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 = $\frac { { \pi }^{ 3 } }{ 384 } l^{ 6 }$