geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, an icosidodecahedron or pentagonal gyrobirotunda is a

polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

with twenty (''icosi-'') triangular faces and twelve (''dodeca-'')

pentagon In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...

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 The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They ...

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 ...

Construction

One way to construct the icosidodecahedron is to start with two pentagonal rotunda by attaching them to their bases. These rotundas cover their decagonal base so that the resulting polyhedron has 32 faces, 30 vertices, and 60 edges. This construction is similar to one of the Johnson solids, the pentagonal orthobirotunda. The difference is that the icosidodecahedron is constructed by twisting its rotundas by 36°, a process known as

gyration In geometry, a gyration is a rotation in a discrete subgroup of symmetries of the Euclidean plane such that the subgroup does not also contain a reflection symmetry whose axis passes through the center of rotational symmetry. In the orbifold ...

, resulting in the pentagonal face connecting to the triangular one. The icosidodecahedron has an alternative name, ''pentagonal gyrobirotunda''. There is another way to construct it, and that is rectification of an

Icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...

or a

Dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...

. Convenient

Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

for the vertices of an icosidodecahedron with unit edges are given by the even permutations of:

(\pm 1, 0, 0), \qquad \tfrac\left(\pm \varphi, \pm \tfrac, \pm 1 \right),

where

\varphi

denotes the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...

Properties

The surface area of an icosidodecahedron can be determined by calculating the area of all pentagonal faces. The volume of an icosidodecahedron can be determined by slicing it off into two pentagonal rotunda, after which summing up their volumes. Therefore, its surface area and volume can be formulated as:

\begin
A &= \left(5\sqrt+3\sqrt\right) a^2 &\approx 29.306a^2 \\
V &= \fraca^3 &\approx 13.836a^3.
\end

The dihedral angle of an icosidodecahedron between pentagon-to-triangle is

\arccos \left(-\sqrt \right) \approx 142.62^\circ,

determined by calculating the angle of a pentagonal rotunda. An icosidodecahedron 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 polyhedr ...

, and its first stellation is the compound of a

dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...

and its dual

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...

, with the vertices of the icosidodecahedron located at the midpoints of the edges of either. The icosidodecahedron is an

, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex. The polygonal faces that meet for every vertex are two equilateral triangles and two regular pentagons, and the

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

of an icosidodecahedron is

(3 \cdot 5)^2 = 3^2 \cdot 5^2

. 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 ...

rhombic triacontahedron The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombus, rhombic face (geometry), faces. It has 60 edge (geometry), edges and 32 vertex ...

, a

Catalan solid The Catalan solids are the dual polyhedron, dual polyhedra of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric vertices. The faces of the Catalan solids correspond by duality to ...

. 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. Discussion Any arc of a great circle is a geodesic of the sphere, so that great circles in spher ...

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

to its edge length; thus its radius is 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 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 triangles which meet at the center, each contributing two radii and an edge.

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 configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vert ...

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 Horton Conway, John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curv ...

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 ...

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

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 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

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 hypersphere is an - dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional and the sphere 2-dimensional because a point ...

from a pair of opposite vertices, are the vertices of an icosidodecahedron. The wireframe figure of the 600-cell consists of 72 flat regular decagons. Six of these are the equatorial decagons to a pair of opposite vertices, and these six form the wireframe figure of an icosidodecahedron. If a

is stereographically projected to 3-space about any vertex and all points are normalised, the

geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...

s upon which edges fall comprise the icosidodecahedron's barycentric subdivision.

Graph

The

skeleton A skeleton is the structural frame that supports the body of most animals. There are several types of skeletons, including the exoskeleton, which is a rigid outer shell that holds up an organism's shape; the endoskeleton, a rigid internal fra ...

of an icosidodecahedron can be represented as the symmetric

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

with 30 vertices and 60 edges, one of the Archimedean graphs. It is quartic, meaning that each of its vertex is connected by four other vertices.

Applications

The icosidodecahedron may appear in structures, as in the geodesic dome or the Hoberman sphere. 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. Thi ...

coat-protein formations. The icosidodecahedron may also found in popular culture. In Star Trek universe, the Vulcan game of logic Kal-Toh has the goal of creating a shape with two nested

holographic Holography is a technique that allows a wavefront to be recorded and later reconstructed. It is best known as a method of generating three-dimensional images, and has a wide range of other uses, including data storage, microscopy, and interfe ...

icosidodecahedra joined at the midpoints of their segments.

References

External links

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