Demipenteract

In five dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a "5-hypercube" (penteract) with alternated vertices deleted.

It was discovered by Thorold Gosset. Since it was the only "semiregular" 5-polytope (made of more than one type of regular hypercell), he called it a "5-ic semi-regular".

Coxeter named this polytope as 1₂₁ from its Coxeter-Dynkin diagram, which has branches of length 2, 1 and 1 with a ringed node on one of the short branches. It exists in the k₂₁ polytope family as 1₂₁ with the Gosset polytopes: 2₂₁, 3₂₁, and 4₂₁.

It is a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 23 uniform polyterons (uniform 5-polytope) that can be constructed from the B₅ symmetry of the demipenteract, 7 of which are unique to this family, and 16 are shared within the penteractic family.

Cartesian coordinates

Cartesian coordinates for the vertices of a demipenteract centered at the origin and edge length 2√2 are alternate halves of the penteract:: (±1,±1,±1,±1,±1)with an odd number of plus signs.

Projected images

See also

* Other 5-polyopes:
** 5-simplex (hexateron) - {3,3,3,3}
** 5-cube (penteract) - {4,3,3,3}
** 5-orthoplex (pentacross) - {3,3,3,4}
* Others in the hypercube family
** 5-polytope
** Demihypercube

References

* T. Gosset: "On the Regular and Semi-Regular Figures in Space of n Dimensions", Messenger of Mathematics, Macmillan, 1900
* John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, "The Symmetry of Things" 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_n1)

External links

*GlossaryForHyperspace | anchor=half | title=Demipenteract
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]