# Great hexacronic icositetrahedron

Great hexacronic icositetrahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramm4/3m3o4*a
Elements
Faces24 kites
Edges24+24
Vertices6+6+8
Vertex figures8 triangles
6 squares
6 octagrams
Measures (edge length 1)
Inradius${\displaystyle {\frac {\sqrt {34(7-4{\sqrt {2}})}}{17}}\approx 0.39751}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {7-4{\sqrt {2}}}{17}}\right)\approx 94.53158^{\circ }}$
Central density4
Number of external pieces48
Related polytopes
DualGreat cubicuboctahedron
ConjugateSmall hexacronic icositetrahedron
Convex coreTriakis octahedron
Abstract & topological properties
Flag count192
Euler characteristic–4
OrientableYes
Genus3
Properties
SymmetryB3, order 48
ConvexNo
NatureTame

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

## Vertex coordinates

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

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