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 ...
, this polyhedron can be seen as either a polyhedral
stellation
In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific el ...
or a
compound
Compound may refer to:
Architecture and built environments
* Compound (enclosure), a cluster of buildings having a shared purpose, usually inside a fence or wall
** Compound (fortification), a version of the above fortified with defensive struct ...
.
As a compound
It can be seen as the
compound
Compound may refer to:
Architecture and built environments
* Compound (enclosure), a cluster of buildings having a shared purpose, usually inside a fence or wall
** Compound (fortification), a version of the above fortified with defensive struct ...
of an
icosahedron
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 ...
and
dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
. It is one of four compounds constructed from a
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
or
Kepler-Poinsot solid, and its
dual.
It has
icosahedral symmetry
In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the ...
(I
''h'') and the same
vertex 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 a
rhombic triacontahedron
In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Cata ...
.
This can be seen as the three-dimensional equivalent of the compound of two pentagons ( "
decagram"); this series continues into the fourth dimension as the
compound of 120-cell and 600-cell and into higher dimensions as compounds of hyperbolic tilings.
As a stellation
This
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on th ...
is the first
stellation
In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific el ...
of the
icosidodecahedron
In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 id ...
, and given as
Wenninger model index 47.
The stellation facets for construction are:
:
In popular culture
In the film ''
Tron
''Tron'' (stylized as ''TRON'') is a 1982 American science fiction action-adventure film written and directed by Steven Lisberger from a story by Lisberger and Bonnie MacBird. The film stars Jeff Bridges as Kevin Flynn, a computer programmer a ...
'' (1982), the character
Bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
took this shape when not speaking.
In the cartoon series ''
Steven Universe
''Steven Universe'' is an American animated series, animated television series created by Rebecca Sugar for Cartoon Network. It tells the coming-of-age story of a young boy, Steven Universe (character), Steven Universe (Zach Callison), who li ...
'' (2013-2019),
Steven
Stephen or Steven is a common English first name. It is particularly significant to Christians, as it belonged to Saint Stephen ( grc-gre, Στέφανος ), an early disciple and deacon who, according to the Book of Acts, was stoned to death; ...
's shield bubble, briefly used in the episode
Change Your Mind, had this shape.
See also
*
Compound of two tetrahedra
In geometry, a compound of two tetrahedra is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra.
Stellated octahedron
There is only one uniform polyhedral compound, the stellated octahedron, which has octahedral ...
*
Compound of cube and octahedron
The compound of cube and octahedron is a polyhedron which can be seen as either a polyhedral stellation or a compound. Construction
The 14 Cartesian coordinates of the vertices of the compound are.
: 6: (±2, 0, 0), ( 0, ±2, 0), ( 0, 0, ±2)
: ...
*
*
Compound of great stellated dodecahedron and great icosahedron
References
*
External links
*{{mathworld , urlname=Dodecahedron-IcosahedronCompound , title=Dodecahedron-Icosahedron Compound
Polyhedral stellation
Polyhedral compounds