The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced

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 ...

, is a

convex polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

with 30 rhombic

faces The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect the ...

. It has 60 edges and 32 vertices of two types. It is 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 ...

, and the

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 ...

of the

icosidodecahedron In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (''icosi-'') triangular faces and twelve (''dodeca-'') pentagonal faces. An icosidodecahedron has 30 identical Vertex (geometry), vertices, with two triang ...

. It is a

zonohedron In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...

and can be seen as a elongated

rhombic icosahedron The rhombic icosahedron is a polyhedron shaped like an Oblate spheroid, oblate sphere. Its 20 faces are Congruence (geometry), congruent golden rhombi; 3, 4, or 5 faces meet at each vertex. It has 5 faces (green on top figure) meeting at each of ...

The ratio of the long diagonal to the short diagonal of each face is exactly equal to 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 ...

, , so that the acute angles on each face measure , or approximately 63.43°. A rhombus so obtained is called a ''

golden rhombus In geometry, a golden rhombus is a rhombus whose diagonals are in the golden ratio: : = \varphi = \approx 1.618~034 Equivalently, it is the Varignon parallelogram formed from the edge midpoints of a golden rectangle. Rhombi with this shape f ...

''. Being the dual of an

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 ...

, the rhombic triacontahedron is ''

face-transitive 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 Face (geometry), faces are the same. More specifically, all faces must be not ...

'', meaning the

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

of the solid acts transitively on the set of faces. This means that for any two faces, and , there is a

rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...

or reflection of the solid that leaves it occupying the same region of space while moving face to face . The rhombic triacontahedron is somewhat special in being one of the nine

edge-transitive In geometry, a polytope (for example, a polygon or a polyhedron) or a Tessellation, tiling is isotoxal () or edge-transitive if its Symmetry, symmetries act Transitive group action, transitively on its Edge (geometry), edges. Informally, this mea ...

convex polyhedra, the others being the five

Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...

s, the

cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a tr ...

, the

, and the

rhombic dodecahedron In geometry, the rhombic dodecahedron is a Polyhedron#Convex_polyhedra, convex polyhedron with 12 congruence (geometry), congruent rhombus, rhombic face (geometry), faces. It has 24 edge (geometry), edges, and 14 vertex (geometry), vertices of 2 ...

. The rhombic triacontahedron is also interesting in that its vertices include the arrangement of four Platonic solids. It contains ten

tetrahedra In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

, five

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

s, 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 ...

and 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 ...

. The centers of the faces contain five

octahedra In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...

. It can be made from a

truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...

by dividing the hexagonal faces into three rhombi: Rhombic triacontahedron in truncated octahedron

Cartesian coordinates

Let be the

. The 12 points given by and cyclic permutations of these coordinates are the vertices of a

regular icosahedron The regular icosahedron (or simply ''icosahedron'') is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with Regular polygon, regular faces to each of its pentagonal faces, or by putting ...

. Its dual

regular dodecahedron A regular dodecahedron or pentagonal dodecahedronStrictly speaking, a pentagonal dodecahedron need not be composed of regular pentagons. The name "pentagonal dodecahedron" therefore covers a wider class of solids than just the Platonic solid, the ...

, 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 .

Dimensions

If the edge length of a rhombic triacontahedron is , surface area, volume, the

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

of an

inscribed sphere image:Circumcentre.svg, An inscribed triangle of a circle In geometry, an inscribed plane (geometry), planar shape or solid (geometry), solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figu ...

(

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

to each of the rhombic triacontahedron's faces) and midradius, which touches the middle of each edge are: :

r_\mathrm &= \left(1+\frac\right)\,a &&\approx 1.44721 a \end

where is the

. 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.

Dissection

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

Orthogonal projections

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 A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of two-dimensional space, the plane by non-overlapping polygons or other shapes, and a tiling is ''aperiodic'' if it does not contain arbitrarily large Perio ...

Stellations

Construction of Rhombic hexecontahedron from Rhombic Triacontahedron

The rhombic triacontahedron has 227 fully supported stellations. One of the stellations of the rhombic triacontahedron is the

compound of five cubes The compound of five cubes is one of the five regular polyhedral compounds. It was first described by Edmund Hess in 1876. Its vertices are those of a regular dodecahedron. Its edges form pentagrams, which are the stellations of the pentag ...

. The total number of stellations of the rhombic triacontahedron is .

Related polyhedra

This polyhedron is a part of a sequence of rhombic polyhedra and tilings with

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...

symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are also rectangles. File:Spherical rhombic triacontahedron.png, Spherical rhombic triacontahedron File:Rhombic tricontahedron cube tetrahedron.png, A rhombic triacontahedron with an inscribed

tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

(red) and

(yellow).
(Click here for rotating model) File:Rhombic tricontahedron icosahedron dodecahedron.png, A rhombic triacontahedron with an inscribed

(blue) and

(purple).
(Click here for rotating model) File:StellaTruncRhombicTriaconta.png, Fully truncated rhombic triacontahedron

Uses

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

Woodworker Jane Kostick builds boxes in the shape of a rhombic triacontahedron.triacontahedron box - KO Sticks LLC
/ref> 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.

References

* (Section 3-9) * (The thirteen semiregular convex polyhedra and their duals, p. 22, Rhombic triacontahedron) * ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss,

(Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, p. 285, Rhombic triacontahedron )

The Wolfram Demonstrations Project The Wolfram Demonstrations Project is an open-source collection of interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown to over 10,000. The site won a Pa ...

A viper drawn on a rhombic triacontahedron
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