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 ...

, an icosidodecahedron is a

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 ...

with twenty (''icosi'')

triangular A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- collinea ...

faces and twelve (''dodeca'')

pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...

al faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the

Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...

s and more particularly, a

quasiregular polyhedron In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the se ...

Geometry

An icosidodecahedron has icosahedral symmetry, and its first stellation is 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 struc ...

of a

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 ...

and its dual

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 ...

, with the vertices of the icosidodecahedron located at the midpoints of the edges of either. Its

dual polyhedron In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. ...

is the

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 Ca ...

. An icosidodecahedron can be split along any of six planes to form a pair of

pentagonal rotunda In geometry, the pentagonal rotunda is one of the Johnson solids (). It can be seen as half of an icosidodecahedron, or as half of a pentagonal orthobirotunda. It has a total of 17 faces. Formulae The following formulae for volume, surface ar ...

e, which belong among the Johnson solids. The icosidodecahedron can be considered a ''pentagonal gyrobirotunda'', as a combination of two rotundae (compare

pentagonal orthobirotunda In geometry, the pentagonal orthobirotunda is one of the Johnson solids (). It can be constructed by joining two pentagonal rotundae () along their decagonal faces, matching like faces. Related polyhedra The pentagonal orthobirotunda is als ...

, one of the Johnson solids). In this form its symmetry is D_5d, 0,2⁺ (2*5), order 20. The wire-frame figure of the icosidodecahedron consists of six flat regular decagons, meeting in pairs at each of the 30 vertices. The icosidodecahedron has 6 central

decagon In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°. A self-intersecting ''regular decagon'' i ...

s. Projected into a sphere, they define 6

great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...

Buckminster Fuller Richard Buckminster Fuller (; July 12, 1895 – July 1, 1983) was an American architect, systems theorist, writer, designer, inventor, philosopher, and futurist. He styled his name as R. Buckminster Fuller in his writings, publishing more t ...

used these 6 great circles, along with 15 and 10 others in two other polyhedra to define his 31 great circles of the spherical icosahedron.

Cartesian coordinates

Convenient Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by the

even permutation In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total ...

s of: *(0, 0, ±''φ'') *(±, ±, ±) where ''φ'' is the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

, . The long radius (center to vertex) of the icosidodecahedron is in the

to its edge length; thus its radius is ''φ'' if its edge length is 1, and its edge length is if its radius is 1. Only a few uniform polytopes have this property, including the four-dimensional

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 " ...

, the three-dimensional icosidodecahedron, and the two-dimensional

. (The icosidodecahedron is the equatorial cross section of the 600-cell, and the decagon is the equatorial cross section of the icosidodecahedron.) These radially golden polytopes can be constructed, with their radii, from

golden triangle Golden Triangle may refer to: Places Asia * Golden Triangle (Southeast Asia), named for its opium production * Golden Triangle (Yangtze), China, named for its rapid economic development * Golden Triangle (India), comprising the popular tourist ...

s which meet at the center, each contributing two radii and an edge.

Orthogonal projections

The icosidodecahedron has four special

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...

s, centered on a vertex, an edge, a triangular face, and a pentagonal face. The last two correspond to the A₂ and H₂

Coxeter plane In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...

Surface area and volume

The surface area ''A'' and the volume ''V'' of the icosidodecahedron of edge length ''a'' are: :

\begin
A &= \left(5\sqrt+3\sqrt\sqrt\right) a^2 &&= \left(5\sqrt+3\sqrt\right) a^2 &&\approx 29.3059828a^2 \\
V &= \frac a^3 &&= \frac a^3 &&\approx 13.8355259a^3.
\end

Spherical tiling

The icosidodecahedron can also be represented as a

spherical tiling In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most c ...

, and projected onto the plane via a

stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...

. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

Related polytopes

The icosidodecahedron is a rectified

and also a rectified

, existing as the full-edge truncation between these regular solids. The icosidodecahedron contains 12 pentagons of the

and 20 triangles of the

: The icosidodecahedron exists in a sequence of symmetries of quasiregular polyhedra and tilings with

vertex configuration In geometry, a vertex configurationCrystallography ...

s (3.''n'')², progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With

orbifold notation In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advanta ...

symmetry of *''n''32 all of these tilings are

wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

within a fundamental domain of symmetry, with generator points at the right angle corner of the domain.

Dissection

The icosidodecahedron is related to the Johnson solid called a

created by two

e connected as mirror images. The ''icosidodecahedron'' can therefore be called a ''pentagonal gyrobirotunda'' with the gyration between top and bottom halves.

Related polyhedra

The

truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edg ...

can be turned into an icosidodecahedron by dividing the octagons into two pentagons and two triangles. It has

pyritohedral symmetry image:tetrahedron.jpg, 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that c ...

. Eight

uniform star polyhedra In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, ...

share 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 ...

. Of these, two also share 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 ...

: the small icosihemidodecahedron (having the triangular faces in common), and the

small dodecahemidodecahedron In geometry, the small dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as . It has 18 faces (12 Pentagon, pentagons and 6 Decagon, decagons), 60 Edge (geometry), edges, and 30 Vertex (geometry), vertices. Its vertex figure alte ...

(having the pentagonal faces in common). The vertex arrangement is also shared with the compounds of five octahedra and of five tetrahemihexahedra.

Related polychora

In four-dimensional geometry the icosidodecahedron appears in the regular

as the equatorial slice that belongs to the vertex-first passage of the 600-cell through 3D space. In other words: the 30 vertices of the 600-cell which lie at arc distances of 90 degrees on its circumscribed

hypersphere In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, cal ...

from a pair of opposite vertices, are the vertices of an icosidodecahedron. The wire frame figure of the 600-cell consists of 72 flat regular decagons. Six of these are the equatorial decagons to a pair of opposite vertices. They are precisely the six decagons which form the wire frame figure of the icosidodecahedron.

Icosidodecahedral graph

In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

, a icosidodecahedral graph is the graph of vertices and edges of the icosidodecahedron, one of the

s. It has 30 vertices and 60 edges, and is a

quartic graph In the mathematical field of graph theory, a quartic graph is a graph where all vertices have degree 4. In other words, a quartic graph is a 4-regular graph. Examples Several well-known graphs are quartic. They include: *The complete graph ''K'' ...

Archimedean graph.

Trivia

In Star Trek Universe, the Vulcan game of logic Kal-Toh has the goal to create a

holographic Holography is a technique that enables a wavefront to be recorded and later re-constructed. Holography is best known as a method of generating real three-dimensional images, but it also has a wide range of other applications. In principle, i ...

icosidodecahedron. In ''The Wrong Stars'', book one of the Axiom series, by Tim Pratt, Elena has an icosidodecahedron machine on either side of her. aperback p 336 The

Hoberman sphere A Hoberman sphere is an isokinetic structure patented by Chuck Hoberman that resembles a geodesic dome, but is capable of folding down to a fraction of its normal size by the scissor-like action of its joints. Colorful plastic versions have becom ...

is an icosidodecahedron. Icosidodecahedra can be found in all eukaryotic cells, including human cells, as Sec13/31

COPII The Coat Protein Complex II, or COPII, is a group of proteins that facilitate the formation of vesicles to transport proteins from the endoplasmic reticulum to the Golgi apparatus or endoplasmic-reticulum–Golgi intermediate compartment. This ...

coat-protein formations.

Notes

References

* (Section 3-9) *

External links

* *
Editable printable net of an icosidodecahedron with interactive 3D viewThe Uniform Polyhedra
The Encyclopedia of Polyhedra {{Polyhedron navigator Archimedean solids Quasiregular polyhedra