# Rectified small ditrigonary hexacosihecatonicosachoron

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.

Rectified small ditrigonary hexacosihecatonicosachoron
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymRissidtixhi
Coxeter diagram (o3x3o3o5/2*b)
Elements
Cells600 octahedra, 120 small ditrigonary icosidodecahedra, 120 great icosidodecahedra
Faces2400+2400 triangles, 1440 pentagrams
Edges7200
Vertices1200
Vertex figureDitrigonal prism, base edge lengths 1 and (5–1)/2, side edge length 1
Measures (edge length 1)
Circumradius${\displaystyle \sqrt3 ≈ 1.73205}$
Hypervolume${\displaystyle 20\left(9\sqrt5-5\right) ≈ 302.49224}$
Dichoral anglesGid–3–oct: ${\displaystyle \arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124^\circ}$
Sidtid–5/2–gid: 144°
Sidtid–3–oct: ${\displaystyle \arccos\left(-\frac{\sqrt{10}}{4}\right) ≈ 142.23876^\circ}$
Central density9
Number of external pieces7560
Related polytopes
ArmyRahi
RegimentRissidtixhi
ConjugateRectified great ditrigonary hexacosihecatonicosachoron
Convex coreSemi-uniform hexacosihecatonicosachoron
Abstract & topological properties
Euler characteristic–600
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

## Vertex coordinates

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

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

along with all even permutations of:

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

## 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