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 icosahedral 120-cell honeycomb is one of four
regular star-
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
icosahedral 120-cell
In geometry, the icosahedral 120-cell, polyicosahedron, faceting, faceted 600-cell or icosaplex is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes.
It is constructed by 5 icosahedron, icosahedra a ...
s around each face. It is
dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual (grammatical ...
to the
great 120-cell honeycomb.
It can be constructed by replacing the
great dodecahedral cells of the great 120-cell honeycomb with their
icosahedral
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
convex hulls, thus replacing the
great 120-cell
In geometry, the great 120-cell or great polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. It is one of the two such polytopes that is self-dual.
Related polytopes
It has ...
s with
icosahedral 120-cell
In geometry, the icosahedral 120-cell, polyicosahedron, faceting, faceted 600-cell or icosaplex is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes.
It is constructed by 5 icosahedron, icosahedra a ...
s. It is thus analogous to the four-dimensional
icosahedral 120-cell
In geometry, the icosahedral 120-cell, polyicosahedron, faceting, faceted 600-cell or icosaplex is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes.
It is constructed by 5 icosahedron, icosahedra a ...
. It has
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
10.
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, ''
Regular Polytopes'', 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
*
Coxeter, ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
{{geometry-stub
Honeycombs (geometry)
5-polytopes