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 o ...

with twenty (''icosi'') triangular 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 sim ...

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 compose ...

s and more particularly, a quasiregular polyhedron.

Geometry

An icosidodecahedron has icosahedral symmetry, and its 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 specif ...

is the compound of a dodecahedron 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 oth ...

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 rotundae, which belong among the

Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnso ...

s. 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 al ...

, one of the

s). 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 decagons. 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 geome ...

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 ...

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 A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...

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 o ...

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, the three-dimensional icosidodecahedron, and the two-dimensional decagon. (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 projections, centered on a vertex, an edge, a triangular face, and a pentagonal face. The last two correspond to the A₂ and H₂ Coxeter planes.

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, 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 (the ''projection plane'') perpendicular to the diameter thro ...

. 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 dodecahedron and also a rectified

, existing as the full-edge truncation between these regular solids. The icosidodecahedron contains 12 pentagons of the dodecahedron and 20 triangles of the

: The icosidodecahedron exists in a sequence of symmetries of quasiregular polyhedra and tilings with vertex configurations (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 adva ...

symmetry of *''n''32 all of these tilings are wythoff construction within a

fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...

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

Dissection

The icosidodecahedron is related to the

called a

created by two pentagonal rotundae 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 can be turned into an icosidodecahedron by dividing the octagons into two pentagons and two triangles. It has pyritohedral symmetry. 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 figure ...

share the same vertex arrangement. Of these, two also share the same edge arrangement: the small icosihemidodecahedron (having the triangular faces in common), and the small dodecahemidodecahedron (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 600-cell 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 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 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 conn ...

, 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 Archimedean graph.

Trivia

Star Trek Universe ''Star Trek'' is an American science fiction media franchise created by Gene Roddenberry, which began with the eponymous 1960s television series and quickly became a worldwide pop-culture phenomenon. The franchise has expanded into vario ...

, the Vulcan game of logic Kal-Toh has the goal to create a holographic 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 bec ...

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