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 A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional ...

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 The pentagonal rotunda is a convex polyhedron with regular polygonal faces. These faces comprise ten equilateral triangles, six regular pentagons, and one regular decagon, making a total of seventeen. The pentagonal rotunda is an example of Johns ...

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 In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two solids with such ...

, the

pentagonal orthobirotunda In geometry, the pentagonal orthobirotunda is a polyhedron constructed by attaching two pentagonal rotundae along their decagonal faces, matching like faces. It is an example of Johnson solid. Construction The pentagonal orthobirotunda is con ...

. 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 Rectification has the following technical meanings: Mathematics * Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points * Rectifiable curve, in mathematics * Recti ...

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 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:

(\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 In geometry, stellation is the process of extending a polygon in two dimensions, a 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 ...

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 (; ) 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

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°. Regular decagon A '' regular decagon'' has a ...

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. 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 In geometry, the 31 great circles of the spherical icosahedron is an arrangement of 31 great circles in icosahedral symmetry. It was first identified by Buckminster Fuller and is used in construction of geodesic domes. Construction The 31 great ...

. 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

. (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 sp ...

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

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

pyritohedral symmetry 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 combine a reflection a ...

. 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, equa ...

. 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 Small means of insignificant size. Small may also refer to: Science and technology * SMALL, an ALGOL-like programming language * ''Small'' (journal), a nano-science publication * <small>, an HTML element that defines smaller text Arts and ...

(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 Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...

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 In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension to simplicial complexes is a canonical method to refining them. Therefore, the barycentric subdivision is an important tool ...

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 graph In the mathematics, mathematical field of graph theory, an Archimedean graph is a Graph (discrete mathematics), graph that forms the skeleton of one of the Archimedean solids. There are 13 Archimedean graphs, and all of them are regular graph, regu ...

s. 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 A Hoberman sphere is a kinetic 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 become ...

. 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 ''Star Trek'' is an American science fiction media franchise created by Gene Roddenberry, which began with the series of the same name and became a worldwide pop-culture phenomenon. Since its creation, the franchise has expanded into var ...

, 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