User:Tomruen/Uniform honeycombs

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

class=wikitable !BSA	Dim	Uniform polytope (vertex figure)	Vertices	Fam	Uniform tessellation	Tessellation, packing and vertex figure	Dual tessellation and facet
BGCOLOR="#e0f0e0"	\|2D	{{CDD\|node_1\|3\|node_1}} t0,1{3} expanded 2-simplex (hexagon)	6	${\tilde{A}}_2$ \|{{CDD\|node_1\|split1\|branch}} ${\tilde{A}}_2$ honeycomb A2 lattice \|rowspan=2\|100px 100px 100px \|rowspan=2\|100px 100px
BGCOLOR="#e0f0e0"	\|2D	{{CDD\|node_1\|6\|node}} {6} hexagon	6	H~2	{{CDD\|node_1\|3\|node\|6\|node}} {3,6} H2 lattice
colspan=7\|
BGCOLOR="#e0f0e0" \|co	3D	{{CDD\|node_1\|3\|node\|3\|node_1}} t0,2{3,3} expanded 3-simplex (cuboctahedron)	12	${\tilde{A}}_3$	{{CDD\|branch_10\|3ab\|branch}} ${\tilde{A}}_3$ honeycomb A3 lattice \|rowspan=2\|100px 100px 100px \|rowspan=2\|100px 100px
BGCOLOR="#e0f0e0" \|co	3D	{{CDD\|node\|3\|node_1\|4\|node}} t1{3,4} rectified 3-orthoplex (cuboctahedron)	12	${\tilde{B}}_3$	{{CDD\|nodes_10\|split2\|node\|3\|4\|node}} h{4,3,4} D3 lattice
colspan=7\|
BGCOLOR="#e0f0e0" \|spid	4D	{{CDD\|node_1\|3\|node\|3\|node\|3\|node_1}} t0,3{3,3,3} expanded 4-simplex	20	${\tilde{A}}_4$	{{CDD\|branch_01\|3ab\|nodes\|split2\|node}} ${\tilde{A}}_4$ honeycomb A4 lattice \|80px
tes	4D	{{CDD\|node\|4\|node_1\|2\|node_1\|4\|node}} (4-4 duoprism (4-cube)	16	C~4	{{CDD\|node\|4\|node\|3\|node_1\|3\|node\|4\|node}} t2{4,3,3,4} \|rowspan=2\|80px
tes	4D	{{CDD\|node_1\|4\|node\|3\|node\|2\|node_1}} {}x{4,3} cube prism (4-cube)	16	F~4	{{CDD\|node\|3\|node_1\|3\|node\|4\|node\|3\|node}} t1{3,3,4,3}
BGCOLOR="#e0f0e0" \|ico	4D	{{CDD\|node\|3\|node_1\|3\|node\|4\|node}} t1{3,3,4} rectified 4-orthoplex	24	${\tilde{D}}_4$	{{CDD\|nodea\|4a\|nodea\|3a\|branch\|3a\|nodea_1}} h{4,3,3,4} D4 lattice \|rowspan=2\|100px 100px \|rowspan=2\|100px 100px
BGCOLOR="#e0f0e0" \|ico	4D	{{CDD\|node_1\|3\|node\|4\|node\|3\|node}} {3,4,3} (24-cell)	24	F~4	{{CDD\|node_1\|3\|node\|3\|node\|4\|node\|3\|node}} {3,3,4,3} F4 lattice
colspan=7\|
BGCOLOR="#e0f0e0" \|scad	5D	{{CDD\|node_1\|3\|node\|3\|node\|3\|node\|3\|node_1}} t0,4{3,3,3,3} expanded 5-simplex	30	${\tilde{A}}_5$	{{CDD\|branch_01\|3ab\|nodes\|3ab\|branch}} ${\tilde{A}}_5$ honeycomb A5 lattice \|80px
squoct	5D	{{CDD\|node\|4\|node_1\|2\|node_1\|3\|node\|4\|node}} {4}x{3,4} duoprism	24	C~5	{{CDD\|node\|4\|node\|3\|node_1\|3\|node\|3\|node\|4\|node}} t2{4,3,3,3,4}
BGCOLOR="#e0f0e0" \|rat	5D	{{CDD\|node\|3\|node_1\|3\|node\|3\|node\|4\|node}} t1{3,3,3,4} rectified 5-orthoplex	40	${\tilde{D}}_5$	{{CDD\|nodea_1\|3a\|branch\|3a\|nodea\|3a\|nodea\|4a\|nodea}} h{4,3,3,3,4} D5 lattice \|80px
colspan=7\|
BGCOLOR="#e0f0e0" \|stef	6D	{{CDD\|node_1\|3\|node\|3\|node\|3\|node\|3\|node\|3\|node_1}} t0,5{3,3,3,3,3} expanded 6-simplex	42	${\tilde{A}}_6$	{{CDD\|branch_01\|3ab\|nodes\|3ab\|nodes\|split2\|node}} ${\tilde{A}}_6$ honeycomb A6 lattice
squahex	6D	{{CDD\|node\|4\|node_1\|2\|node_1\|3\|node\|3\|node\|4\|node}} {4}x{3,3,4} duoprism	32	${\tilde{C}}_6$	{{CDD\|node\|4\|node\|3\|node_1\|3\|node\|3\|node\|3\|node\|4\|node}} t2{4,3,3,3,3,4}
octdip	6D	{{CDD\|node\|4\|node\|3\|node_1\|2\|node_1\|3\|node\|4\|node}} {3,4}x{3,4} duoprism	36	${\tilde{C}}_6$	{{CDD\|node\|4\|node\|3\|node\|3\|node_1\|3\|node\|3\|node\|4\|node}} t3{4,3,3,3,3,4}
BGCOLOR="#e0f0e0" \|rag	6D	{{CDD\|node\|3\|node_1\|3\|node\|3\|node\|3\|node\|4\|node}} t1{3,3,3,3,4} rectified 6-orthoplex	60	${\tilde{D}}_6$	{{CDD\|nodea_1\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea\|4a\|nodea}} h{4,3,3,3,3,4} D6 lattice \|80px
trittip	6D	{{CDD\|node_1\|3\|node\|2\|node_1\|3\|node\|2\|node_1\|3\|node}} {3}x{3}x{3} triaprism	27	${\tilde{E}}_6$	{{CDD\|node\|3\|node\|3\|branch_10\|3ab\|nodes\|3\|node}} t2(2_22) honeycomb
nodeip	6D	{{CDD\|node_1\|2\|node\|3\|node\|3\|node_1\|3\|node\|3\|node}} {}xt2{3,3,3,3} birectified 5-simplex prism	40	${\tilde{E}}_6$	{{CDD\|node\|3\|nodes_10\|split2\|node\|3ab\|nodes\|3\|node}} t1(2_22) honeycomb
BGCOLOR="#e0f0e0" \|mo	6D	{{CDD\|nodea\|3a\|nodea\|3a\|branch_01\|3a\|nodea\|3a\|nodea}} 1_22	72	${\tilde{E}}_6$	{{CDD\|node_1\|3\|node\|split1\|nodes\|3ab\|nodes\|3ab\|nodes}} Gosset 2_22 honeycomb E6 lattice \|80px
colspan=7\|
BGCOLOR="#e0f0e0"	\|7D	{{CDD\|node_1\|3\|node\|3\|node\|3\|node\|3\|node\|3\|node\|3\|node_1}} t0,6{3,3,3,3,3,3} expanded 7-simplex	56	${\tilde{A}}_7$	{{CDD\|branch_01\|3ab\|nodes\|3ab\|nodes\|3ab\|branch}} ${\tilde{A}}_7$ honeycomb A7 lattice
	7D	{{CDD\|node\|4\|node_1\|2\|node_1\|3\|node\|3\|node\|3\|node\|4\|node}} {4}x{3,3,3,4} duoprism	40	${\tilde{C}}_7$	{{CDD\|node\|4\|node\|3\|node_1\|3\|node\|3\|node\|3\|node\|3\|node\|4\|node}} t2{4,3,3,3,3,3,4}
	7D	{{CDD\|node\|4\|node\|3\|node_1\|2\|node_1\|3\|node\|3\|node\|4\|node}} {3,4}x{3,3,4} duoprism	48	${\tilde{C}}_7$	{{CDD\|node\|4\|node\|3\|node\|3\|node_1\|3\|node\|3\|node\|3\|node\|4\|node}} t3{4,3,3,3,3,3,4}
BGCOLOR="#e0f0e0" \|rez	7D	{{CDD\|node\|3\|node_1\|3\|node\|3\|node\|3\|node\|3\|node\|4\|node}} t1{3,3,3,3,3,4} rectified 7-orthoplex	84	${\tilde{D}}_7$	{{CDD\|nodea_1\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|4a\|nodea}} h{4,3,3,3,3,3,4} D7 lattice \|100px
he	7D	{{CDD\|node\|3\|node\|3\|node\|3\|node_1\|3\|node\|3\|node\|3\|node}} 0_33 trirectified 7-simplex	70	${\tilde{E}}_7$	{{CDD\|nodea\|3a\|nodea\|3a\|nodea\|3a\|branch_01\|3a\|nodea\|3a\|nodea\|3a\|nodea}} Gosset 1_33 honeycomb \|100px
	7D	{}x(1_31)		${\tilde{E}}_7$	{{CDD\|nodea\|3a\|nodea_1\|3a\|nodea\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea}} t1(3₃₁) honeycomb \|
	7D	{3}x(0_31)		${\tilde{E}}_7$	{{CDD\|nodea\|3a\|nodea\|3a\|nodea_1\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea}} t2(3₃₁) honeycomb \|
	7D	{3,3}x{3}x{}		${\tilde{E}}_7$	{{CDD\|nodea\|3a\|nodea\|3a\|nodea\|3a\|branch_10\|3a\|nodea\|3a\|nodea\|3a\|nodea}} t3(3₃₁) honeycomb \|
BGCOLOR="#e0f0e0" \|laq	7D	{{CDD\|nodea_1\|3a\|nodea\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea}} 2_31	126	${\tilde{E}}_7$	{{CDD\|nodea_1\|3a\|nodea\|3a\|nodea\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea}} Gosset 3_31 honeycomb E7 lattice \|100px
colspan=7\|
BGCOLOR="#e0f0e0"	\|8D	{{CDD\|node_1\|3\|node\|3\|node\|3\|node\|3\|node\|3\|node\|3\|node\|3\|node_1}} t0,7{3,3,3,3,3,3,3} expanded 8-simplex	72	${\tilde{A}}_8$	{{CDD\|branch_01\|3ab\|nodes\|3ab\|nodes\|3ab\|nodes\|split2\|node}} ${\tilde{A}}_8$ honeycomb A8 lattice
	8D	{{CDD\|node\|4\|node_1\|2\|node_1\|3\|node\|3\|node\|3\|node\|3\|node\|4\|node}} {4}x{3,3,3,3,4} duoprism	48	${\tilde{C}}_7$	{{CDD\|node\|4\|node\|3\|node_1\|3\|node\|3\|node\|3\|node\|3\|node\|3\|node\|4\|node}} t2{4,3,3,3,3,3,3,4}
	8D	{{CDD\|node\|4\|node\|3\|node_1\|2\|node_1\|3\|node\|3\|node\|3\|node\|4\|node}} {3,4}x{3,3,3,4} duoprism	60	${\tilde{C}}_7$	{{CDD\|node\|4\|node\|3\|node\|3\|node_1\|3\|node\|3\|node\|3\|node\|3\|node\|4\|node}} t3{4,3,3,3,3,3,3,4}
	8D	{{CDD\|node\|4\|node\|3\|node\|3\|node_1\|2\|node_1\|3\|node\|3\|node\|4\|node}} {3,3,4}x{3,3,4} duoprism	64	${\tilde{C}}_7$	{{CDD\|node\|4\|node\|3\|node\|3\|node\|3\|node_1\|3\|node\|3\|node\|3\|node\|4\|node}} t4{4,3,3,3,3,3,3,4}
BGCOLOR="#e0f0e0" \|rek	8D	{{CDD\|node\|3\|node_1\|3\|node\|3\|node\|3\|node\|3\|node\|3\|node\|4\|node}} t1{3,3,3,3,3,3,4} rectified 8-orthoplex	112	${\tilde{D}}_8$	{{CDD\|nodea_1\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|4a\|nodea}} h{4,3,3,3,3,3,3,4} D8 lattice \|120px
rene	8D	{{CDD\|node\|3\|node\|3\|node_1\|3\|node\|3\|node\|3\|node\|3\|node\|3\|node}} 0_52 birectified 8-simplex	84	${\tilde{E}}_8$	{{CDD\|nodea\|3a\|nodea\|3a\|branch_01\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea}} Gosset 1_52 honeycomb \|120px
roc prism	8D	{} x 0_51	56	${\tilde{E}}_8$	{{CDD\|nodea\|3a\|nodea_1\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea}} t1(2₅₁) honeycomb \|
hocto	8D	{{CDD\|nodea_1\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea}} 1_51 8-demicube	128	${\tilde{E}}_8$	{{CDD\|nodea_1\|3a\|nodea\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea}} Gosset 2_51 honeycomb \|120px
	8D	{} x (3_21)	112	${\tilde{E}}_8$	{{CDD\|nodea\|3a\|nodea\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea_1\|3a\|nodea}} t1(5₂₁) honeycomb \|
	8D	{3} x (2_21)	81	${\tilde{E}}_8$	{{CDD\|nodea\|3a\|nodea\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea_1\|3a\|nodea\|3a\|nodea}} t2(5₂₁) honeycomb \|
	8D	{3,3} x (1_21)	48	${\tilde{E}}_8$	{{CDD\|nodea\|3a\|nodea\|3a\|branch\|3a\|nodea\|3a\|nodea_1\|3a\|nodea\|3a\|nodea\|3a\|nodea}} t3(5₂₁) honeycomb \|
	8D	{3,3,3} x (0_21)	50	${\tilde{E}}_8$	{{CDD\|nodea\|3a\|nodea\|3a\|branch\|3a\|nodea_1\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea}} t4(5₂₁) honeycomb \|
	8D	{3,3,3,3} x {3} x {}	30	${\tilde{E}}_8$	{{CDD\|nodea\|3a\|nodea\|3a\|branch_10\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea}} t5(5₂₁) honeycomb \|
BGCOLOR="#e0f0e0" \|fy	8D	{{CDD\|nodea\|3a\|nodea\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea_1}} 4_21 polytope	240	${\tilde{E}}_8$	{{CDD\|nodea\|3a\|nodea\|3a\|branch\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea_1}} Gosset 5_21 honeycomb E8 lattice \|120px

Families =

Infinite Coxeter groups

Families of convex uniform tessellations are defined by Coxeter groups.

class="wikitable"

!height=30|n

! ${\tilde{A}}_2+$
eα_n

! ${\tilde{B}}_3+$
Alternated n-cubicCoxeter, The beauty of Geometry, Wythoff's construction of uniform polytopes, Page 47, hδ_n
hδ_n

! ${\tilde{C}}_2+$
n-cubicCoxeter, The beauty of Geometry, Wythoff's construction of uniform polytopes, Page 47, δ_n
δ_n

! ${\tilde{D}}_4+$
Quartered n-cubicCoxeter, The beauty of Geometry, Wythoff's construction of uniform polytopes, Page 47, qδ_n
qδ_n

! ${\tilde{E}}_6$ - ${\tilde{E}}_9$

! ${\tilde{F}}_4$

! ${\tilde{G}}_2$
Hexagonal

! ${\tilde{I}}_1$
Aperigonal

|{4,3,4}

|{3^[5]}

|{4,3²,4}

|q{4,3²,4}

|{3,4,3,3}

|{3^[6]}

|h{4,3³,4}

|{4,3³,4}

|q{4,3³,4}

|{3^[7]}

|h{4,3⁴,4}

|{4,3⁴,4}

|q{4,3⁴,4}

|{3^2,2,2}

{{CDD|node_1|3|node|3|branch|3ab|nodes|3|node}}
E6 lattice

|{3^[8]}

|h{4,3⁵,4}

|{4,3⁵,4}

|q{4,3⁵,4}

{{CDD|nodes_10|split2|node|3|node|3|node|3|node|split1|nodes_10}}

|{3^3,3,1}

{3^1,3,3}

{{CDD|node|3|node|3|node|3|branch_01|3|node|3|node|3|node}}
E7 lattice

|{3^[9]}

|h{4,3⁶,4}

{{CDD|nodes_10|split2|node|3|node|3|node|3|node|3|node|3|node|4|node}}

|{4,3⁶,4}

{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}

|q{4,3⁶,4}

{{CDD|nodes_10|split2|node|3|node|3|node|3|node|3|node|split1|nodes_10}}

|{3^5,2,1}

{{CDD|node|3|node|3|branch|3|node|3|node|3|node|3|node|3|node_1}}
E8 lattice

{3^2,5,1}

{{CDD|node_1|3|node|3|branch|3|node|3|node|3|node|3|node|3|node}}

{3^1,5,2}

{{CDD|node|3|node|3|branch_01|3|node|3|node|3|node|3|node|3|node}}

|{3^[10]}

|h{4,3⁷,4}

{{CDD|nodes_10|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}

|{4,3⁷,4}

{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}

|q{4,3⁷,4}

{{CDD|nodes_10|split2|node|3|node|3|node|3|node|3|node|3|node|split1|nodes_10}}

|{3^6,2,1}

{{CDD|node|3|node|3|branch|3|node|3|node|3|node|3|node|3|node|3|node_1}}
E9 lattice

{3^2,6,1}

{{CDD|node_1|3|node|3|branch|3|node|3|node|3|node|3|node|3|node|3|node}}

{3^1,6,2}

{{CDD|nodea|3a|nodea|3a|branch_01|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

|...

= Notes=

http://books.google.com/books?id=fUm5Mwfx8rAC&printsec=frontcover&dq=Coxeter&hl=en&ei=0jjOS-yZOpKuNvi79BU&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDsQ6AEwAw#v=onepage&q=lattice&f=false