Stericated 5-cube

In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations ( sterication) of the regular 5-cube.

There are eight degrees of sterication for the 5-cube, including permutations of runcination, cantellation, and truncation. The simple stericated 5-cube is also called an expanded 5-cube, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-cube. The highest form, the steriruncicantitruncated 5-cube, is more simply called an omnitruncated 5-cube with all of the nodes ringed.

Stericated 5-cube

Alternate names

* Stericated penteract / Stericated 5-orthoplex / Stericated pentacross
* Expanded penteract / Expanded 5-orthoplex / Expanded pentacross
* Small cellated penteractitriacontaditeron (Acronym: scant) (Jonathan Bowers)

Coordinates

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

:$\left(\pm1,\ \pm1,\ \pm1,\ \pm1,\ \pm(1+\sqrt)\right)$

Images

The stericated 5-cube is constructed by a sterication operation applied to the 5-cube.

Dissections

The stericated 5-cube can be dissected into two tesseractic cupolae and a runcinated tesseract between them. This dissection can be seen as analogous to the 4D runcinated tesseract being dissected into two cubic cupolae and a central rhombicuboctahedral prism between them, and also the 3D rhombicuboctahedron being dissected into two square cupolae with a central octagonal prism between them.

Steritruncated 5-cube

Alternate names

* Steritruncated penteract
* Celliprismated triacontaditeron (Acronym: capt) (Jonathan Bowers)

Construction and coordinates

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

:$\left(\pm1,\ \pm(1+\sqrt),\ \pm(1+\sqrt),\ \pm(1+\sqrt),\ \pm(1+2\sqrt)\right)$

Images

Stericantellated 5-cube

Alternate names

* Stericantellated penteract
* Stericantellated 5-orthoplex, stericantellated pentacross
* Cellirhombated penteractitriacontiditeron (Acronym: carnit) (Jonathan Bowers)

Coordinates

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

:$\left(\pm1,\ \pm1,\ \pm1,\ \pm(1+\sqrt),\ \pm(1+2\sqrt)\right)$

Images

Stericantitruncated 5-cube

Alternate names

* Stericantitruncated penteract
* Steriruncicantellated triacontiditeron / Biruncicantitruncated pentacross
* Celligreatorhombated penteract (cogrin) (Jonathan Bowers)

Coordinates

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

:$\left(1,\ 1+\sqrt,\ 1+2\sqrt,\ 1+2\sqrt,\ 1+3\sqrt\right)$

Images

Steriruncitruncated 5-cube

Alternate names

* Steriruncitruncated penteract / Steriruncitruncated 5-orthoplex / Steriruncitruncated pentacross
* Celliprismatotruncated penteractitriacontiditeron (captint) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of a steriruncitruncated penteract having an edge length of 2 are given by all permutations of coordinates and sign of:

:$\left(1,\ 1+\sqrt,\ 1+1\sqrt,\ 1+2\sqrt,\ 1+3\sqrt\right)$

Images

Steritruncated 5-orthoplex

Alternate names

* Steritruncated pentacross
* Celliprismated penteract (Acronym: cappin) (Jonathan Bowers)

Coordinates

Cartesian coordinates for the vertices of a steritruncated 5-orthoplex, centered at the origin, are all permutations of
:$\left(\pm1,\ \pm1,\ \pm1,\ \pm1,\ \pm(1+\sqrt)\right)$

Images

Stericantitruncated 5-orthoplex

Alternate names

* Stericantitruncated pentacross
* Celligreatorhombated triacontaditeron (cogart) (Jonathan Bowers)

Coordinates

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

:$\left(1,\ 1,\ 1+\sqrt,\ 1+2\sqrt,\ 1+3\sqrt\right)$

Images

Omnitruncated 5-cube

Alternate names

* Steriruncicantitruncated 5-cube (Full expansion of omnitruncation for 5-polytopes by Johnson)
* Omnitruncated penteract
* Omnitruncated triacontiditeron / omnitruncated pentacross
* Great cellated penteractitriacontiditeron (Jonathan Bowers)Klitzing, (x3x3x3x4x - gacnet)

Coordinates

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

:$\left(1,\ 1+\sqrt,\ 1+2\sqrt,\ 1+3\sqrt,\ 1+4\sqrt\right)$

Images

Full snub 5-cube

The full snub 5-cube or omnisnub 5-cube, defined as an alternation of the omnitruncated 5-cube is not uniform, but it can be given Coxeter diagram and symmetry ,3,3,3sup>+, and constructed from 10 snub tesseracts, 32 snub 5-cells, 40 snub cubic antiprisms, 80 snub tetrahedral antiprisms, 80 3-4 duoantiprisms, and 1920 irregular 5-cells filling the gaps at the deleted vertices.

Related polytopes

This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.

Notes

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, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995
wiley.com

*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380–407, MR 2,10*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559–591*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3–45* Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D.
* x3o3o3o4x - scan, x3o3o3x4x - capt, x3o3x3o4x - carnit, x3o3x3x4x - cogrin, x3x3o3x4x - captint, x3x3x3x4x - gacnet, x3x3x3o4x - cogart

External links

*

Polytopes of Various Dimensions
Jonathan Bowers



{{Polytopes

5-polytopes