Polyteron

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.^{[original research?]} A proposed name polyteron, (plural polytera),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.

Definition

A 5-polytope, or polyteron, is a closed five-dimensional figure with vertices, edges, faces, and cells, and hypercells (or terons). 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.

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 - 5-polytope
Polypeton - 6-polytope
Polyexon - 7-polytope
Polyzetton - 8-polytope
Polyyotton - 9-polytope

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 polyterons:

{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)

Each of these regular forms can generate 30 uniform polyterons using truncation, cantellation, runcination, and sterication operations.

Prisms and pyramids

Prismatic polyterons, or 5-prisms, can be generated by two similar polychorons in parallel hyperplanes connected by hyperprisms.

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.

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.

References

T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900

External links

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