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, indexed as U₃₆. It is the

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 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 penta ...

(and that of its dual, the small stellated dodecahedron). It was discovered independently by , and . The edges of this model form 10 central

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

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 geometry ...

s. These 10, along with the great circles from projections of two other polyhedra, form the

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 ...

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 ...

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 defin ...

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 mor ...

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 Rhombic may refer to: * Rhombus, a quadrilateral whose four sides all have the same length (often called a diamond) *Rhombic antenna, a broadband directional antenna most commonly used on shortwave frequencies * polyhedra formed from rhombuses, suc ...

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 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 ...

. It also shares its edge arrangement with the small dodecahemicosahedron (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 Pentagon, pentagons and 10 Hexagon, hexagons), 60 edges, and 30 vertices. Its vertex figure is a q ...

(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 small stellated dodecahedron 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 pentag ...

on the surface, but it has 24 faces: 12 pentagons 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), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party in Germany to descr ...

polyhedron (that looks like a small rhombidodecahedron with polygons filling up the dodecahedral set of holes), but it has a uniform quasitruncation, the truncated dodecadodecahedron. It is topologically equivalent to a

quotient space Quotient space may refer to a quotient set when the sets under consideration are considered as spaces. In particular: *Quotient space (topology), in case of topological spaces * Quotient space (linear algebra), in case of vector spaces *Quotient ...

of the hyperbolic order-4 pentagonal tiling, 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 aro ...

s back into regular

pentagon 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 ...

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 equiv ...

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

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 ...

. It is the

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

of the dodecadodecahedron. It has 30 intersecting

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 Catal ...

, 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 hyperbolic order-5 square tiling, by distorting the rhombi into squares. 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 order-4 pentagonal tiling, and a quotient space of the order-4 pentagonal tiling is topologically equivalent to the dual of the medial rhombic triacontahedron, the dodecadodecahedron.

References

* * * *

External links

* *
Uniform polyhedra and duals
{{Nonconvex polyhedron navigator Uniform polyhedra