Runcinated 5-cubes

5-cube	Runcinated 5-cube	Runcinated 5-orthoplex
Runcitruncated 5-cube	Runcicantellated 5-cube	Runcicantitruncated 5-cube
Runcitruncated 5-orthoplex	Runcicantellated 5-orthoplex	Runcicantitruncated 5-orthoplex
Orthogonal projections in B5 Coxeter plane

In five-dimensional geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination (a 3rd order truncation) of the regular 5-cube.

There are 8 unique degrees of runcinations of the 5-cube, along with permutations of truncations and cantellations. Four are more simply constructed relative to the 5-orthoplex.

Runcinated 5-cube

Runcinated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	t0,3{4,3,3,3}
Coxeter diagram
4-faces	202	10 80 36px|link=3-4 duoprism 80 36px|link=Octahedral prism 32
Cells	1240	40 240 25px|link=Cube 320 25px|link=Triangular prism 160 25px|link=Tetrahedron 320 25px|link=Triangular prism 160
Faces	2160	240 20px|link=Square 960 20px|link=Square 640 20px|link=Triangle 320 20px|link=Triangle
Edges	1440	480+960
Vertices	320
Vertex figure
Coxeter group	B5 [4,3,3,3]
Properties	convex

Alternate names

• Small prismated penteract (Acronym: span) (Jonathan Bowers)

The Cartesian coordinates of the vertices of a runcinated 5-cube having edge length 2 are all permutations of:

[math]\displaystyle{ \left(\pm1,\ \pm1,\ \pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2})\right) }[/math]

Images

orthographic projections

Coxeter plane	B5	B4 / D5	B3 / D4 / A2
Graph
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B2	A3
Graph
Dihedral symmetry	[4]	[4]

Runcitruncated 5-cube

Runcitruncated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	t0,1,3{4,3,3,3}
Coxeter-Dynkin diagrams
4-faces	202	10 80 36px|link=3-8 duoprism 80 36px|link=Octahedral prism 32
Cells	1560	40 240 25px|link=Octagonal prism 320 25px|link=Triangular prism 320 25px|link=Triangular prism 160 25px|link=Cuboctahedron 320 25px|link=Triangular prism 160
Faces	3760	240 960 20px|link=Square 320 20px|link=Equilateral triangle 960 20px|link=Square 640 20px|link=Equilateral triangle 640 20px|link=Equilateral triangle
Edges	3360	480+960+1920
Vertices	960
Vertex figure
Coxeter group	B5, [3,3,3,4]
Properties	convex

Alternate names

• Runcitruncated penteract
• Prismatotruncated penteract (Acronym: pattin) (Jonathan Bowers)

Construction and coordinates

The Cartesian coordinates of the vertices of a runcitruncated 5-cube having edge length 2 are all permutations of:

[math]\displaystyle{ \left(\pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2}),\ \pm(1+2\sqrt{2})\right) }[/math]

Images

orthographic projections

Coxeter plane	B5	B4 / D5	B3 / D4 / A2
Graph
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B2	A3
Graph
Dihedral symmetry	[4]	[4]

Runcicantellated 5-cube

Runcicantellated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	t0,2,3{4,3,3,3}
Coxeter-Dynkin diagram
4-faces	202	10 80 36px|link=3-4 duoprism 80 36px|link=Truncated tetrahedral prism 32
Cells	1240	40 240 25px|link=Cube 320 25px|link=Hexagonal prism 320 25px|link=Triangular prism 160 25px|link=Truncated tetrahedron 160 25px|link=Truncated tetrahedron
Faces	2960	240 20px|link=Square 480 20px|link=Square 960 20px|link=Square 320 20px|link=Equilateral triangle 640 20px|link=Hexagon 320 20px|link=Equilateral triangle
Edges	2880	960+960+960
Vertices	960
Vertex figure
Coxeter group	B5 [4,3,3,3]
Properties	convex

Alternate names

• Runcicantellated penteract
• Prismatorhombated penteract (Acronym: prin) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of a runcicantellated 5-cube having edge length 2 are all permutations of:

[math]\displaystyle{ \left(\pm1,\ \pm1,\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2}),\ \pm(1+2\sqrt{2})\right) }[/math]

Images

orthographic projections

Coxeter plane	B5	B4 / D5	B3 / D4 / A2
Graph
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B2	A3
Graph
Dihedral symmetry	[4]	[4]

Runcicantitruncated 5-cube

Runcicantitruncated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	t0,1,2,3{4,3,3,3}
Coxeter-Dynkin diagram
4-faces	202
Cells	1560
Faces	4240
Edges	4800
Vertices	1920
Vertex figure	Irregular 5-cell
Coxeter group	B5 [4,3,3,3]
Properties	convex, isogonal

Alternate names

• Runcicantitruncated penteract
• Biruncicantitruncated pentacross
• great prismated penteract (gippin) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of a runcicantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:

[math]\displaystyle{ \left(1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2},\ 1+3\sqrt{2}\right) }[/math]

Images

orthographic projections

Coxeter plane	B5	B4 / D5	B3 / D4 / A2
Graph
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B2	A3
Graph
Dihedral symmetry	[4]	[4]

Related polytopes

These polytopes are a part of a set of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.

References

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN:978-0-471-01003-6 [1]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "5D uniform polytopes (polytera)". https://bendwavy.org/klitzing/dimensions/polytera.htm.  o3x3o3o4x - span, o3x3o3x4x - pattin, o3x3x3o4x - prin, o3x3x3x4x - gippin

External links

• Glossary for hyperspace, George Olshevsky.
• Polytopes of Various Dimensions, Jonathan Bowers
  • Runcinated uniform polytera (spid), Jonathan Bowers
• Multi-dimensional Glossary