Rectified small ditrigonary hexacosihecatonicosachoron

Rectified small ditrigonary hexacosihecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Rissidtixhi
Coxeter diagram	(o3x3o3o5/2*b)
Elements
Cells	600 octahedra, 120 small ditrigonary icosidodecahedra, 120 great icosidodecahedra
Faces	2400+2400 triangles, 1440 pentagrams
Edges	7200
Vertices	1200
Vertex figure	Ditrigonal prism, base edge lengths 1 and (√5–1)/2, side edge length 1
Measures (edge length 1)
Circumradius	$\sqrt3 ≈ 1.73205$
Hypervolume	$20\left(9\sqrt5-5\right) ≈ 302.49224$
Dichoral angles	Gid–3–oct: $\arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124^\circ$
	Sidtid–5/2–gid: 144°
	Sidtid–3–oct: $\arccos\left(-\frac{\sqrt{10}}{4}\right) ≈ 142.23876^\circ$
Central density	9
Number of external pieces	7560
Related polytopes
Army	Rahi
Regiment	Rissidtixhi
Conjugate	Rectified great ditrigonary hexacosihecatonicosachoron
Convex core	Semi-uniform hexacosihecatonicosachoron
Abstract & topological properties
Euler characteristic	–600
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The rectified small ditrigonary hexacosihecatonicosachoron, or rissidtixhi, is a nonconvex uniform polychoron that consists of 600 regular octahedra, 120 small ditrigonary icosidodecahedra, and 120 great icosidodecahedra. 2 small ditrigonary icosidodecahedra, 3 great icosidodecahedra, and 3 octahedra join at each ditrigonal prismatic vertex. As the name suggests, it can be obtained by rectifying the small ditrigonary hexacosihecatonicosachoron.

Vertex coordinates

The vertices of a rectified small ditrigonary hexacosihecatonicosachoron of edge length 1 are given by all permutations of:

• $\left(0,\,0,\,±\frac{\sqrt5-1}{2},\,±\frac{1+\sqrt5}{2}\right),$
• $\left(0,\,±1,\,±1,\,±1\right),$
• $\left(±\frac12,\,±\frac12,\,±\frac12,\,±\frac32\right),$
• $\left(±\frac12,\,±\frac12,\,±\frac{\sqrt5}{2},\,±\frac{\sqrt5}{2}\right),$

along with all even permutations of:

• $\left(0,\,±\frac{3-\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,±\frac{\sqrt5}{2}\right),$
• $\left(0,\,±\frac{\sqrt5-1}{4},\,±\frac32,\,±\frac{1+\sqrt5}{4}\right),$
• $\left(±\frac{3-\sqrt5}{4},\,±\frac{\sqrt5-1}{4},\,±\frac{1+\sqrt5}{2},\,±\frac12\right),$
• $\left(±\frac{3-\sqrt5}{4},\,±\frac12,\,±\frac{3+\sqrt5}{4},\,±1\right),$
• $\left(±\frac{\sqrt5-1}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{\sqrt5}{2},\,±1\right),$
• $\left(±\frac12,\,±\frac{\sqrt5-1}{2},\,±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4}\right).$

Related polychora

The rectified small ditrigonary hexacosihecatonicosachoron is the colonel of the largest non-snub regiment of uniform polychora, containing a total of 157 members, plus three compounds and three fissaries. Jonathan Bowers divides the rissidtixhi regiment into twenty subcategories.

The rectified small ditrigonary hexacosihecatonicosachoron contains the vertices and edges of the decagonal-decagrammic duoprism, the quasitruncated dodecadodecahedral prism, and the rectified icositetrachoron.

Rissidtixhi regiment

Index	Name	OBSA	Company	Subcategory
1147	Rectified small ditrigonary hexacosihecatonicosachoron	Rissidtixhi	Rissidtixhi	Main nine
1148	Rectified ditrigonary dishecatonicosachoron	Ridtidohi	Ridtidohi	Main nine
1149	Rectified great ditrigonary hexacosihecatonicosachoron	Riggidtixhi	Riggidtixhi	Main nine
1150	Small ditrigonary dishecatonicosachoron	Sidditdy	Sidditdy	Main nine
1151	Retroditrigonary dishecatonicosachoron	Ridditdy	Ridditdy	Main nine
1152	Great ditrigonary dishecatonicosachoron	Gidditdy	Gidditdy	Main nine
1153	Ditrigonary hexacosidishecatonicosachoron	Dittixdy		Main nine
1154	Ditrigonary trishecatonicosachoron	Dittathi		Main nine
1155	Toroidal ditrigonary hexacosidishecatonicosachoron	Todtixady		Main nine
1156	Small ditrigonary dishecatonicosihecatonicosachoron	Sidtid hihy		Sixteen trapezics
1157	Ditrigonary hecatonicosidishecatonicosachoron	Dithidy		Sixteen trapezics

External links

• Bowers, Jonathan. "Category 23: Rissidtixhi Regiment" (#1147).

• Bowers, Jonathan. "How to Make Rissidtixhi".

• Klitzing, Richard. "rissidtixhi".