7-2 step prism

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7-2 step prism
7-2 step prism.png
SymmetryI2(7)+×2×I, order 14
Vertex figureBilaterally-symmetric bi-apiculated tetrahedron
Cells7+7 phyllic disphenoids
Faces7+7+14 scalene triangles
Measures (circumradius , based on a uniform duoprism)
Edge lengths5-valence (7):
 4-valence (7):
 3-valence (7):
Central density1
Euler characteristic0
Related polytopes
Dual7-2 gyrochoron

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 1: ≈ 1:1.34236.

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:

  • (a*sin(2πk/7), a*cos(2πk/7), b*sin(4πk/7), b*cos(4πk/7)),

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 1: ≈ 1:1.34236.

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 884736x3–1613472x–823543, equivalent to 49/(192cos(2π/7)-96)-49/96.

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]