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

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

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: : First stellation of icosidodecahedron

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.

References

External links

*{{mathworld , urlname=Dodecahedron-IcosahedronCompound , title=Dodecahedron-Icosahedron Compound Polyhedral stellation Polyhedral compounds

As a compound

As a stellation

In popular culture

See also

References

External links