6 Dimensional Shapes

Skeleton graph of the heptapeton

Heptapeton

A heptapeton is a 6-dimensional simplex. It is also called the pyropeton under the elemental naming scheme.

Properties

Hypervolumes

• vertex count = $7$
• edge length = $21l$
• surface area = ${\frac {35{\sqrt {3}}}{4}}{l}^{2}$
• surcell volume = ${\frac {35{\sqrt {3}}}{12}}{l}^{3}$
• surteron bulk = ${\frac {7{\sqrt {5}}}{32}}{l}^{4}$
• surpeton pentavolume = ${\frac {7{\sqrt {3}}}{480}}{l}^{5}$
• surexon hexavolume = ${\frac {\sqrt {7}}{5760}}{l}^{6}$

Subfacets

• 7 points (0D)
• 21 lines (1D)
• 35 triangles (2D)
• 35 tetrahedra (3D)
• 21 pentachora (4D)
• 7 hexatera (5D)

Hexeract

A Hexeract is a 6-dimensional hypercube.

 Hypervolumes

• vertex count =$64$
• edge length =$192l$
• surface area =$240{l}^{2}$
• surcell volume =$160{l}^{3}$
• surteron bulk =$60{l}^{4}$
• surpeton pentavolume =$12{l}^{5}$
• surecton hexavolume =${l}^{6}$

Subfacets

• 64 points (0D)
• 192 line segment (1D)
• 240 squares (2D)
• 160 cubes (3D)
• 60 tesseracts (4D)
• 12 penteracts (5D)

6-orthoplex

An icosahedrally symmetric projection of the 6-orthoplex down to three dimensions

A 6-orthoplex (Hexarss) is the six-dimensional cross polytope. It is the dual of the hexeract and has a schläfli symbol of $\{3,3,3,3,4\}$.

Subfacets

• 12 points (0D)
• 60 line segments (1D)
• 160 triangles (2D)
• 240 tetrahedrons (3D)
• 192 tetrarsses (4D)
• 64 pentarsses (5D)

Hexaphere

A hexaphere is the 6-dimensional hypersphere.

  Hypervolumes

• vertex count = $N/A$
• edge length = $N/A$
• surface area = $N/A$
• surcell volume = $N/A$
• surteron bulk = $N/A$
• surpeton pentavolume = ${\frac {{\pi }^{3}}{32}}{l}^{5}$
• surecton hexavolume = ${\frac {{\pi }^{3}}{384}}l^{6}$