# Swirlprismatodiminished rectified hexacosichoron

The swirlprismatodiminished rectified hexacosichoron or spidrox, also known as the prismantiprismoidal transitional didecafold icosidodecaswirlchoron, is a convex scaliform polychoron. It consists of 120 pentagonal prisms, 120 pentagonal antiprisms, and 600 square pyramids. 2 pentagonal antiprisms, 2 pentagonal prisms, and 5 square pyramids join at each vertex.

Swirlprismatodiminished rectified hexacosichoron
Rank4
TypeScaliform
SpaceSpherical
Notation
Bowers style acronymSpidrox
Elements
Cells600 square pyramids, 120 pentagonal prisms, 120 pentagonal antiprisms
Faces600+1200 triangles, 600 squares, 240 pentagons
Edges600+600+1200
Vertices600
Vertex figureParabidiminished pentagonal prism, edge lengths 1, 2, and (1+5)/2
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{5+2\sqrt5} ≈ 3.07768}$
Hypervolume${\displaystyle 5\frac{155+72\sqrt5}{4} ≈ 394.99612}$
Dichoral anglesPip–4–squippy: ${\displaystyle \arccos\left(-\sqrt{\frac{5+2\sqrt5}{10}}\right) ≈ 166.71747°}$
Squippy–3–squippy: ${\displaystyle \arccos\left(-\frac{1+3\sqrt5}{8}\right) ≈ 164.47751°}$
Pap–5–pip: 162°
Pap–3–squippy: ${\displaystyle \arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124°}$
Central density1
Related polytopes
ArmySpidrox
RegimentSpidrox
DualSwirlprismatostellated joined hecatonicosachoron
ConjugateSwirlprismatoreplenished rectified grand hexacosichoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryH3●I2(10), order 1200
ConvexYes
NatureTame

It can be constructed by diminishing the rectified hexacosichoron, specifically by removing the 120 vertices of an inscribed hexacosichoron. As a result every icosahedral cell of the rectified hexacosichoron gets diminished down to a pentagonal antiprism, while every octahedral cell gets diminished down to a square pyramid. The pentagonal prism cells are the vertex figures under the removed vertices.

## Vertex coordinates

A swirlprismatodiminished rectified hexacosichoron of edge length 1 has vertex coordinates given by:

• ${\displaystyle \left(0,\,0,\,±\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(0,\,0,\,±\frac{3+\sqrt5}{2},\,±\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(0,\,±\frac{1+\sqrt5}{2},\,0,\,±\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(0,\,-\frac{1+\sqrt5}{2},\,-\frac{3+\sqrt5}{2},\,0\right),}$
• ${\displaystyle \left(0,\,\frac{1+\sqrt5}{2},\,\frac{3+\sqrt5}{2},\,0\right),}$
• ${\displaystyle \left(0,\,±\frac{3+\sqrt5}{2},\,0,\,±\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(0,\,-\frac{3+\sqrt5}{2},\,\frac{1+\sqrt5}{2},\,0\right),}$
• ${\displaystyle \left(0,\,\frac{3+\sqrt5}{2},\,-\frac{1+\sqrt5}{2},\,0\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{2},\,0,\,0,\,\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{2},\,0,\,0,\,-\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,0,\,±\frac{3+\sqrt5}{2},\,0\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{2},\,0,\,0\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{2},\,0,\,0,\,-\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{2},\,0,\,0,\,\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{2},\,0,\,±\frac{1+\sqrt5}{2},\,0\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{2},\,±\frac{1+\sqrt5}{2},\,0,\,0\right),}$
• ${\displaystyle \left(0,\,-\frac12,\,-\frac{1+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4}\right),}$
• ${\displaystyle \left(0,\,\frac12,\,\frac{1+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4}\right),}$
• ${\displaystyle \left(0,\,±\frac{1+\sqrt5}{4},\,±\frac{5+3\sqrt5}{4},\,±\frac12\right),}$
• ${\displaystyle \left(0,\,±\frac{5+3\sqrt5}{4},\,±\frac12,\,±\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac12,\,0,\,±\frac{5+3\sqrt5}{4},\,\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac12,\,0,\,±\frac{5+3\sqrt5}{4},\,-\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{1+\sqrt5}{4},\,0,\,±\frac{5+3\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{5+3\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,0\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4}),\,0,\,±\frac12,\,±\frac{5+3\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4}),\,±\frac12,\,±\frac{5+3\sqrt5}{4},\,0\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{4}),\,±\frac{5+3\sqrt5}{4},\,0,\,-\frac12\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{4}),\,±\frac{5+3\sqrt5}{4},\,0,\,\frac12\right),}$
• ${\displaystyle \left(±\frac{5+3\sqrt5}{4},\,0,\,±\frac{1+\sqrt5}{4},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{5+3\sqrt5}{4},\,±\frac12,\,0,\,±\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{5+3\sqrt5}{4},\,-\frac{1+\sqrt5}{4},\,\frac12,\,0\right),}$
• ${\displaystyle \left(±\frac{5+3\sqrt5}{4},\,\frac{1+\sqrt5}{4},\,-\frac12,\,0\right),}$
• ${\displaystyle \left(0,\,-\frac{3+\sqrt5}{4},\,-\frac{2+\sqrt5}{2},\,±\frac{5+\sqrt5}{4}\right),}$
• ${\displaystyle \left(0,\,\frac{3+\sqrt5}{4},\,\frac{2+\sqrt5}{2},\,±\frac{5+\sqrt5}{4}\right),}$
• ${\displaystyle \left(0,\,±\frac{5+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,±\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(0,\,±\frac{2+\sqrt5}{2},\,±\frac{5+\sqrt5}{4},\,±\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{4},\,0,\,±\frac{5+\sqrt5}{4},\,\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{4},\,0,\,±\frac{5+\sqrt5}{4},\,-\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,±\frac{2+\sqrt5}{2},\,0\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{2+\sqrt5}{2},\,0,\,±\frac{5+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{5+\sqrt5}{4},\,0,\,±\frac{2+\sqrt5}{2},\,±\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{5+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,0,\,±\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{5+\sqrt5}{4},\,-\frac{2+\sqrt5}{2},\,\frac{3+\sqrt5}{4},\,0\right),}$
• ${\displaystyle \left(±\frac{5+\sqrt5}{4},\,\frac{2+\sqrt5}{2},\,-\frac{3+\sqrt5}{4},\,0\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt5}{2},\,0,\,±\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt5}{2},\,±\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,0\right),}$
• ${\displaystyle \left(-\frac{2+\sqrt5}{2},\,±\frac{5+\sqrt5}{4},\,0,\,-\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{2+\sqrt5}{2},\,±\frac{5+\sqrt5}{4},\,0,\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac12,\,-\frac12,\,±\frac{2+\sqrt5}{2},\,-\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac12,\,\frac12,\,-\frac{2+\sqrt5}{2},\,±\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac12,\,\frac12,\,\frac{2+\sqrt5}{2},\,-\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac12,\,±\frac12,\,-\frac{2+\sqrt5}{2},\,\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac12,\,-\frac12,\,\frac{2+\sqrt5}{2},\,-\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac12,\,\frac12,\,-\frac{2+\sqrt5}{2},\,-\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac12,\,\frac12,\,\frac{2+\sqrt5}{2},\,\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,-\frac{2+\sqrt5}{2},\,-\frac12,\,±\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,\frac{2+\sqrt5}{2},\,\frac12,\,\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac12,\,±\frac{2+\sqrt5}{2},\,\frac12,\,-\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac12,\,\frac{2+\sqrt5}{2},\,-\frac12,\,-\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac12,\,-\frac{2+\sqrt5}{2},\,\frac12,\,\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac12,\,\frac{2+\sqrt5}{2},\,-\frac12,\,\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac12,\,\frac{2+\sqrt5}{2},\,\frac12,\,-\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{2+\sqrt5}{2},\,±\frac{2+\sqrt5}{2},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt5}{2},\,±\frac12,\,±\frac12,\,±\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt5}{2},\,-\frac12,\,\frac{2+\sqrt5}{2},\,-\frac12\right),}$
• ${\displaystyle \left(-\frac{2+\sqrt5}{2},\,-\frac12,\,±\frac{2+\sqrt5}{2},\,\frac12\right),}$
• ${\displaystyle \left(-\frac{2+\sqrt5}{2},\,\frac12,\,-\frac{2+\sqrt5}{2},\,±\frac12\right),}$
• ${\displaystyle \left(-\frac{2+\sqrt5}{2},\,\frac12,\,\frac{2+\sqrt5}{2},\,\frac12\right),}$
• ${\displaystyle \left(\frac{2+\sqrt5}{2},\,±\frac12,\,-\frac{2+\sqrt5}{2},\,-\frac12\right),}$
• ${\displaystyle \left(\frac{2+\sqrt5}{2},\,-\frac12,\,\frac{2+\sqrt5}{2},\,\frac12\right),}$
• ${\displaystyle \left(\frac{2+\sqrt5}{2},\,\frac12,\,-\frac{2+\sqrt5}{2},\,\frac12\right),}$
• ${\displaystyle \left(\frac{2+\sqrt5}{2},\,\frac12,\,\frac{2+\sqrt5}{2},\,-\frac12\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt5}{2},\,-\frac{2+\sqrt5}{2},\,-\frac12,\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt5}{2},\,\frac{2+\sqrt5}{2},\,\frac12,\,-\frac12\right),}$
• ${\displaystyle \left(-\frac{2+\sqrt5}{2},\,±\frac{2+\sqrt5}{2},\,\frac12,\,\frac12\right),}$
• ${\displaystyle \left(-\frac{2+\sqrt5}{2},\,\frac{2+\sqrt5}{2},\,-\frac12,\,\frac12\right),}$
• ${\displaystyle \left(\frac{2+\sqrt5}{2},\,-\frac{2+\sqrt5}{2},\,\frac12,\,-\frac12\right),}$
• ${\displaystyle \left(\frac{2+\sqrt5}{2},\,\frac{2+\sqrt5}{2},\,-\frac12,\,-\frac12\right),}$
• ${\displaystyle \left(\frac{2+\sqrt5}{2},\,\frac{2+\sqrt5}{2},\,\frac12,\,\frac12\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{2},\,±\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,-\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4},\,-\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac12,\,-\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac12,\,\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac12,\,\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac12,\,±\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{4},\,-\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac12,\,-\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac12,\,\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac12,\,\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4},\,-\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,-\frac{3+\sqrt5}{2},\,-\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,\frac{3+\sqrt5}{2},\,\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac12,\,±\frac{3+\sqrt5}{2},\,\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac12,\,\frac{3+\sqrt5}{2},\,-\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac12,\,-\frac{3+\sqrt5}{2},\,\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac12,\,\frac{3+\sqrt5}{2},\,-\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac12,\,\frac{3+\sqrt5}{2},\,\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4}),\,±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4}),\,-\frac{3+\sqrt5}{4},\,\frac{3+\sqrt5}{2},\,\frac12\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{4}),\,-\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{2},\,-\frac12\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{4}),\,\frac{3+\sqrt5}{4},\,-\frac{3+\sqrt5}{2},\,±\frac12\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{4}),\,\frac{3+\sqrt5}{4},\,\frac{3+\sqrt5}{2},\,-\frac12\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{4}),\,±\frac{3+\sqrt5}{4},\,-\frac{3+\sqrt5}{2},\,\frac12\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{4}),\,-\frac{3+\sqrt5}{4},\,\frac{3+\sqrt5}{2},\,-\frac12\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{4}),\,\frac{3+\sqrt5}{4},\,-\frac{3+\sqrt5}{2},\,-\frac12\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{4}),\,\frac{3+\sqrt5}{4},\,\frac{3+\sqrt5}{2},\,\frac12\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4}),\,-\frac{3+\sqrt5}{2},\,\frac12,\,-\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{4}),\,-\frac{3+\sqrt5}{2},\,±\frac12,\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{4}),\,\frac{3+\sqrt5}{2},\,-\frac12,\,±\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{4}),\,\frac{3+\sqrt5}{2},\,\frac12,\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{4}),\,±\frac{3+\sqrt5}{2},\,-\frac12,\,-\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{4}),\,-\frac{3+\sqrt5}{2},\,\frac12,\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{4}),\,\frac{3+\sqrt5}{2},\,-\frac12,\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{4}),\,\frac{3+\sqrt5}{2},\,\frac12,\,-\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,-\frac12,\,-\frac{3+\sqrt5}{2},\,±\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,\frac12,\,\frac{3+\sqrt5}{2},\,\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{4},\,±\frac12,\,\frac{3+\sqrt5}{2},\,-\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{4},\,\frac12,\,-\frac{3+\sqrt5}{2},\,-\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{4},\,-\frac12,\,\frac{3+\sqrt5}{2},\,\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{4},\,\frac12,\,-\frac{3+\sqrt5}{2},\,\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{4},\,\frac12,\,\frac{3+\sqrt5}{2},\,-\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{4},\,\frac12,\,\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{4},\,±\frac12,\,-\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4},\,-\frac12,\,±\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4},\,\frac12,\,-\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,-\frac12,\,\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{4},\,\frac12,\,-\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4},\,-\frac12,\,-\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4},\,\frac12,\,\frac{3+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{2},\,±\frac{1+\sqrt5}{4},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{2},\,-\frac12,\,-\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{2},\,\frac12,\,\frac{1+\sqrt5}{4},\,-\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{2},\,±\frac12,\,\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{2},\,\frac12,\,-\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{2},\,-\frac12,\,\frac{1+\sqrt5}{4},\,-\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{2},\,\frac12,\,-\frac{1+\sqrt5}{4},\,-\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{2},\,\frac12,\,\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{2},\,-\frac{1+\sqrt5}{4},\,-\frac{3+\sqrt5}{4},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{2},\,\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{4},\,\frac12\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{2},\,±\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{4},\,-\frac12\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{2},\,\frac{1+\sqrt5}{4},\,-\frac{3+\sqrt5}{4},\,-\frac12\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{2},\,-\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{4},\,\frac12\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{2},\,\frac{1+\sqrt5}{4},\,-\frac{3+\sqrt5}{4},\,\frac12\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{2},\,\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{4},\,-\frac12\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{2},\,±\frac{3+\sqrt5}{4},\,±\frac12,\,±\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4}),\,±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4}),\,-\frac{1+\sqrt5}{2},\,\frac{2+\sqrt5}{2},\,-\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{4}),\,-\frac{1+\sqrt5}{2},\,±\frac{2+\sqrt5}{2},\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{4}),\,\frac{1+\sqrt5}{2},\,-\frac{2+\sqrt5}{2},\,±\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{4}),\,\frac{1+\sqrt5}{2},\,\frac{2+\sqrt5}{2},\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{4}),\,±\frac{1+\sqrt5}{2},\,-\frac{2+\sqrt5}{2},\,-\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{4}),\,-\frac{1+\sqrt5}{2},\,\frac{2+\sqrt5}{2},\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{4}),\,\frac{1+\sqrt5}{2},\,-\frac{2+\sqrt5}{2},\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{4}),\,\frac{1+\sqrt5}{2},\,\frac{2+\sqrt5}{2},\,-\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4}),\,-\frac{2+\sqrt5}{2},\,\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{4}),\,-\frac{2+\sqrt5}{2},\,±\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{4}),\,\frac{2+\sqrt5}{2},\,-\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{4}),\,\frac{2+\sqrt5}{2},\,\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{4}),\,±\frac{2+\sqrt5}{2},\,-\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{4}),\,-\frac{2+\sqrt5}{2},\,\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{4}),\,\frac{2+\sqrt5}{2},\,-\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{4}),\,\frac{2+\sqrt5}{2},\,\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{2+\sqrt5}{2},\,±\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{2},\,-\frac{1+\sqrt5}{4},\,±\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{2},\,\frac{1+\sqrt5}{4},\,-\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,\frac{1+\sqrt5}{4},\,\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{2},\,-\frac{1+\sqrt5}{4},\,\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{2},\,\frac{1+\sqrt5}{4},\,-\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{2},\,-\frac{1+\sqrt5}{4},\,-\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{2},\,\frac{1+\sqrt5}{4},\,\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,-\frac{2+\sqrt5}{2},\,-\frac{1+\sqrt5}{2},\,±\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,\frac{2+\sqrt5}{2},\,\frac{1+\sqrt5}{2},\,\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{4},\,±\frac{2+\sqrt5}{2},\,\frac{1+\sqrt5}{2},\,-\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{3+\sqrt5}{4},\,\frac{2+\sqrt5}{2},\,-\frac{1+\sqrt5}{2},\,-\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{4},\,-\frac{2+\sqrt5}{2},\,\frac{1+\sqrt5}{2},\,\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{4},\,\frac{2+\sqrt5}{2},\,-\frac{1+\sqrt5}{2},\,\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{3+\sqrt5}{4},\,\frac{2+\sqrt5}{2},\,\frac{1+\sqrt5}{2},\,-\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,-\frac{1+\sqrt5}{4},\,-\frac{3+\sqrt5}{4},\,±\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{4},\,\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{2},\,±\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{4},\,-\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{2},\,\frac{1+\sqrt5}{4},\,-\frac{3+\sqrt5}{4},\,-\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{2},\,-\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{4},\,\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{2},\,\frac{1+\sqrt5}{4},\,-\frac{3+\sqrt5}{4},\,\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{2},\,\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{4},\,-\frac{2+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,-\frac{3+\sqrt5}{4},\,-\frac{2+\sqrt5}{2},\,±\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,\frac{3+\sqrt5}{4},\,\frac{2+\sqrt5}{2},\,-\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{4},\,\frac{2+\sqrt5}{2},\,\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{1+\sqrt5}{2},\,\frac{3+\sqrt5}{4},\,-\frac{2+\sqrt5}{2},\,\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{2},\,-\frac{3+\sqrt5}{4},\,\frac{2+\sqrt5}{2},\,-\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{2},\,\frac{3+\sqrt5}{4},\,-\frac{2+\sqrt5}{2},\,-\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{1+\sqrt5}{2},\,\frac{3+\sqrt5}{4},\,\frac{2+\sqrt5}{2},\,\frac{1+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,±\frac{2+\sqrt5}{2},\,±\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt5}{2},\,-\frac{1+\sqrt5}{4},\,\frac{1+\sqrt5}{2},\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{2+\sqrt5}{2},\,-\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,-\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{2+\sqrt5}{2},\,\frac{1+\sqrt5}{4},\,-\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(-\frac{2+\sqrt5}{2},\,\frac{1+\sqrt5}{4},\,\frac{1+\sqrt5}{2},\,-\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{2+\sqrt5}{2},\,±\frac{1+\sqrt5}{4},\,-\frac{1+\sqrt5}{2},\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{2+\sqrt5}{2},\,-\frac{1+\sqrt5}{4},\,\frac{1+\sqrt5}{2},\,-\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{2+\sqrt5}{2},\,\frac{1+\sqrt5}{4},\,-\frac{1+\sqrt5}{2},\,-\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(\frac{2+\sqrt5}{2},\,\frac{1+\sqrt5}{4},\,\frac{1+\sqrt5}{2},\,\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt5}{2},\,-\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4},\,-\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac{2+\sqrt5}{2},\,-\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac{2+\sqrt5}{2},\,\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(-\frac{2+\sqrt5}{2},\,\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4},\,\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{2+\sqrt5}{2},\,±\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{4},\,-\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{2+\sqrt5}{2},\,-\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4},\,\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{2+\sqrt5}{2},\,\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{4},\,\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(\frac{2+\sqrt5}{2},\,\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4},\,-\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac{2+\sqrt5}{2},\,±\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4}\right).}$

These are derived by removing 120 vertices from the rectified hexacosichoron.