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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

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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
2D
{6}
hexagon
6 H~2
{3,6}
H2 lattice
 
co 3D
t0,2{3,3}
expanded 3-simplex (cuboctahedron)
12
honeycomb
A3 lattice
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
tes 4D
(4-4 duoprism (4-cube)
16 C~4
t2{4,3,3,4}
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
File:24-cell.svg
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
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
 
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
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
 
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
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
7D {3}x(0_31)
t2(331) honeycomb
7D {3,3}x{3}x{}
t3(331) honeycomb
laq 7D
2_31
126
Gosset 3_31 honeycomb
E7 lattice
 
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
rene 8D
0_52
birectified 8-simplex
84
Gosset 1_52 honeycomb
roc prism 8D {} x 0_51 56
t1(251) honeycomb
hocto 8D
1_51
8-demicube
128
Gosset 2_51 honeycomb
8D {} x (3_21) 112
t1(521) honeycomb
8D {3} x (2_21) 81
t2(521) honeycomb
8D {3,3} x (1_21) 48
t3(521) honeycomb
8D {3,3,3} x (0_21) 50
t4(521) honeycomb
8D {3,3,3,3} x {3} x {} 30
t5(521) honeycomb
fy 8D
4_21 polytope
240
Gosset 5_21 honeycomb
E8 lattice

Families =

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Infinite Coxeter groups

[edit]

Families of convex uniform tessellations are defined by Coxeter groups.

n
n

Alternated n-cubic[1]
n

n-cubic[2]
δn

Quartered n-cubic[3]
n
-

Hexagonal

Aperigonal
1               {∞}

2 {6,3}

  {4,4}

      {6,3}

 
3 q{4,3,4}

h{4,3,4}

{4,3,4}

         
4 {3[5]}

{4,32,4}

{4,32,4}

q{4,32,4}

  {3,4,3,3}

   
5 {3[6]}

h{4,33,4}

{4,33,4}

q{4,33,4}

       
6 {3[7]}

h{4,34,4}

{4,34,4}

q{4,34,4}

{32,2,2}


E6 lattice

     
7 {3[8]}

h{4,35,4}

{4,35,4}

q{4,35,4}

{33,3,1}


{31,3,3}


E7 lattice

     
8 {3[9]}

h{4,36,4}

{4,36,4}

q{4,36,4}

{35,2,1}


E8 lattice
{32,5,1}


{31,5,2}

     
9 {3[10]}

h{4,37,4}

{4,37,4}

q{4,37,4}

{36,2,1}


E9 lattice
{32,6,1}


{31,6,2}

     
10 ... ... ... ...        

Notes

[edit]
  1. ^ Coxeter, The beauty of Geometry, Wythoff's construction of uniform polytopes, Page 47, hδn
  2. ^ Coxeter, The beauty of Geometry, Wythoff's construction of uniform polytopes, Page 47, δn
  3. ^ Coxeter, The beauty of Geometry, Wythoff's construction of uniform polytopes, Page 47, qδn