Icosahedron atop dodecahedron

Icosahedron atop dodecahedron
(OFF file)
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Ikadoe
Coxeter diagram	xo5oo3ox&#x
Elements
Cells	20+30 tetrahedra, 12 pentagonal pyramids, 1 icosahedron, 1 dodecahedron
Faces	20+30+30 triangles, 12 pentagons
Edges	30+30+60
Vertices	12+20
Vertex figures	12 pentagonal antiprisms, edge length 1
	20 triangular antipodiums, edge lengths (1+√5)/2 (large base) and 1 (small base and sides)
Measures (edge length 1)
Circumradius	${\frac {1+{\sqrt {5}}}{2}}\approx 1.61803$
Hypervolume	${\frac {135+61{\sqrt {5}}}{48}}\approx 5.65417$
Dichoral angles	Tet–3–tet: $\arccos \left(-{\frac {1+3{\sqrt {5}}}{8}}\right)\approx 164.47751^{\circ }$
	Tet–3–ike: $\arccos \left(-{\frac {\sqrt {10}}{4}}\right)\approx 142.23876^{\circ }$
	Tet–3–peppy: $\arccos \left(-{\frac {\sqrt {10}}{4}}\right)\approx 142.23876^{\circ }$
	Doe–5–peppy: 72°
Height	${\frac {1}{2}}=0.5$
Central density	1
Related polytopes
Army	Ikadoe
Regiment	Ikadoe
Dual	Dodecahedral-icosahedral tegmoid
Conjugate	Great icosahedron atop great stellated dodecahedron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H3×I, order 120
Convex	Yes
Nature	Tame

The icosahedron atop dodecahedron, or ikadoe, is a CRF segmentochoron (designated K-4.78 on Richard Klitzing's list). As the name suggests, it consists of a dodecahedron and an icosahedron as bases, connected by 12 pentagonal pyramids and 20+30 tetrahedra.

It is also commonly referred to as a dodecahedral or icosahedral antiprism, as the two bases are a pair of dual polyhedra.

The icosahedron atop dodecahedron can also be obtained from the hexacosichoron as a monostratic stack. This is more readily seen from the hexacosichoron's vertex-first projection (where the two bases are concentric) or its edge-first projection (where the two bases are flattened).

Segmentochoron display

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Vertex coordinates

Coordinates for the vertices of an icosahedron atop dodecahedron of edge length 1 are given by:

• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}}\right)$ and all permutations of the first three coordinates,
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0,\,0\right)$ and all permutations of the first three coordinates,
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,0\right).$

External links

• Klitzing, Richard. "ikadoe".
• Hi.gher.Space Wiki Contributors. "Rhodohedral antiprism".