In
geometry, the icosahedral 120-cell, polyicosahedron,
faceted Faceted may refer to an object containing a facet.
Faceted may also refer to:
* Faceted classification, organizational system allowing multiple characteristics or attributes of each item
*Faceted search, technique for accessing information via fac ...
600-cell or icosaplex is a regular
star 4-polytope
In mathematics, a regular 4-polytope 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 regular polygons in two dim ...
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 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 ...
s.
It is constructed by 5
icosahedra around each edge in a
pentagrammic figure. The
vertex figure is a
great dodecahedron.
Related polytopes
It has the same
edge arrangement as the
600-cell,
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 ...
and
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 ...
, and shares its vertices with all other
Schläfli–Hess 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 ...
s except the
great grand stellated 120-cell (another stellation of the
120-cell
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, heca ...
).
As a faceted 600-cell, replacing the
simplicial cells of the 600-cell with
icosahedral pentagonal polytope cells, it could be seen as a four-dimensional analogue of the
great dodecahedron, which replaces the triangular faces of the icosahedron with pentagonal faces. Indeed, the icosahedral 120-cell is dual to the
small stellated 120-cell, which could be taken as a 4D analogue of the
small stellated dodecahedron, dual of the great dodecahedron.
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
*
Kepler-Poinsot solids - regular
star polyhedron
*
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, (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