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 great dirhombicosidodecahedron (or great snub disicosidisdodecahedron) 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 last as . It has 124 faces (40

triangles A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear ...

, 60 squares, and 24 pentagrams), 240 edges, and 60 vertices. This is the only non-degenerate uniform polyhedron with more than six faces meeting at a vertex. Each vertex has 4 squares which pass through the vertex central axis (and thus through the centre of the figure), alternating with two triangles and two pentagrams. Another unusual feature is that the faces all occur in coplanar pairs. This is also the only uniform polyhedron that cannot be made by the Wythoff construction from a spherical triangle. It has a special Wythoff symbol relating it to a spherical

quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

. This symbol suggests that it is a sort of

snub polyhedron In geometry, a snub polyhedron is a polyhedron obtained by performing a snub operation: alternating a corresponding omnitruncated or truncated polyhedron, depending on the definition. Some, but not all, authors include antiprisms as snub poly ...

, except that instead of the non-snub faces being surrounded by snub triangles as in most snub polyhedra, they are surrounded by snub squares. It has been nicknamed "Miller's monster" (after J. C. P. Miller, who with

H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

and M. S. Longuet-Higgins enumerated the uniform polyhedra in 1954).

Related polyhedra

If the definition of a uniform polyhedron is relaxed to allow any even number of faces adjacent to an edge, then this definition gives rise to one further polyhedron: the

great disnub dirhombidodecahedron In geometry, the great disnub dirhombidodecahedron, also called ''Skilling's figure'', is a degenerate uniform star polyhedron. It was proven in 1970 that there are only 75 uniform polyhedra other than the infinite families of prisms and antipri ...

which has the same vertices and edges but with a different arrangement of triangular faces. The vertices and edges are also shared with the uniform compounds of 20 octahedra or 20 tetrahemihexahedra. 180 of the 240 edges are shared with the

great snub dodecicosidodecahedron In geometry, the great snub dodecicosidodecahedron (or great snub dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U64. It has 104 faces (80 triangles and 24 pentagrams), 180 edges, and 60 vertices. It has Coxeter diagram, . ...

. This polyhedron is related to the

nonconvex great rhombicosidodecahedron In geometry, the nonconvex great rhombicosidodecahedron is a nonconvex uniform polyhedron, indexed as U67. It has 62 faces (20 triangles, 30 squares and 12 pentagrams), 120 edges, and 60 vertices. It is also called the quasirhombicosidodecahedron ...

(quasirhombicosidodecahedron) by a branched cover: there is a function from the great dirhombicosidodecahedron to the quasirhombicosidodecahedron that is 2-to-1 everywhere, except for the vertices.

Cartesian coordinates

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...

for the vertices of a great dirhombicosidodecahedron are all the even permutations of :

\left(0, \pm\frac, \pm\frac\right),\left(\pm\left(-1+\frac\right), \pm\left(\frac-\frac\right), \pm\left(\frac+\sqrt\right)\right),

\left(\pm\left(\frac+\sqrt\right), \pm\left(-1-\frac\right), \pm\left(\frac+\frac\right)\right),

where τ = (1+)/2 is the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

(sometimes written φ). These vertices result in an edge length of 2.

Gallery

References

* * * Har'El, Z
''Uniform Solution for Uniform Polyhedra.''
Geometriae Dedicata 47, 57-110, 1993
Zvi Har’ElKaleido software
*
Mäder, R. E.

Uniform Polyhedra.
' Mathematica J. 3, 48-57, 1993. *

External links

* {{mathworld , urlname = GreatDirhombicosidodecahedron, title =Great dirhombicosidodecahedron * http://www.mathconsult.ch/showroom/unipoly/75.html * http://www.software3d.com/MillersMonster.php Uniform polyhedra