Tetradyakis hexahedron

Tetradyakis hexahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Notation
Coxeter diagram	m4/3m3m4*a
Elements
Faces	48 scalene triangles
Edges	24+24+24
Vertices	6+6+8
Vertex figures	8 hexagons
	6 octagons
	6 octagrams
Measures (edge length 1)
Inradius	${\frac {3{\sqrt {7}}}{7}}\approx 1.13389$
Dihedral angle	$\arccos \left(-{\frac {5}{7}}\right)\approx 135.58469^{\circ }$
Central density	4
Number of external pieces	96
Related polytopes
Dual	Cuboctatruncated cuboctahedron
Conjugate	Tetradyakis hexahedron
Convex core	Non-Catalan disdyakis dodecahedron
Abstract & topological properties
Flag count	288
Euler characteristic	–4
Orientable	Yes
Genus	3
Properties
Symmetry	B3, order 48
Convex	No
Nature	Tame

The tetradyakis hexahedron is a uniform dual polyhedron. It consists of 48 scalene triangles.

If its dual, the cuboctatruncated cuboctahedron, has an edge length of 1, then the short edges of the triangles will measure $2\left({\sqrt {6}}-{\sqrt {3}}\right)\approx 1.43488$, the medium edges will be $3{\sqrt {6}}\approx 7.34847$, and the long edges will be $2\left({\sqrt {6}}+{\sqrt {3}}\right)\approx 8.36308$. The triangles have one interior angle of $\arccos \left({\frac {3}{4}}\right)\approx 41.40962^{\circ }$, one of $\arccos \left({\frac {2+7{\sqrt {2}}}{12}}\right)\approx 7.42069^{\circ }$, and one of $\arccos \left({\frac {2-7{\sqrt {2}}}{12}}\right)\approx 131.16968^{\circ }$.

Vertex coordinates

A tetradyakis hexahedron with dual edge length 1 has vertex coordinates given by all permutations of:

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

External links

• Wikipedia contributors. "Tetradyakis hexahedron".
• McCooey, David. "Tetradyakis Hexahedron"