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 ...

, the dodecadodecahedron is a

nonconvex uniform polyhedron 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, ...

, indexed as U₃₆. It is the rectification of the

great dodecahedron In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagon ...

(and that of its dual, the

small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeti ...

). It was discovered independently by , and . The edges of this model form 10 central

hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...

s, and these, projected onto a

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

, become 10

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 ...

s. These 10, along with the great circles from projections of two other polyhedra, form the 31 great circles of the spherical icosahedron used in construction of

geodesic dome A geodesic dome is a hemispherical thin-shell structure (lattice-shell) based on a geodesic polyhedron. The triangular elements of the dome are structurally rigid and distribute the structural stress throughout the structure, making geodesic dom ...

Wythoff constructions

It has four

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 ...

s between four

Schwarz triangle In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in . These can be defined mor ...

families: 2 , 5 5/2, 2 , 5 5/3, 2 , 5/2 5/4, 2 , 5/3 5/4, but represent identical results. Similarly it can be given four extended

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

s: r, r, r, and r or as Coxeter-Dynkin diagrams: , , , and .

Net

A shape with the same exterior appearance as the dodecadodecahedron can be constructed by folding up these nets: Dodecadodecahedron net

12 pentagrams and 20 rhombic clusters are necessary. However, this construction replaces the crossing pentagonal faces of the dodecadodecahedron with non-crossing sets of rhombi, so it does not produce the same internal structure.

Related polyhedra

Its

convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

is 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 ...

. It also shares its

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 ...

with the

small dodecahemicosahedron In geometry, the small dodecahemicosahedron (or great dodecahemiicosahedron) is a nonconvex uniform polyhedron, indexed as U62. It has 22 faces (12 pentagrams and 10 hexagons), 60 edges, and 30 vertices. Its vertex figure is a crossed quadrilate ...

(having the pentagrammic faces in common), and with the

great dodecahemicosahedron In geometry, the great dodecahemicosahedron (or small dodecahemiicosahedron) is a nonconvex uniform polyhedron, indexed as U65. It has 22 faces (12 pentagons and 10 hexagons), 60 edges, and 30 vertices. Its vertex figure is a crossed quadrilater ...

(having the pentagonal faces in common). Small stellated dodecahedron truncations

This polyhedron can be considered a rectified

. It is center of a truncation sequence between a

and

: The truncated small stellated dodecahedron looks like 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 ...

on the surface, but it has 24 faces: 12

pentagons In geometry, a pentagon (from the Greek language, 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 polygon, simple pentagon is ...

from the truncated vertices and 12 overlapping as (truncated pentagrams). The truncation of the dodecadodecahedron itself is not uniform and attempting to make it uniform results in a

degenerate Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party i ...

polyhedron (that looks like a

small rhombidodecahedron In geometry, the small rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U39. It has 42 faces (30 squares and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral. Related polyhedra It shares ...

with polygons filling up the dodecahedral set of holes), but it has a uniform quasitruncation, the

truncated dodecadodecahedron In geometry, the truncated dodecadodecahedron (or stellatruncated dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U59. It is given a Schläfli symbol t0,1,2. It has 54 faces (30 squares, 12 decagons, and 12 decagrams), 180 edges ...

. It is topologically equivalent to a quotient space of the

hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

order-4 pentagonal tiling In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of . It can also be called a pentapentagonal tiling in a bicolored quasiregular form. Symmetry This tiling represents a hyperbolic ...

, by distorting the

pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle aroun ...

s back into regular

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 ...

s. As such, it is topologically a

regular polyhedron A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equival ...

of index two: Uniform tiling 552-t1

The colours in the above image correspond to the red pentagrams and yellow pentagons of the dodecadodecahedron at the top of this article.

Medial rhombic triacontahedron

The medial rhombic triacontahedron is a nonconvex

isohedral In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...

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 ...

. It is the dual of the dodecadodecahedron. It has 30 intersecting rhombic faces. It can also be called the small stellated triacontahedron.

Stellation

The ''medial rhombic triacontahedron'' is a

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 ...

of 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 Cata ...

, which is the dual of the icosidodecahedron, the convex hull of the dodecadodecahedron (dual to the original medial rhombic triacontahedron).

Related hyperbolic tiling

It is topologically equivalent to a quotient space of the

order-5 square tiling In geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of . Related polyhedra and tiling This tiling is topologically related as a part of sequence of regular polyhedra and tilings with ver ...

, by distorting the rhombi into

squares In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

. As such, it is topologically a

of index two:The Regular Polyhedra (of index two)
David A. Richter Uniform tiling 45-t0

Note that the order-5 square tiling is dual to the

, and a quotient space of the order-4 pentagonal tiling is topologically equivalent to the dual of the medial rhombic triacontahedron, the dodecadodecahedron.

References

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External links

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Uniform polyhedra and duals
{{Nonconvex polyhedron navigator Uniform polyhedra