# 7-2 step prism

7-2 step prism
Rank4
TypeIsogonal
SpaceSpherical
Elements
Cells7+7 phyllic disphenoids
Faces14 scalene triangles, 7+7 isosceles triangles
Edges7+7+7
Vertices7
Vertex figureBilaterally-symmetric bi-apiculated tetrahedron
Measures (circumradius ${\displaystyle \sqrt2}$, based on a uniform duoprism)
Edge lengths5-valence (7): ${\displaystyle 2\sin\frac\pi7\sqrt{3+2\cos\frac{2\pi}{7}} ≈ 1.78831}$
4-valence (7): ${\displaystyle 2\sqrt{\sin^2\frac\pi7+\cos^2\frac{\pi}{14}} ≈ 2.13423}$
3-valence (7): ${\displaystyle \sqrt{4+2\cos\frac\pi7+2\sin\frac{\pi}{14}} ≈ 2.49940}$
Central density1
Related polytopes
Dual7-2 gyrochoron
Abstract properties
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryS2(I2(7)-2), order 14
ConvexYes
NatureTame
Discovered by{{{discoverer}}}

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:${\displaystyle \sqrt{2\cos\frac\pi7}}$ ≈ 1:1.34236.

## Vertex coordinates

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:${\displaystyle \sqrt{2\cos\frac\pi7}}$ ≈ 1:1.34236.

## Measures

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

${\displaystyle V=a^2b^2\xi,}$

where ξ ≈ 1.55622 is the largest real root of 884736x3–1613472x–823543, equivalent to 49/(192cos(2π/7)-96)-49/96.

## Isogonal derivatives

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