Rectified pentachoron

Rectified pentachoron
Rank4
TypeUniform
Notation
Bowers style acronymRap
Coxeter diagramo3x3o3o ()
Elements
Cells5 tetrahedra, 5 octahedra
Faces10+20 triangles
Edges30
Vertices10
Vertex figureTriangular prism, edge length 1
Edge figuretet 3 oct 3 oct 3
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {15}}{5}}\approx 0.77460}$
Hypervolume${\displaystyle {\frac {11{\sqrt {5}}}{96}}\approx 0.25622}$
Dichoral anglesOct–3–tet: ${\displaystyle \arccos \left(-{\frac {1}{4}}\right)\approx 104.47751^{\circ }}$
Oct-3-oct: ${\displaystyle \arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }}$
Height${\displaystyle {\frac {\sqrt {10}}{4}}\approx 0.79057}$
Central density1
Number of external pieces10
Level of complexity3
Related polytopes
ArmyRap
RegimentRap
DualJoined pentachoron
ConjugateNone
Abstract & topological properties
Flag count360
Euler characteristic0
OrientableYes
Properties
SymmetryA4, order 120
Flag orbits3
ConvexYes
NatureTame

The rectified pentachoron, or rap, also commonly called the rectified 5-cell, is a convex uniform polychoron that consists of 5 regular tetrahedra and 5 regular octahedra. Two tetrahedra and three octahedra join at each triangular prismatic vertex. As the name suggests, it can be constructed by rectification of the pentachoron.

It is the vertex figure of the demipenteract.

It is also a convex segmentochoron (designated K-4.5 in Richard Klitzing's list), formed as a tetrahedron atop an octahedron. If the edge lengths of this subsymmetrical variation are changed, the result is various non-uniform tetrahedron atop octahedron polychora with 4 triangle antipodiums and 4 triangle pyramids as sides.

It is also isogonal under the 5-2 step prism subsymmetry, and can be considered to be a 5-2 double step prism.

Vertex coordinates

The vertices of a rectified pentachoron of edge length 1 are given by:

• ${\displaystyle \left(-{\frac {3{\sqrt {10}}}{20}},\,-{\frac {\sqrt {6}}{4}},\,0,\,0\right)}$,
• ${\displaystyle \left(-{\frac {3{\sqrt {10}}}{20}},\,{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {3}}{3}},\,0\right)}$,
• ${\displaystyle \left(-{\frac {3{\sqrt {10}}}{20}},\,{\frac {\sqrt {6}}{12}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left({\frac {\sqrt {10}}{10}},\,{\frac {\sqrt {6}}{6}},\,{\frac {\sqrt {3}}{3}},\,0\right)}$,
• ${\displaystyle \left({\frac {\sqrt {10}}{10}},\,-{\frac {\sqrt {6}}{6}},\,-{\frac {\sqrt {3}}{3}},\,0\right)}$,
• ${\displaystyle \left({\frac {\sqrt {10}}{10}},\,{\frac {\sqrt {6}}{6}},\,-{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left({\frac {\sqrt {10}}{10}},\,-{\frac {\sqrt {6}}{6}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right)}$.

Simpler coordinates can be given by all even sign changes of first 3 coordinates of:

• ${\displaystyle \left({\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {3{\sqrt {10}}}{20}}\right)}$,

and all permutations of the first three coordinates of:

• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,0,\,0,\,-{\frac {\sqrt {10}}{10}}\right)}$.

Much simpler coordinates can be given in five dimensions, as all permutations of:

• ${\displaystyle \left({\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {2}}{2}},\,0,\,0,\,0\right)}$.

Representations

A rectified pentachoron has the following Coxeter diagrams:

• o3x3o3o () (full symmetry)
• xo3ox3oo&#x (A3 axial, as tetrahedron atop octahedron)
• oxo oxo3oox&#xt (A2×A1 axial, vertex-first)
• oxoo3xoxo&#xr (A2 axial)
• oxoox oxoxo&#xr (A1×A1 axial)

Related polychora

The rectified pentachoron is the colonel of a three-member regiment that also includes the facetorectified pentachoron and the prismatointercepted pentachoron.

The rectified pentachoron can be diminished by cutting off triangular prismatic pyramids, with each removing 2 tetrahedra and diminishing the octahedra down to square pyramids. If one pyramid is removed, the result is the triangular antifastegium. If two non-adjacent pyramids are removed, such that one of the octahedra gets reduced down to an equatorial square only, the result is the bidiminished rectified pentachoron.

An octahedral pyramid can be attached to one of the rectified pentachoron's octahedral cells to create the augmented rectified pentachoron, notable as a non-uniform Blind polytope.

Uniform polychoron compounds composed of rectified pentachora include:

Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora: