In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the great 120-cell or great polydodecahedron is a
regular star 4-polytope with
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
. It is one of 10 regular
Schläfli-Hess polytope
In mathematics, a regular 4-polytope or regular polychoron is a regular polytope, regular 4-polytope, four-dimensional polytope. They are the four-dimensional analogues of the Regular polyhedron, regular polyhedra in three dimensions and the regul ...
s. It is one of the two such polytopes that is self-dual.
Related polytopes
It has the same
edge arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
as the
600-cell
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol .
It is also known as the C600, hexacosichoron and hexacosihedroid.
It is also called a tetraplex (abbreviated from ...
,
icosahedral 120-cell as well as the same
face arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
as the
grand 120-cell.
Due to its self-duality, it does not have a good three-dimensional analogue, but (like all other star polyhedra and polychora) is analogous to the two-dimensional
pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around ...
.
See also
*
List of regular polytopes
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces.
Overview
This table shows a summary of regular polytope counts by rank.
There are no Euclidean regular star tessellations in any number of dimensions.
...
*
Convex regular 4-polytope
In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
There are six co ...
*
Kepler-Poinsot solids regular
star polyhedron
In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvex polygon, nonconvexity giving it a star-like visual quality.
There are two general kinds of star polyhedron:
*Polyhedra which self-intersect in a repetit ...
*
Star polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, Decagram (geometry)#Related figures, certain notable ones can ...
regular star polygons
References
*
Edmund Hess
Edmund Hess (17 February 1843 – 24 December 1903) was a German mathematician who discovered several regular polytopes.
Publications
*''Über die zugleich gleicheckigen und gleichflächigen Polyeder.'' In: Sitzungsberichte der Gesellscha ...
, (1883) ''Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder'
*
Coxeter, H. S. M. Coxeter, ''Regular Polytopes'', 3rd. ed., Dover Publications, 1973. .
*
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, (Chapter 26, Regular Star-polytopes, pp. 404–408)
*
External links
Regular polychora
Reguläre Polytope
Regular 4-polytopes
{{polychora-stub