Hecatonicosadiminished hecatonicosachoron

Hecatonicosadiminished hecatonicosachoron
(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Hidhi
Elements
Cells	120 tetrahedra, 120 propello tetrahedra
Faces	480 triangles, 720 isosceles trapezoids
Edges	720+720
Vertices	480
Vertex figure	Triangular frustum
Measures (based on hecatonicosachoron of edge length 1)
Edge lengths	Short edges (720): 1
	Long edges (720): ${\frac {1+{\sqrt {5}}}{2}}$ $\approx 1.61803$
Circumradius	${\frac {3{\sqrt {2}}+{\sqrt {10}}}{2}}\approx 3.70246$
Central density	1
Related polytopes
Army	Hidhi
Regiment	Hidhi
Dual	Triangular-tegmatic tetracosioctacontachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A3●H3, order 1440
Convex	Yes
Nature	Tame

The hecatonicosadiminished hecatonicosachoron or hidhi is a convex isogonal polychoron that consists of 120 propello tetrahedra and 120 tetrahedra. 4 propello tetrahedra and 1 tetrahedron join at each vertex.

It can be constructed by diminishing the 120 vertices of an inscribed hexacosichoron of edge length ${\sqrt {3+{\sqrt {5}}}}$ from a hecatonicosachoron. In doing so, each dodecahedral cell of the hecatonicosachoron has 4 vertices corresponding to a tetrahedron diminished, while the tetrahedra come in as the hecatonicosachoron's vertex figures.

The ratio between the longest and shortest edges is 1:${\frac {1+{\sqrt {5}}}{2}}$ ≈ 1:1.61803.

Vertex coordinates

Vertex coordinates for a hecatonicosadiminished hecatonicosachoron, created from the vertices of a hecatonicosachoron of edge length 1, are given by all even permutations and all sign changes of:

• $\left({\frac {7+3{\sqrt {5}}}{4}},\,{\frac {3+{\sqrt {5}}}{4}},\,{\frac {1}{2}},\,0\right),$
• $\left({\frac {2+{\sqrt {5}}}{2}},\,{\frac {5+3{\sqrt {5}}}{4}},\,0,\,{\frac {1+{\sqrt {5}}}{4}}\right),$
• $\left({\frac {2+{\sqrt {5}}}{2}},\,{\frac {3+{\sqrt {5}}}{4}},\,{\frac {3+{\sqrt {5}}}{2}},\,{\frac {1+{\sqrt {5}}}{4}}\right),$

as well as all permutations and even sign changes of:

• $\left({\frac {2+{\sqrt {5}}}{2}},\,{\frac {2+{\sqrt {5}}}{2}},\,{\frac {2+{\sqrt {5}}}{2}},\,{\frac {1}{2}}\right),$
• $\left({\frac {7+3{\sqrt {5}}}{4}},\,{\frac {1+{\sqrt {5}}}{4}},\,{\frac {1+{\sqrt {5}}}{4}},\,{\frac {1+{\sqrt {5}}}{4}}\right),$

as well as all permutations and odd sign changes of:

• $\left({\frac {5+3{\sqrt {5}}}{4}},\,{\frac {3+{\sqrt {5}}}{4}},\,{\frac {3+{\sqrt {5}}}{4}},\,{\frac {3+{\sqrt {5}}}{4}}\right).$

External links

• Klitzing, Richard. "hidhi".