# Great dirhombicosidodecahedron

(Redirected from Gidrid)
Great dirhombicosidodecahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymGidrid
Coxeter diagrams3s5/2s3/2s5/3*aØ*c *bØ*d
Elements
Faces40 triangles, 60 squares, 24 pentagrams
Edges120+120
Vertices60
Vertex figureIrregular octagon, edge lengths 1, 2, (5–1)/2, 2, 1, 2, (5–1)/2, 2
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt2}{2} ≈ 0.70711}$
Volume0
Dihedral angles5/2–4: ${\displaystyle \arccos\left(\sqrt{\frac{5-2\sqrt5}{5}}\right) ≈ 71.03929^\circ}$
3–4: ${\displaystyle \arccos\left(\frac{\sqrt3}{3}\right) ≈ 54.73561^\circ}$
Central density0
Number of external pieces3000
Level of complexity164
Related polytopes
ArmySemi-uniform srid, edge lengths ${\displaystyle \sqrt{\frac{3-\sqrt5-\sqrt{10\sqrt5-22}}{2}}}$ (pentagons), ${\displaystyle \sqrt{\frac{\sqrt5-1-2\sqrt{\sqrt5-2}}{2}}}$ (triangles)
RegimentGidrid
DualGreat dirhombicosidodecacron
ConjugateGreat dirhombicosidodecahedron
Abstract & topological properties
Flag count960
Euler characteristic–56
OrientableYes
Genus29
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The great dirhombicosidodecahedron, or gidrid, is the most complex uniform hemipolyhedron. It consists of 40 triangles, 60 snub squares, and 24 pentagrams, all of which fall into coinciding planes in pairs. Two triangles, four squares, and two pentagrams meet at each vertex.

## Vertex coordinates

A great dirhombicosidodecahedron of edge length 1 has vertex coordinates given by all even permutations of:

• ${\displaystyle \left(±\sqrt{\frac{\sqrt5-1-2\sqrt{\sqrt5-2}}{8}},\,±\sqrt{\frac{3-\sqrt5-\sqrt{10\sqrt5-22}}{8}},\,±\sqrt{\frac{2+\sqrt{2\sqrt5-2}}{8}}\right),}$
• ${\displaystyle \left(0,\,±\frac{\sqrt{3-\sqrt5}}{2},\,±\frac{\sqrt{\sqrt5-1}}{2}\right),}$
• ${\displaystyle \left(±\sqrt{\frac{3-\sqrt5+\sqrt{10\sqrt5-22}}{8}},\,±\sqrt{\frac{2-\sqrt{2\sqrt5-2}}{8}},\,±\sqrt{\frac{\sqrt5-1+2\sqrt{\sqrt5-2}}{8}}\right).}$

## Related polyhedra

The great dirhombicosidodecahedron has the same edges as two uniform polyhedron compounds: the disnub icosahedron, the compound of 20 octahedra, and the snub pseudosnub rhombicosahedron, the compound of 20 tetrahemihexahedra. It also has the same vertices of the great snub dodecicosidodecahedron, which uses a subset of its edges.

The degenerate great disnub dirhombidodecahedron can be constructed by blending this polyhedron with the disnub icosahedron.