Rectified great faceted hexacosichoron

Rectified great faceted hexacosichoron
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Rigfix
Coxeter diagram	o5o5/2x3o ()
Elements
Cells	120 small stellated dodecahedra 120 great icosidodecahedra
Faces	1440 pentagrams 1200 triangles
Edges	3600
Vertices	720
Vertex figure	Semi-uniform pentagonal prism, edge lengths (√5–1)/2 (base) and 1 (side)
Measures (edge length 1)
Circumradius	${\sqrt {5-2{\sqrt {5}}}}\approx 0.72654$
Hypervolume	$5{\frac {545-237{\sqrt {5}}}{4}}\approx 18.81486$
Dichoral angle	Gid–3–gid: 120° Sissid–5/2–gid: 108°
Central density	76
Number of external pieces	5040
Level of complexity	24
Related polytopes
Army	Rox, edge length ${\sqrt {5}}-2$
Regiment	Rigfix
Conjugate	Rectified faceted hexacosichoron
Convex core	Hecatonicosachoron
Abstract & topological properties
Flag count	43200
Euler characteristic	–480
Orientable	Yes
Properties
Symmetry	H4, order 14400
Flag orbits	3
Convex	No
Nature	Tame

The rectified great faceted hexacosichoron, or rigfix, is a nonconvex uniform polychoron that consists of 120 small stellated dodecahedra and 120 great icosidodecahedra. Two small stellated dodecahedra and five great icosidodecahedra join at each pentagonal prismatic vertex. As the name suggests, it can be obtained by rectifying the great faceted hexacosichoron.

Vertex coordinates

The vertices of a rectified great faceted hexacosichoron of edge length 1 are given by all permutations of:

• $\left(0,\,0,\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {3-{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {{\sqrt {5}}-2}{2}}\right)$,

along with even permutations of:

• $\left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {3{\sqrt {5}}-5}{4}}\right)$,
• $\left(0,\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {5-{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {3-{\sqrt {5}}}{2}},\,\pm {\frac {3-{\sqrt {5}}}{4}}\right)$,
• $\left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {{\sqrt {5}}-2}{2}}\right)$.

Gallery

Related polychora

The rectified great faceted hexacosichoron is the colonel of a regiment with 15 members. Of these, one other besideds the colonel itself is Wythoffian (the rectified grand hexacosichoron), two are hemi-Wythoffian (the great pentagrammal antiprismatoverted dishecatonicosachoron and quasiprismatohecatonicosachoron), and one is noble (the grand retropental hecatonicosachoron).

Uniform polychoron compounds composed of rectified great faceted hexacosichora include:

• Rectified great faceted chirotetrahedral dishexacosichoron (2)
• Rectified great faceted snub pentishecatonicosachoron (5)
• Rectified great faceted snub decahecatonicosachoron (10)

External links

• Bowers, Jonathan. "Category 5: Pentagonal Rectates" (#118).

• Klitzing, Richard. "rigfix".