In
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 ...
, 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 regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
. 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 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, equ ...
as the
grand 120-cell
In geometry, the grand 120-cell or grand polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes.
It is one of four ''regular star 4-polytopes'' discovered by Ludwig Schläfli. It is n ...
.
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 aroun ...
.
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 ' ...
*
Convex regular 4-polytope
In mathematics, a regular 4-polytope 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 convex and ten star regu ...
*
Kepler-Poinsot solids regular
star polyhedron
In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality.
There are two general kinds of star polyhedron:
*Polyhedra which self-intersect in a repetitive way.
*Concave p ...
*
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, certain notable ones can arise through truncation operations ...
regular star polygons
References
*
Edmund Hess
Edmund Hess (17 February 1843 – 24 December 1903) was a German mathematician who discovered several regular polytopes.
See also
* Schläfli–Hess polychoron
* Hess polytope
References
* ''Regular Polytopes
In mathematics, a regu ...
, (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
John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...
, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 26, Regular Star-polytopes, pp. 404–408)
*
External links
Regular polychora
Reguläre Polytope
4-polytopes
{{polychora-stub