Square-great rhombicuboctahedral duoprism

Square-great rhombicuboctahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Squagirco
Coxeter diagram	x4o x4x3x
Elements
Tera	12 tesseracts, 8 square-hexagonal duoprisms, 6 square-octagonal duoprisms, 4 great rhombicuboctahedral prisms
Cells	24+24+24+48 cubes, 32 hexagonal prisms, 24 octagonal prisms, 4 great rhombicuboctahedra
Faces	48+48+96+96+96 squares, 32 hexagons, 24 octagons
Edges	96+96+96+192
Vertices	192
Vertex figure	Mirror-symmetric pentachoron, edge lengths √2, √3, √2+√2 (base triangle), √2 (top and side edges)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {15+6{\sqrt {2}}}}{2}}\approx 2.42308$
Hypervolume	$2(11+7{\sqrt {2}})\approx 41.79899$
Diteral angles	Tes–cube–shiddip: $\arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }$
	Tes–cube–sodip: 135°
	Shiddip–cube–sodip: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	Gircope–girco–gircope: 90°
	Tes–cube–gircope: 90°
	Shiddip–hip–gircope: 90°
	Sodip–op–gircope: 90°
Height	1
Central density	1
Number of external pieces	30
Level of complexity	60
Related polytopes
Army	Squagirco
Regiment	Squagirco
Dual	Square-disdyakis dodecahedral duotegum
Conjugate	Square-quasitruncated cuboctahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B3×B2, order 384
Convex	Yes
Nature	Tame

The square-great rhombicuboctahedral duoprism or squagirco is a convex uniform duoprism that consists of 4 great rhombicuboctahedral prisms, 6 square-octagonal duoprisms, 12 tesseracts, and 8 square-hexagonal duoprisms. Each vertex joins 2 great rhombicuboctahedral prisms, 1 tesseract, 1 square-hexagonal duoprism, and 1 square-octagonal duoprism. It is a duoprism based on a square and a great rhombicuboctahedron, which makes it a convex segmentoteron.

The square-great rhombicuboctahedral duoprism can be vertex-inscribed into a celliprismated penteract.

This polyteron can be alternated into a digonal-snub cubic duoantiprism, although it cannot be made uniform. The great rhombicuboctahedra can also be edge-snubbed to create a digonal-pyritohedral prismantiprismoid, which is also nonuniform.

Vertex coordinates

The vertices of a square-great rhombicuboctahedral duoprism of edge length 1 are given by all permutations of the last three coordinates of:

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

Representations

A square-great rhombicuboctahedral duoprism has the following Coxeter diagrams:

• x4o x4x3x (full symmetry)
• x x x4x3x (great rhombicuboctahedral prismatic prism)

External links

• Klitzing, Richard. "squagirco".