5-polytope

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In geometry, a five-dimensional polytope, or 5-polytope, is a polytope in 5-dimensional space. Each polyhedral cell being shared by exactly two polychoron facets.

A proposed name polyteron, has been advocated (plural: polytera), from the Greek root poly, meaning "many", a shortened tetra meaning "four" and suffix -on. Four refers to the dimension of the 5-polytope facets.

1 Definition
2 Regular forms
3 Prisms and pyramids
4 Semiregular forms
5 A note on generality of terms for n-polytopes and elements
6 See also
7 References
8 External links

[edit] Definition

A 5-polytope, or polyteron, is a closed five-dimensional figure with vertices, edges, faces, and cells, and hypercells. A vertex is a point where five or more edges meet. An edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A cell is the a polyhedron, and a hypercell is a polychoron. Furthermore, the following requirements must be met:

Each cell must join exactly two hypercells.
Adjacent hypercells are not in the same four-dimensional hyperplane.
The figure is not a compound of other figures which meet the requirements.

[edit] Regular forms

Regular polytera can be represented by the Schläfli symbol {p,q,r,s}, with s {p,q,r} polychoral facets around each face.

There are 3 finite regular 5-polytopes:

{3,3,3,3} - hexa-5-tope or 5-simplex (Elements: 6 facets {3,3,3}, cells=15 {3,3}, faces=20 {3}, edges=15, vertices=6)
{4,3,3,3} - deca-5-tope or penteract or 5-measure polytope (Elements: 10 facets {4,3,3}, C=40 {4,3}, F=80 {4}, E=80, V=32)
{3,3,3,4} - triacontadi-5-tope or pentacross or 5-cross-polytope (Elements: 32 facets {3,3,3}, C=80 {3,3}, F=80 {3}, E=40, V=10)

The self-dual {3,3,3,3} regular polytope can generate 18 uniform polytopes from special points of a Wythoff construction. Similarly the {3,3,3,4} and {4,3,3,3} dual pair can generate another 29 uniform polytopes.

[edit] Prisms and pyramids

There are 3 categorical uniform prismatic forms:

{} x {p,q,r} - polychoron prisms
{p} x {q,r} - polygonal-polyhedral duoprisms
{} x {p} x {q} - duoprism prisms

Pyramidal polyterons, or 5-pyramids, can be generated by a polychoron base in a 4-space hyperplane connected to a point off the hyperplane. The 5-simplex is the simplest example with a 4-simplex base.

[edit] Semiregular forms

There is a one semiregular polytope from a set of semiregular n-polytopes called a half measure polytope, discovered by Thorold Gosset in his complete enumeration of semiregular polytopes. They are all formed by half the vertices of a measure polytope (alternatingly truncated).

This one is called a demipenteract. It has 16 vertices, with 10 16-cells, and 16 5-cells.

[edit] A note on generality of terms for n-polytopes and elements

A 5-polytope, or polyteron, follows from the lower dimensional polytopes: 2: polygon, 3: polyhedron, and 4: polychoron.

In more generality, although there is no agreed upon standard for higher polytopes, following a SI prefix sequencing, a proposed sequence of higher polytopes may be called:

Polyteron as a name for 5-polytope (tera for 4D faceted polytope), and terons for 4-face element.
Polypeton as a name for 6-polytope (peta for 5D faceted polytope), and petons for 5-face elements.
Polyexon as a name for 7-polytope (exa for 6D faceted polytope), and exons for 6-face elements.
Polyzetton as a name for 8-polytope (zetta for 7D faceted polytope), and zettons for 7-face elements.
Polyyotton as a name for 9-polytope (yotta for 8D faceted polytope), and yottons for 8-face elements.