7-2 step prism

From Polytope Wiki
Jump to navigation Jump to search
7-2 step prism
Rank4
TypeIsogonal
Elements
Cells7+7 phyllic disphenoids
Faces14 scalene triangles, 7+7 isosceles triangles
Edges7+7+7
Vertices7
Vertex figureBilaterally-symmetric bi-apiculated tetrahedron
Measures (circumradius , based on a uniform duoprism)
Edge lengths5-valence (7):
 4-valence (7):
 3-valence (7):
Central density1
Related polytopes
Dual7-2 gyrochoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(7)-2), order 14
ConvexYes
NatureTame

The 7-2 step prism is a convex isogonal polychoron and a member of the step prism family. It has 14 phyllic disphenoids of two different types as cells, with 8 joining at each vertex. It can also be constructed as the 7-3 step prism.

It is the simplest step prism, excluding the pentachoron and the triangular duotegum, which are part of more specific families, as well as the only isogonal polychoron with 7 vertices. It is also the triangular funk tegum.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is .

Vertex coordinates[edit | edit source]

Coordinates for the vertices of a 7-2 step prism inscribed in a heptagonal duoprism with base lengths a  and b  are given by:

  • ,

where k  is an integer from 0 to 6. If the edge length differences are to be minimized, the ratio of a :b  must be equivalent to .

Measures[edit | edit source]

The hypervolume of a 7-2 step prism inscribed in a heptagonal-heptagonal duoprism with base lengths a  and b  is given by:

where ξ  ≈ 1.55622 is the largest real root of , equivalent to .

Isogonal derivatives[edit | edit source]

Substitution by vertices of these following elements will produce these convex isogonal polychora:

  • Phyllic disphenoid (7): 7-2 step prism
  • Scalene triangle (7): 7-2 step prism
  • Scalene triangle (14): 14-2 step prism
  • Edge (7): 7-2 step prism

External links[edit | edit source]