Truncated 24-cell honeycomb - AbsoluteAstronomy.com

Truncated 24-cell honeycomb
(No image)
Type	Uniform honeycomb
Schläfli symbol	t_0,1{3,4,3,3} t_0,1,2{3,3,4,3} t_1,2,3{4,3,3,4} t_1,2,3{4,3,3^1,1} o{3^1,1,1,1}
Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... s
4-face type	tesseract Tesseract In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8... truncated 24-cell Truncated 24-cell In geometry, a truncated 5-cell is a uniform polychoron formed as the truncation of the regular 5-cell.There are two degrees of trunctions, including a bitruncation.- Truncated 5-cell:...
Cell type	cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... truncated octahedron Truncated octahedron In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....
Face type	Square Square (geometry) In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles... Triangle Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...	Regular tetrahedral pyramid
Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... s	, [3,4,3,3] , [4,3,3^1,1] , [4,3,3,4] , [3^1,1,1,1]
Properties	Vertex transitive

In four-dimensional Euclidean geometry

Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, the truncated 24-cell honeycomb is a uniform space-filling honeycomb

Honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....

. It can be seen as a truncation

Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.- Uniform truncation :...

of the regular 24-cell honeycomb, containing tesseract

Tesseract

In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...

and truncated 24-cell

Truncated 24-cell

In geometry, a truncated 5-cell is a uniform polychoron formed as the truncation of the regular 5-cell.There are two degrees of trunctions, including a bitruncation.- Truncated 5-cell:...

cells.

It has a uniform alternation, called the snub 24-cell honeycomb

Snub 24-cell honeycomb

In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 paper of semiregular polytopes...

. It is a true snub from the D4 construction. This truncated 24-cell is an omnitruncation, o{3^1,1,1,1}, and its snub

Snub (geometry)

In geometry, an alternation is an operation on a polyhedron or tiling that removes alternate vertices. Only even-sided polyhedra can be alternated, for example the zonohedra. Every 2n-sided face becomes n-sided...

is represented as s{3^1,1,1,1}.

Symmetry constructions

There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored truncated 24-cell

Truncated 24-cell

In geometry, a truncated 5-cell is a uniform polychoron formed as the truncation of the regular 5-cell.There are two degrees of trunctions, including a bitruncation.- Truncated 5-cell:...

facets. In all cases, four truncated 24-cells, and one tesseract

Tesseract

meet at each vertex, but the vertex figures have different symmetry generators.

Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...	Full symmetry group	Facets
= [3,4,3,3]	[3,4,3,3]	4: 1:
= [3,3,4,3]	[3,3,4,3]	3: 1: 1:
= [4,3,3,4]	4,3,3,4	2,2: 1:
= [3^1,1,3,4]	<[3^1,1,3,4]> = [4,3,3,4]	1,1: 2: 1:
= [3^1,1,1,1]	[3,3[3^1,1,1,1]] = [3,3,4,3]	1,1,1,1: 1:

Symmetry constructions

See also