Icosahedron atop icosidodecahedron

Icosahedron atop icosidodecahedron
(OFF file)
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Ikaid
Coxeter diagram	oo5ox3xo&#x
Elements
Cells	12 pentagonal pyramids, 20 octahedra, 1 icosahedron, 1 icosidodecahedron
Faces	20+20+30+60 triangles, 12 pentagons
Edges	30+60+60
Vertices	12+30
Vertex figures	12 pentagonal prisms, edge length 1
	30 wedges, edge lengths (1+√5)/2 (two base edges) and 1 (remaining edges)
Measures (edge length 1)
Circumradius	${\sqrt {5+2{\sqrt {5}}}}\approx 3.07768$
Hypervolume	${\frac {25+21{\sqrt {5}}}{32}}\approx 2.24867$
Dichoral angles	Oct–3–oct: $\arccos \left(-{\frac {1+3{\sqrt {5}}}{8}}\right)\approx 164.47751^{\circ }$
	Ike–3–oct: $\arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }$
	Peppy–3–oct: $\arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }$
	Id–5–peppy: 36°
	Id–3-oct: $\arccos \left({\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 22.23876^{\circ }$
Height	${\frac {{\sqrt {5}}-1}{4}}\approx 0.30902$
Central density	1
Related polytopes
Army	Ikaid
Regiment	Ikaid
Dual	Dodecahedral-rhombic triacontahedral tegmoid
Conjugate	Great icosahedron atop great icosidodecahedron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H3×I, order 120
Convex	Yes
Nature	Tame

Icosahedron atop icosidodecahedron, or ikaid, is a CRF segmentochoron (designated K-4.137 on Richard Klitzing's list). As the name suggests, it consists of an icosahedron and an icosidodecahedron as bases, connected by 20 octahedra and 12 pentagonal pyramids.

It can also be seen as a rectification of the CRF icosahedral pyramid.

It is also the cap of the rectified hexacosichoron in icosahedron-first orientation.

Vertex coordinates

The vertices of an icosahedron atop icosidodecahedron segmentochoron of edge length 1 are given by:

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

External links

• Klitzing, Richard. "ikaid".