() ![]() ![]() ![]() {3,6} H2 lattice | |||
co | 3D | ![]() ![]() ![]() ![]() ![]() t0,2{3,3} expanded 3-simplex (cuboctahedron) |
12 | ![]() ![]() ![]() honeycomb A3 lattice |
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co | 3D | ![]() ![]() ![]() ![]() ![]() t1{3,4} rectified 3-orthoplex (cuboctahedron) |
12 | ![]() ![]() ![]() ![]() ![]() ![]() h{4,3,4} D3 lattice | |||
spid | 4D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,3{3,3,3} expanded 4-simplex |
20 | ![]() ![]() ![]() ![]() ![]() honeycomb A4 lattice |
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tes | 4D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() (4-4 duoprism (4-cube) |
16 | C~4 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t2{4,3,3,4} |
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tes | 4D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() {}x{4,3} cube prism (4-cube) |
16 | F~4 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1{3,3,4,3} | ||
ico | 4D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1{3,3,4} rectified 4-orthoplex |
24 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() h{4,3,3,4} D4 lattice |
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ico | 4D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() {3,4,3} (24-cell) |
24 | F~4 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() {3,3,4,3} F4 lattice | ||
scad | 5D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,4{3,3,3,3} expanded 5-simplex |
30 | ![]() ![]() ![]() ![]() ![]() honeycomb A5 lattice |
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squoct | 5D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() {4}x{3,4} duoprism |
24 | C~5 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t2{4,3,3,3,4} | ||
rat | 5D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1{3,3,3,4} rectified 5-orthoplex |
40 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() h{4,3,3,3,4} D5 lattice |
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stef | 6D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,5{3,3,3,3,3} expanded 6-simplex |
42 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() honeycomb A6 lattice | |||
squahex | 6D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() {4}x{3,3,4} duoprism |
32 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t2{4,3,3,3,3,4} | |||
octdip | 6D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() {3,4}x{3,4} duoprism |
36 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t3{4,3,3,3,3,4} | |||
rag | 6D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1{3,3,3,3,4} rectified 6-orthoplex |
60 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() h{4,3,3,3,3,4} D6 lattice |
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trittip | 6D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() {3}x{3}x{3} triaprism |
27 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t2(2_22) honeycomb | |||
nodeip | 6D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() {}xt2{3,3,3,3} birectified 5-simplex prism |
40 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1(2_22) honeycomb | |||
mo | 6D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() 1_22 |
72 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Gosset 2_22 honeycomb E6 lattice |
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7D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,6{3,3,3,3,3,3} expanded 7-simplex |
56 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() honeycomb A7 lattice | ||||
7D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() {4}x{3,3,3,4} duoprism |
40 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t2{4,3,3,3,3,3,4} | ||||
7D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() {3,4}x{3,3,4} duoprism |
48 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t3{4,3,3,3,3,3,4} | ||||
rez | 7D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1{3,3,3,3,3,4} rectified 7-orthoplex |
84 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() h{4,3,3,3,3,3,4} D7 lattice |
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he | 7D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() 0_33 trirectified 7-simplex |
70 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Gosset 1_33 honeycomb |
File:7-simplex t3.svg.png | ||
7D | {}x(1_31) | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1(331) honeycomb |
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7D | {3}x(0_31) | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t2(331) honeycomb |
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7D | {3,3}x{3}x{} | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t3(331) honeycomb |
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laq | 7D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() 2_31 |
126 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Gosset 3_31 honeycomb E7 lattice |
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8D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t0,7{3,3,3,3,3,3,3} expanded 8-simplex |
72 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() honeycomb A8 lattice | ||||
8D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() {4}x{3,3,3,3,4} duoprism |
48 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t2{4,3,3,3,3,3,3,4} | ||||
8D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() {3,4}x{3,3,3,4} duoprism |
60 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t3{4,3,3,3,3,3,3,4} | ||||
8D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() {3,3,4}x{3,3,4} duoprism |
64 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t4{4,3,3,3,3,3,3,4} | ||||
rek | 8D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1{3,3,3,3,3,3,4} rectified 8-orthoplex |
112 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() h{4,3,3,3,3,3,3,4} D8 lattice |
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rene | 8D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() 0_52 birectified 8-simplex |
84 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Gosset 1_52 honeycomb |
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roc prism | 8D | {} x 0_51 | 56 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1(251) honeycomb |
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hocto | 8D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() 1_51 8-demicube |
128 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Gosset 2_51 honeycomb |
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8D | {} x (3_21) | 112 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t1(521) honeycomb |
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8D | {3} x (2_21) | 81 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t2(521) honeycomb |
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8D | {3,3} x (1_21) | 48 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t3(521) honeycomb |
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8D | {3,3,3} x (0_21) | 50 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t4(521) honeycomb |
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8D | {3,3,3,3} x {3} x {} | 30 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() t5(521) honeycomb |
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fy | 8D | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() 4_21 polytope |
240 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Gosset 5_21 honeycomb E8 lattice |
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Families =
[edit]Infinite Coxeter groups
[edit]Families of convex uniform tessellations are defined by Coxeter groups.
n | eαn |
Alternated n-cubic[1] hδn |
n-cubic[2] δn |
Quartered n-cubic[3] qδn |
- | Hexagonal |
Aperigonal | |
---|---|---|---|---|---|---|---|---|
1 | {∞}
| |||||||
2 | {6,3}
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{4,4}
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{6,3}
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3 | q{4,3,4}
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h{4,3,4}
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{4,3,4}
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4 | {3[5]}
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{4,32,4}
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{4,32,4}
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q{4,32,4}
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{3,4,3,3}
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5 | {3[6]}
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h{4,33,4}
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{4,33,4}
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q{4,33,4}
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6 | {3[7]}
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h{4,34,4}
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{4,34,4}
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q{4,34,4}
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{32,2,2} | |||
7 | {3[8]}
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h{4,35,4}
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{4,35,4}
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q{4,35,4}
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{33,3,1}
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8 | {3[9]}
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h{4,36,4}
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{4,36,4}
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q{4,36,4}
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{35,2,1}
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9 | {3[10]}
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h{4,37,4}
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{4,37,4}
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q{4,37,4}
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{36,2,1}
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10 | ... | ... | ... | ... |