5-simplex honeycomb
(No image)
Type	Uniform 5-honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3[6]}
Coxeter diagram
5-face types	{34} , t1{34} t2{34}
4-face types	{33} , t1{33}
Cell types	{3,3} , t1{3,3}
Face types	{3}
Vertex figure	t0,4{34}
Coxeter groups	${\displaystyle {\tilde {A))_{5))$×2, <[3[6]]>
Properties	vertex-transitive

In five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb or pentacomb). Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes. These facet types occur in proportions of 2:2:1 respectively in the whole honeycomb.

A5 lattice

This vertex arrangement is called the A₅ lattice or 5-simplex lattice. The 30 vertices of the stericated 5-simplex vertex figure represent the 30 roots of the ${\displaystyle {\tilde {A))_{5))$ Coxeter group.^[1] It is the 5-dimensional case of a simplectic honeycomb.

The A²
₅ lattice is the union of two A₅ lattices:

∪

The A³
₅ is the union of three A₅ lattices:

∪ ∪ .

The A^*
₅ lattice (also called A⁶
₅) is the union of six A₅ lattices, and is the dual vertex arrangement to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-simplex.

∪ ∪ ∪ ∪ ∪ = dual of

Related polytopes and honeycombs

This honeycomb is one of 12 unique uniform honeycombs^[2] constructed by the ${\displaystyle {\tilde {A))_{5))$ Coxeter group. The extended symmetry of the hexagonal diagram of the ${\displaystyle {\tilde {A))_{5))$ Coxeter group allows for automorphisms that map diagram nodes (mirrors) on to each other. So the various 12 honeycombs represent higher symmetries based on the ring arrangement symmetry in the diagrams:

A5 honeycombs
Hexagon symmetry	Extended symmetry	Extended diagram	Extended group	Honeycomb diagrams
a1	[3[6]]		${\displaystyle {\tilde {A))_{5))$
d2	<[3[6]]>		${\displaystyle {\tilde {A))_{5))$×21	1, , , ,
p2	[[3[6]]]		${\displaystyle {\tilde {A))_{5))$×22	2,
i4	[<[3[6]]>]		${\displaystyle {\tilde {A))_{5))$×21×22	,
d6	<3[3[6]]>		${\displaystyle {\tilde {A))_{5))$×61
r12	[6[3[6]]]		${\displaystyle {\tilde {A))_{5))$×12	3

Projection by folding

The 5-simplex honeycomb can be projected into the 3-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

${\displaystyle {\tilde {A))_{5))$
${\displaystyle {\tilde {C))_{3))$