Rhombic triacontahedron

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Rhombic triacontahedron
(Click here for rotating model)
Type Catalan solid
Coxeter diagram CDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png
Conway notation jD
Face type V3.5.3.5
DU24 facets.png

Faces 30
Edges 60
Vertices 32
Vertices by type 20{3}+12{5}
Symmetry group Ih, H3, [5,3], (*532)
Rotation group I, [5,3]+, (532)
Dihedral angle 144°
Properties convex, face-transitive isohedral, isotoxal, zonohedron
(dual polyhedron)
Rhombic triacontahedron Net
3D model of a 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 Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.

A face of the rhombic triacontahedron. The lengths
of the diagonals are in the golden ratio.
This animation shows a transformation from a cube to a rhombic triacontahedron by dividing the square faces into 4 squares and splitting middle edges into new rhombic faces.

The ratio of the long diagonal to the short diagonal of each face is exactly equal to the golden ratio, φ, so that the acute angles on each face measure 2 tan−1(1/φ) = tan−1(2), or approximately 63.43°. A rhombus so obtained is called a golden rhombus.

Being the dual of an Archimedean solid, the rhombic triacontahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces. This means that for any two faces, A and B, there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B.

The rhombic triacontahedron is somewhat special in being one of the nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron, and the rhombic dodecahedron.

The rhombic triacontahedron is also interesting in that its vertices include the arrangement of four Platonic solids. It contains ten tetrahedra, five cubes, an icosahedron and a dodecahedron. The centers of the faces contain five octahedra.

It can be made from a truncated octahedron by dividing the hexagonal faces into 3 rhombi:

A topological rhombic triacontahedron in truncated octahedron

Cartesian coordinates[edit]

Let be the golden ratio. The 12 points given by and cyclic permutations of these coordinates are the vertices of a regular icosahedron. Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the 8 points together with the 12 points and cyclic permutations of these coordinates. All 32 points together are the vertices of a rhombic triacontahedron centered at the origin. The length of its edges is . Its faces have diagonals with lengths and .


If the edge length of a rhombic triacontahedron is a, surface area, volume, the radius of an inscribed sphere (tangent to each of the rhombic triacontahedron's faces) and midradius, which touches the middle of each edge are:[1]

where φ is the golden ratio.

The insphere is tangent to the faces at their face centroids. Short diagonals belong only to the edges of the inscribed regular dodecahedron, while long diagonals are included only in edges of the inscribed icosahedron.


The rhombic triacontahedron can be dissected into 20 golden rhombohedra: 10 acute ones and 10 obtuse ones.[2][3]

10 10
Acute golden rhombohedron.png
Acute form
Flat golden rhombohedron.png
Obtuse form

Orthogonal projections[edit]

The rhombic triacontahedron has four symmetry positions, two centered on vertices, one mid-face, and one mid-edge. Embedded in projection "10" are the "fat" rhombus and "skinny" rhombus which tile together to produce the non-periodic tessellation often referred to as Penrose tiling.

Orthogonal projections
[2] [2] [6] [10]
Image Dual dodecahedron t1 v.png Dual dodecahedron t1 e.png Dual dodecahedron t1 A2.png Dual dodecahedron t1 H3.png
Dodecahedron t1 v.png Dodecahedron t1 e.png Dodecahedron t1 A2.png Dodecahedron t1 H3.png


Rhombic hexecontahedron
An example of stellations of the rhombic triacontahedron.

The rhombic triacontahedron has 227 fully supported stellations.[4][5] Another stellation of the Rhombic triacontahedron is the compound of five cubes. The total number of stellations of the rhombic triacontahedron is 358,833,097.

Related polyhedra[edit]

Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
Uniform polyhedron-53-t0.svg Uniform polyhedron-53-t01.svg Uniform polyhedron-53-t1.svg Uniform polyhedron-53-t12.svg Uniform polyhedron-53-t2.svg Uniform polyhedron-53-t02.png Uniform polyhedron-53-t012.png Uniform polyhedron-53-s012.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
{5,3} t{5,3} r{5,3} t{3,5} {3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
Icosahedron.jpg Triakisicosahedron.jpg Rhombictriacontahedron.jpg Pentakisdodecahedron.jpg Dodecahedron.jpg Deltoidalhexecontahedron.jpg Disdyakistriacontahedron.jpg Pentagonalhexecontahedronccw.jpg
V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3. V3.4.5.4 V4.6.10 V3.

This polyhedron is a part of a sequence of rhombic polyhedra and tilings with [n,3] Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are also rectangles.

Symmetry mutations of dual quasiregular tilings: V(3.n)2
*n32 Spherical Euclidean Hyperbolic
*332 *432 *532 *632 *732 *832... *∞32
Tiling Uniform tiling 432-t0.png Spherical rhombic dodecahedron.png Spherical rhombic triacontahedron.png Rhombic star tiling.png 7-3 rhombille tiling.svg H2-8-3-rhombic.svg Ord3infin qreg rhombic til.png
Conf. V(3.3)2 V(3.4)2 V(3.5)2 V(3.6)2 V(3.7)2 V(3.8)2 V(3.∞)2


The rhombic triacontahedron forms a 32 vertex convex hull of one projection of a 6-cube to three dimensions.

The 3D basis vectors [u,v,w] are:
u = (1, φ, 0, -1, φ, 0)
v = (φ, 0, 1, φ, 0, -1)
w = (0, 1, φ, 0, -1, φ)
Shown with inner edges hidden
20 of 32 interior vertices form a dodecahedron, and the remaining 12 form an icosahedron.


An example of the use of a rhombic triacontahedron in the design of a lamp

Danish designer Holger Strøm used the rhombic triacontahedron as a basis for the design of his buildable lamp IQ-light (IQ for "Interlocking Quadrilaterals").

STL model of a rhombic triacontahedral box made of six panels around a cubic hole – zoom into the model to see the hole from the inside

Woodworker Jane Kostick builds boxes in the shape of a rhombic triacontahedron.[6] The simple construction is based on the less than obvious relationship between the rhombic triacontahedron and the cube.

Roger von Oech's "Ball of Whacks" comes in the shape of a rhombic triacontahedron.

The rhombic triacontahedron is used as the "d30" thirty-sided die, sometimes useful in some roleplaying games or other places.

Christopher Bird, co-author of The Secret Life of Plants wrote an article for New Age Journal in May, 1975, popularizing the dual icosahedron and dodecahedron as "the crystalline structure of the Earth," a model of the "Earth (telluric) energy Grid." The EarthStar Globe by Bill Becker and Bethe A. Hagens purports to show "the natural geometry of the Earth, and the geometric relationship between sacred places such as the Great Pyramid, the Bermuda Triangle, and Easter Island." It is printed as a rhombic triacontahedron, on 30 diamonds, and folds up into a globe.[7]

See also[edit]


  1. ^ Stephen Wolfram, "[1]" from Wolfram Alpha. Retrieved 7 January 2013.
  2. ^ "How to make golden rhombohedra out of paper".
  3. ^ Dissection of the rhombic triacontahedron
  4. ^ Pawley, G. S. (1975). "The 227 triacontahedra". Geometriae Dedicata. Kluwer Academic Publishers. 4 (2–4): 221–232. doi:10.1007/BF00148756. ISSN 1572-9168. S2CID 123506315.
  5. ^ Messer, P. W. (1995). "Stellations of the rhombic triacontahedron and Beyond". Structural Topology. 21: 25–46.
  6. ^ triacontahedron box - KO Sticks LLC
  7. ^ "History of the World Grid Theory".

External links[edit]