Excavated truncated rhombicuboctahedron

Excavated truncated rhombicuboctahedron
(3D model)
Rank	3
Type	Quasi-convex Stewart toroid
Elements
Faces	8 triangles, 6+24+24+24+48 squares, 8 hexagons, 6 octagons
Edges	24+24+24+24+24+24+24+48+48+48
Vertices	24+24+48+48
Measures (edge length 1)
Volume	$8+12{\sqrt {2}}+6{\sqrt {3}}\approx 35.36287$
Surface area	$138+12{\sqrt {2}}+14{\sqrt {3}}\approx 179.21927$
Dihedral angles	4-4 (op lacing, narrow): $90^{\circ }+\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 234.73561^{\circ }$
	4-4 (op lacing, wide): $90^{\circ }+\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 215.26439^{\circ }$
	4-6 (tricu to trip): $90^{\circ }+\arccos \left({\frac {1}{3}}\right)\approx 160.52878^{\circ }$
	4-8 (squacu to trip): $90^{\circ }+\arccos \left({\frac {\sqrt {3}}{3}}\right)\approx 144.73561^{\circ }$
	4-4 (in squacu): 135°
	3-4 (in tricu): $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	4-4 (trip lacings): 60°
	4–6 (in tricu): $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 54.73561^{\circ }$
	4-8 (in squacu): 45°
Related polytopes
Convex hull	Truncated rhombicuboctahedron
Abstract & topological properties
Flag count	1248
Euler characteristic	–20
Orientable	Yes
Genus	11
Properties
Symmetry	B3, order 48
Flag orbits	26
Convex	No
Nature	Tame
History
Discovered by	Alex Doskey
First discovered	2004

The excavated truncated rhombicuboctahedron is a quasi-convex Stewart toroid. It can be obtained by excavating 12 unit-edge-length irregular octagonal prisms from a truncated rhombicuboctahedron, then excavating the central polyhedron as well (blending it with the remaining irregular-octagon faces of the prisms and opening up that space). Since the irregular faces are all blended away, the toroid is regular-faced. Its convex hull is an equilateral truncated rhombicuboctahedron, with 6 regular octagons, 12 rectangular-symmetric octagons, 8 hexagons, and 24 squares.

The central polyhedron is a truncation of the rhombic dodecahedron that cuts away both types of vertices, with 6 squares, 8 triangles, and 12 rectangular-symmetric octagons.

The excavated truncated rhombicuboctahedron can also be obtained by outer-blending six square cupolae, eight triangular cupolae, and twenty-four triangular prisms together.

Related polyhedra

If the cupolae are "contracted" into pyramids (by removing the side squares and pulling the side triangle faces together, like the opposite of a partial expansion), an excavated expanded cuboctahedron will be formed.

If the triangular prisms are removed (and the cupolae brought together), a truncated cuboctahedron excavated with cubes and a central small rhombicuboctahedron will be formed.

External links

• Wikipedia contributors. "Excavated truncated rhombicuboctahedron".
• Doskey, Alex. "Prism Expansions".