In the
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
of
hyperbolic 4-space, the order-5 tesseractic honeycomb is one of five compact
regular space-filling
tessellation
A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...
s (or
honeycombs). With
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
, it has five
8-cell
In geometry, a tesseract is the four-dimensional analogue 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 six square faces, the hypersurface of the tesseract consists of eigh ...
s (also known as tesseracts) around each face. Its
dual is the
order-4 120-cell honeycomb, .
Related polytopes and honeycombs
It is related to the Euclidean 4-space (order-4)
tesseractic honeycomb
In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol , and constructed by a 4-dimensional packing of tesseract facets.
Its verte ...
, , and the
5-cube
In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.
It is represented by Schläfli symbol or , constructed as 3 tesseracts, ...
, in Euclidean 5-space. The ''5-cube'' can also be seen as an ''order-3 tesseractic honeycomb'' on the surface of a
4-sphere.
It is analogous to the
order-5 cubic honeycomb
In hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol it has five cubes around each edge, and 20 cubes around each vertex. It ...
and
order-5 square tiling
In geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of .
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with ver ...
.
See also
*
List of regular polytopes
This article lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces.
The Schläfli symbol describes every regular tessellation of an ' ...
References
*
Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
, ''
Regular Polytopes
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
'', 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
*
Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
, ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999 {{isbn, 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
Honeycombs (geometry)