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 ...

, a rhombic hexecontahedron is a

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 ...

of the

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 ...

. It is nonconvex with 60 golden rhombic faces with

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 ...

. It was described mathematically in 1940 by Helmut Unkelbach. It is topologically identical to the convex

deltoidal hexecontahedron In geometry, a deltoidal hexecontahedron (also sometimes called a ''trapezoidal hexecontahedron'', a ''strombic hexecontahedron'', or a ''tetragonal hexacontahedron'') is a Catalan solid which is the dual polyhedron of the rhombicosidodecahedron, ...

which has

kite A kite is a tethered heavier than air flight, heavier-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. Kites often have ...

faces.

Dissection

The rhombic hexecontahedron can be dissected into 20 acute golden rhombohedra meeting at a central point. This gives the volume of a hexecontahedron of side length ''a'' to be

V = (10 + 2\sqrt 5)a^3

and the area to be

A = (24\sqrt 5)a^2

. :

Construction

A rhombic hexecontahedron can be constructed from a

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 ...

, by taking its vertices, its face centers and its edge centers and scaling them in or out from the body center to different extents. Thus, if the 20 vertices of a dodecahedron are pulled out to increase the circumradius by a factor of ≈ 1.309, the 12 face centers are pushed in to decrease the

inradius In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

to ≈ 0.691 of its original value, and the 30 edge centers are left unchanged, then a rhombic hexecontahedron is formed. (The circumradius is increased by 30.9% and the inradius is decreased by the same 30.9%.) Scaling the points by different amounts results in hexecontahedra with kite-shaped faces or other polyhedra. Every golden rhombic face has a face center, a vertex, and two edge centers of the original dodecahedron, with the edge centers forming the short diagonal. Each edge center is connected to two vertices and two face centers. Each face center is connected to five edge centers, and each vertex is connected to three edge centers.

Stellation

The rhombic hexecontahedron is one of 227 self-supporting stellations of the rhombic triacontahedron. Its

stellation diagram In geometry, a stellation diagram or stellation pattern is a two-dimensional diagram in the plane of some face of a polyhedron, showing lines where other face planes intersect with this one. The lines cause 2D space to be divided up into regions ...

looks like this, with the original rhombic triacontahedron faces as the central

rhombus In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...

. :

Rhombic hexecontahedron stellation diagram

Related polyhedra

The

great rhombic triacontahedron In geometry, the great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron. It is the dual of the great icosidodecahedron (U54). Like the convex rhombic triacontahedron it has 30 rhombic faces, 60 edges and 32 vertices (also ...

contains the 30 larger intersecting rhombic faces: : DU54 great rhombic triacontahedron

In popular culture

In Brazilian culture, handcrafted rhombic hexecontahedra used to be made from colored fabric and cardboard, called ("world turners" in Portuguese) or happiness stars, sewn by mothers and given as wedding gifts to their daughters. The custom got lost with the urbanization of

Brazil Brazil, officially the Federative Republic of Brazil, is the largest country in South America. It is the world's List of countries and dependencies by area, fifth-largest country by area and the List of countries and dependencies by population ...

, though the technique for producing the handicrafts was still taught in Brazilian rural schools up until the first half of the twentieth century. The logo of the

WolframAlpha WolframAlpha ( ) is an answer engine developed by Wolfram Research. It is offered as an online service that answers factual queries by computing answers from externally sourced data. History Launch preparations for WolframAlpha began on Ma ...

website is a red rhombic hexecontahedron and was inspired by the logo of the related

Mathematica Wolfram (previously known as Mathematica and Wolfram Mathematica) is a software system with built-in libraries for several areas of technical computing that allows machine learning, statistics, symbolic computation, data manipulation, network ...

software.

References

Bibliography

* Unkelbach, H. "Die kantensymmetrischen, gleichkantigen Polyeder". Deutsche Math. 5, 306-316, 1940. * * * * Grünbaum, B. ''Parallelogram-Faced Isohedra with Edges in Mirror-Planes.'' Discrete Math. 221, 93–100, 2000.

External links

* {{MathWorld, title=Rhombic hexecontahedron, urlname=RhombicHexecontahedron * http://www.georgehart.com/virtual-polyhedra/zonohedra-info.html
The Bilinski dodecahedron, and assorted parallelohedra, zonohedra, monohedra, isozonohedra and otherhedra.

Branko Grünbaum Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentZonohedra Polyhedral stellation