Great dirhombicosidodecahedron
From Wikipedia, the free encyclopedia
| Great dirhombicosidodecahedron | |
|---|---|
 |
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| Type | Uniform polyhedron |
| Elements | F=124, E=240, V=60 (χ=-56) |
| Faces by sides | 40{3}+60{4}+24{5/2} |
| Wythoff symbol | |3/2 5/3 3 5/2 |
| Symmetry group | Ih |
| Index references | U75, C92, W119 |
 (4.5/3.4.3.4. 5/2.4.3/2)/2 (Vertex figure) |
 Great dirhombicosadodecacron (dual polyhedron) |
In geometry, the great dirhombicosidodecahedron is a nonconvex uniform polyhedron, indexed last as U75.
This is the only 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.
This is also the only uniform polyhedron that cannot be made by Wythoff construction. It has a special Wythoff symbol | 3/2 5/3 3 5/2.
It has been nicknamed "Miller's monster" (after J.C.P. Miller, who with H. S. M. Coxeter and M. S. Longuet-Higgins enumerated the uniform polyhedra in 1954).
If the definition of a uniform polyhedron is relaxed to allow an even number of faces adjacent to an edge then this polyhedron gives rise to one further polyhedron the Great disnub dirhombidodecahedron 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 tetrahemihexahedra. 180 of the edges are shared with the great snub dodecicosidodecahedron.
The shape is also significant mathematically. At a 1972 Pasedena mathematics conference Dr. Steven McHarty, a professor of mathematics at Princeton showed using number theory that the Wythoff Conversion allows for inverse magnitutes of infinite series. This means that there are more points along any edge than are contained within the surface area of the shape, and more points in the surface area than are contained within the volume of the object.
[edit] Cartesian coordinates
Cartesian coordinates for the vertices of a great dirhombicosidodecahedron are all the even permutations of
- (0, ±2/τ, ±2/√τ)
- (±(−1+1/√τ3), ±(1/τ2−1/√τ), ±(1/τ+√τ))
- (±(−1/τ+√τ), ±(−1−1/√τ3, ±(1/τ2+1/√τ))
where τ = (1+√5)/2 is the golden ratio (sometimes written φ). These vertices result in an edge length of 2√2.
