Great hexacronic icositetrahedron

Great hexacronic icositetrahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Notation
Coxeter diagram	m4/3m3o4*a
Elements
Faces	24 kites
Edges	24+24
Vertices	6+6+8
Vertex figures	8 triangles
	6 squares
	6 octagrams
Measures (edge length 1)
Inradius	${\frac {\sqrt {34(7-4{\sqrt {2}})}}{17}}\approx 0.39751$
Dihedral angle	$\arccos \left(-{\frac {7-4{\sqrt {2}}}{17}}\right)\approx 94.53158^{\circ }$
Central density	4
Number of external pieces	48
Related polytopes
Dual	Great cubicuboctahedron
Conjugate	Small hexacronic icositetrahedron
Convex core	Triakis octahedron
Abstract & topological properties
Flag count	192
Euler characteristic	–4
Orientable	Yes
Genus	3
Properties
Symmetry	B3, order 48
Convex	No
Nature	Tame

The great hexacronic icositetrahedron is a uniform dual polyhedron. It consists of 24 kites.

If its dual, the great cubicuboctahedron, has an edge length of 1, then the short edges of the kites will measure $2{\frac {\sqrt {2\left(26-17{\sqrt {2}}\right)}}{7}}\approx 0.56545$, and the long edges will be $2{\sqrt {2-{\sqrt {2}}}}\approx 1.53073$. The kite faces will have length $2{\frac {\sqrt {31+8{\sqrt {2}}}}{7}}\approx 1.85854$, and width $2\left({\sqrt {2}}-1\right)\approx 0.82843$. The kites have two interior angles of $\arccos \left({\frac {1-2{\sqrt {2}}}{4}}\right)\approx 117.20057^{\circ }$, one of $\arccos \left({\frac {-2+{\sqrt {2}}}{8}}\right)\approx 94.19914^{\circ }$, and one of $\arccos \left({\frac {2+{\sqrt {2}}}{4}}\right)\approx 31.39971^{\circ }$.

Vertex coordinates

A great hexacronic icositetrahedron with dual edge length 1 has vertex coordinates given by all permutations of:

• $\left(\pm \left(2-{\sqrt {2}}\right),\,0,\,0\right),$
• $\left(\pm {\sqrt {2}},\,0,\,0\right),$
• $\left(\pm {\frac {4-{\sqrt {2}}}{7}},\,\pm {\frac {4-{\sqrt {2}}}{7}},\,\pm {\frac {4-{\sqrt {2}}}{7}}\right).$

External links

• Wikipedia contributors. "Great hexacronic icositetrahedron".
• McCooey, David. "Great Hexacronic Icositetrahedron"