Icosidodecahedron atop truncated icosahedron

Icosidodecahedron atop truncated icosahedron
(OFF file)
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Idati
Coxeter diagram	oo5xx3ox&#x
Elements
Cells	12 pentagonal prisms, 20 triangular cupolas, 1 icosidodecahedron, 1 truncated icosahedron
Faces	20+30 triangles, 60 squares, 12+12 pentagons, 20 hexagons
Edges	30+60+60+60
Vertices	30+60
Vertex figures	30 wedges, edge lengths 1 (2 base edges and top edges), √2 (sides), and (1+√5)/2 (other base edges)
	60 sphenoids, edge lengths 1 (1), √2 (2), (1+√5)/2 (1), and √3 (2)
Measures (edge length 1)
Circumradius	${\sqrt {19+8{\sqrt {5}}}}\approx 6.07359$
Hypervolume	${\frac {325+277{\sqrt {5}}}{96}}\approx 9.83740$
Dichoral angles	Pip–4–tricu: $\arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{10}}}\right)\approx 166.71747^{\circ }$
	Tricu–3–tricu: $\arccos \left(-{\frac {1+3{\sqrt {5}}}{8}}\right)\approx 164.47751^{\circ }$
	Id–5–pip: 162°
	Id–3–tricu: $\arccos \left(-{\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 157.76124^{\circ }$
	Ti–6–tricu: $\arccos \left({\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 22.23876^{\circ }$
	Ti–5–pip: 18°
Height	${\frac {{\sqrt {5}}-1}{4}}\approx 0.30902$
Central density	1
Related polytopes
Army	Idati
Regiment	Idati
Dual	Rhombic triacontahedral-pentakis dodecahedral tegmoid
Conjugate	Great icosidodecahedron atop truncated great icosahedron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H3×I, order 120
Convex	Yes
Nature	Tame

Icosidodecahedron atop truncated icosahedron, or idati, is a CRF segmentochoron (designated K-4.158 on Richard Klitzing's list). As the name suggests, it consists of an icosidodecahedron and a truncated icosahedron as bases, connected by 12 pentagonal prisms and 20 triangular cupolas.

It can be obtained as an icosidodecahedron-first cap of the small rhombated hexacosichoron.

Vertex coordinates

The vertices of an icosidodecahedron atop truncated icosahedron segmentochoron of edge length 1 are given by all permutations of the first three coordinates of:

• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,0,\,{\frac {{\sqrt {5}}-1}{4}}\right),$

Plus all even permutations of the first three coordinates of:

• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,{\frac {{\sqrt {5}}-1}{4}}\right),$
• $\left(0,\,\pm {\frac {1}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,0\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {2+{\sqrt {5}}}{2}},\,0\right).$

External links

• Klitzing, Richard. "idati".