Rectified tetracontoctachoron

Rectified tetracontoctachoron
(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Notation
Bowers style acronym	Recont
Coxeter diagram	xo3oK4Ko3ox&#zk
Elements
Cells	288 tetragonal disphenoids, 48 rectified truncated cubes
Faces	1152 isosceles triangles, 192 triangles, 144 octagons
Edges	576+1152
Vertices	576
Vertex figure	Wedge
Measures (short edge length 1)
Edge lengths	Edges of triangles (576): 1
	Lacing edges (1152): $\sqrt{2+\sqrt2} ≈ 1.84776$
Circumradius	$\sqrt{23+16\sqrt2} ≈ 6.75491$
Central density	1
Related polytopes
Army	Recont
Regiment	Recont
Dual	Joined bitetracontoctachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	F4×2, order 2304
Convex	Yes
Nature	Tame

The rectified tetracontoctachoron or recont is a convex isogonal polychoron that consists of 48 rectified truncated cubes and 288 tetragonal disphenoids. 3 rectified truncated cubes and 2 tetragonal disphenoids join at each vertex. It can be formed by rectifying the tetracontoctachoron.

It can also be formed as the convex hull of 2 oppositely oriented semi-uniform variants of the small rhombated icositetrachoron, where the edges of the cuboctahedra are $2+\sqrt2 ≈ 3.41421$ times the length of the other edges. It is one of five polychora (including two transitional cases) formed from two small rhombated icositetrachora, and is the transitional point between the small birhombatotetracontoctachoron and great birhombatotetracontoctachoron.

The ratio between the longest and shortest edges is 1:$\sqrt{2+\sqrt2}$ ≈ 1:1.84776.

Vertex coordinates

The vertices of a rectified tetracontoctachoron with triangles of edge length 1, centered at the origin, are given by:

• $\left(0,\,±(2+\sqrt2),\,±\frac{4+3\sqrt2}{2},\,±\frac{4+3\sqrt2}{2}\right),$
• $\left(±\frac{2+\sqrt2}{2},\,±\frac{2+\sqrt2}{2},\,±(1+\sqrt2),\,±(3+2\sqrt2)\right),$
• $\left(0,\,±(1+\sqrt2),\,±(1+\sqrt2),\,±(3+2\sqrt2)\right),$
• $\left(±\frac12,\,±\frac{3+\sqrt2}{2},\,±\frac{3+2\sqrt2}{2},\,±\frac{5+4\sqrt2}{2}\right).$

External links

• Klitzing, Richard. "recont".