# Truncated 24-cell honeycomb

In four-dimensional Euclidean geometry, the truncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a truncation of the regular 24-cell honeycomb, containing tesseract and truncated 24-cell cells.

Truncated 24-cell honeycomb
(No image)
TypeUniform 4-honeycomb
Schläfli symbolt{3,4,3,3}
tr{3,3,4,3}
t2r{4,3,3,4}
t2r{4,3,31,1}
t{31,1,1,1}
Coxeter-Dynkin diagrams

4-face typeTesseract
Truncated 24-cell
Cell typeCube
Truncated octahedron
Face typeSquare
Triangle
Vertex figure
Tetrahedral pyramid
Coxeter groups${\tilde {F}}_{4}$ , [3,4,3,3]
${\tilde {B}}_{4}$ , [4,3,31,1]
${\tilde {C}}_{4}$ , [4,3,3,4]
${\tilde {D}}_{4}$ , [31,1,1,1]
PropertiesVertex transitive

It has a uniform alternation, called the snub 24-cell honeycomb. It is a snub from the ${\tilde {D}}_{4}$ construction. This truncated 24-cell has Schläfli symbol t{31,1,1,1}, and its snub is represented as s{31,1,1,1}.

## Alternate names

• Truncated icositetrachoric tetracomb
• Truncated icositetrachoric honeycomb
• Cantitruncated 16-cell honeycomb
• Bicantitruncated tesseractic honeycomb

## Symmetry constructions

There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored truncated 24-cell facets. In all cases, four truncated 24-cells, and one tesseract meet at each vertex, but the vertex figures have different symmetry generators.

Coxeter group Coxeter
diagram
Facets Vertex figure Vertex
figure
symmetry
(order)
${\tilde {F}}_{4}$ = [3,4,3,3]
4:
1:
, [3,3]
(24)
${\tilde {F}}_{4}$ = [3,3,4,3]
3:
1:
1:
, 
(6)
${\tilde {C}}_{4}$ = [4,3,3,4]
2,2:
1:
, 
(4)
${\tilde {B}}_{4}$ = [31,1,3,4]
1,1:
2:
1:
, [ ]
(2)
${\tilde {D}}_{4}$ = [31,1,1,1]
1,1,1,1:

1:
[ ]+
(1)

## See also

Regular and uniform honeycombs in 4-space:

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