Grand ditetrahedronary hexacosidishecatonicosachoron

Grand ditetrahedronary hexacosidishecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Gadtaxady
Coxeter diagram	x3o3o3o5/2*a ()
Elements
Cells	600 tetrahedra, 120 great stellated dodecahedra, 120 great icosidodecahedra
Faces	2400 triangles, 1440 pentagrams
Edges	3600
Vertices	600
Vertex figure	Rhombitetratetrahedron, edge lengths 1 and (√5–1)/2
Measures (edge length 1)
Circumradius	$\frac{\sqrt{10}-\sqrt2}{2} ≈ 0.87403$
Hypervolume	$5\frac{194\sqrt5-405}{4} ≈ 35.99648$
Dichoral angles	Gid–3–tet: $\arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124^\circ$
	Gissid–5/2–gid: 144°
Central density	34
Number of pieces	8520
Level of complexity	26
Related polytopes
Army	Hi
Regiment	Gadtaxady
Company	Gadtaxady
Conjugate	Small retroditetrahedronary hexacosidishecatonicosachoron
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The grand ditetrahedronary hexacosidishecatonicosachoron, or gadtaxady, is a nonconvex uniform polychoron that consists of 600 regular tetrahedra, 120 great stellated dodecahedra, and 120 great icosidodecahedra. 4 great stellated dodecahedra, 4 tetrahedra, and 6 great icosidodecahedra join at each vertex, with a small rhombitetratetrahedron as the vertex figure.

The grand ditetrahedronary hexacosidishecatonicosachoron contains the vertices of a great dodecicosidodecahedral prism and decagrammic duoprism.

Cross-sections

Vertex coordinates

The vertices of a grand ditetrahedronary hexacosidishecatonicosachoron of edge length 1, centered at the origin, are given by all permutations of:

• $\left(±\frac{\sqrt5-1}{2},\,±\frac{\sqrt5-1}{2},\,0,\,0\right),$
• $\left(±\frac{5-\sqrt5}{4},\,±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4}\right),$
• $\left(±\frac{3-\sqrt5}{4},\,±\frac{3-\sqrt5}{4},\,±\frac{3-\sqrt5}{4},\,±\frac{1+\sqrt5}{4}\right),$
• $\left(±\frac{\sqrt5-2}{2},\,±\frac12,\,±\frac12,\,±\frac12\right),$

together with all the even permutations of:

• $\left(±\frac{\sqrt5-2}{2},\,±\frac{\sqrt5-1}{4},\,±\frac{1+\sqrt5}{4},\,0\right),$
• $\left(±\frac{3-\sqrt5}{4},\,±\frac{5-\sqrt5}{4},\,0,\,±\frac12\right),$
• $\left(±\frac{3-\sqrt5}{4},\,±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{2},\,±\frac12\right).$

Related polychora

The grand ditetrahedronary hexacosidishecatonicosachoron is the colonel of a regiment with 37 members, plus 5 fissaries and three compounds (two are subsymmetric), as well as 11 scaliform members plus 54 scaliform fissaries and 2 scaliform compounds.

External links

• Bowers, Jonathan. "Category 18: Ditetrahedrals" (#852).

• Bowers, Jonathan. "How to Make Gadtaxady".

• Klitzing, Richard. "gadtaxady".