Small hexacronic icositetrahedron

Small hexacronic icositetrahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Notation
Coxeter diagram	m4/3o3m4*a ()
Elements
Faces	24 darts
Edges	24+24
Vertices	6+6+8
Vertex figures	8 triangles
	6 squares
	6 octagons
Measures (edge length 1)
Inradius	${\frac {\sqrt {34(7+4{\sqrt {2}})}}{17}}\approx 1.22026$
Dihedral angle	$\arccos \left(-{\frac {7+4{\sqrt {2}}}{17}}\right)\approx 138.11796^{\circ }$
Central density	2
Number of external pieces	48
Related polytopes
Dual	Small cubicuboctahedron
Conjugate	Great hexacronic icositetrahedron
Convex core	Deltoidal icositetrahedron
Abstract & topological properties
Flag count	192
Euler characteristic	–4
Orientable	Yes
Genus	3
Properties
Symmetry	B3, order 48
Convex	No
Nature	Tame

The small hexacronic icositetrahedron is a uniform dual polyhedron. It consists of 24 darts.

It appears the same as the small rhombihexacron.

If its dual, the small cubicuboctahedron, has an edge length of 1, then the short edges of the darts will measure $2{\frac {\sqrt {2\left(26+17{\sqrt {2}}\right)}}{7}}\approx 2.85833$, and the long edges will be $2{\sqrt {2+{\sqrt {2}}}}\approx 3.69552$. ​The dart faces will have length $2{\frac {\sqrt {31-8{\sqrt {2}}}}{7}}\approx 1.26769$, and width $2\left(1+{\sqrt {2}}\right)\approx 4.82843$. ​The darts have two interior angles of $\arccos \left({\frac {1+2{\sqrt {2}}}{4}}\right)\approx 16.84212^{\circ }$, one of $\arccos \left({\frac {2-{\sqrt {2}}}{4}}\right)\approx 81.57894^{\circ }$, and one of $360^{\circ }-\arccos \left(-{\frac {2+{\sqrt {2}}}{8}}\right)\approx 244.73683^{\circ }$.

Vertex coordinates

A small 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. "Small hexacronic icositetrahedron".
• McCooey, David. "Small Hexacronic Icositetrahedron"