Bitetrahedral diacositetracontachoron

Bitetrahedral diacositetracontachoron
File:Bitetrahedral diacositetracontachoron.png (OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Notation
Bowers style acronym	Bittid
Elements
Cells	1440 phyllic disphenoids, 720 rhombic disphenoids
Faces	2880 scalene triangles, 1440 isosceles triangles
Edges	480+480+1440
Vertices	240
Vertex figure	Chiral ditriakis tetrahedron
Measures (based on 2 hexacosichora of edge length 1)
Edge lengths	12-valence lacing (480: $\frac{\sqrt{12-7\sqrt2+4\sqrt5-3\sqrt{10}}}{2} ≈ 0.62408$
	Edges of hexacosichora (1440): 1
	6-valence lacing (480): $\frac{\sqrt{12-5\sqrt2+4\sqrt5-3\sqrt{10}}}{2} ≈ 1.04718$
Circumradius	$\frac{1+\sqrt5}{2} ≈ 1.61803$
Central density	1
Related polytopes
Army	Bittid
Regiment	Bittid
Dual	Ditruncated-tetrahedral diacositetracontachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B3●H3, order 2880
Convex	Yes
Nature	Tame

The bitetrahedral diacositetracontachoron or bittid, also known as the octafold icosidodecaswirlchoron, is a convex isogonal polychoron that consists of 720 rhombic disphenoids and 1440 phyllic disphenoids. 12 rhombic disphenoids and 24 phyllic disphenoids join at each vertex. However, it cannot be made uniform. It is the second in a series of isogonal icosidodecahedral swirlchora.

It can be formed as the convex hull of two hexacosichora, oriented such that the hull of 2 icositetrachoric sets of 24 vertices from each is a bitetracontoctachoron.

The ratio between the longest and shortest edges is 1:$\frac{\sqrt{54+6\sqrt{37+8\sqrt{10}}}}{6}$ ≈ 1:1.67794.

Vertex coordinates

Vertex coordinates for a bitetrahedral diacositetracontachoron, created from the vertices of a hexacosichoron of edge length 1, are given by all permutations of:

• $\left(0,\,0,\,0,\,±\frac{1+\sqrt5}{2}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4}\right),$
• $\left(0,\,0,\,±\frac{\sqrt2+\sqrt{10}}{4},\,±\frac{\sqrt2+\sqrt{10}}{4}\right),$

as well as all even permutations of:

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

as well as all permutations and even sign changes of:

• $\left(\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{2\sqrt2+\sqrt{10}}{4}\right),$
• $\left(\frac{\sqrt{10}-\sqrt2}{8},\,\frac{3\sqrt2+\sqrt{10}}{8},\,\frac{3\sqrt2+\sqrt{10}}{8},\,\frac{3\sqrt2+\sqrt{10}}{8}\right),$

as well as all permutations and odd sign changes of:

• $\left(\frac{\sqrt2+\sqrt{10}}{8},\,\frac{\sqrt2+\sqrt{10}}{8},\,\frac{\sqrt2+\sqrt{10}}{8},\,\frac{5\sqrt2+\sqrt{10}}{8}\right).$

Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora:

• Rhombic disphenoid (720): Snub bitetrahedral diacositetracontachoron