# 7-2 step prism

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.

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 $\sqrt2$ , based on a uniform duoprism)
Edge lengths5-valence (7): $2\sin\frac\pi7\sqrt{3+2\cos\frac{2\pi}{7}} ≈ 1.78831$ 4-valence (7): $2\sqrt{\sin^2\frac\pi7+\cos^2\frac{\pi}{14}} ≈ 2.13423$ 3-valence (7): $\sqrt{4+2\cos\frac\pi7+2\sin\frac{\pi}{14}} ≈ 2.49940$ Central density1
Related polytopes
Dual7-2 gyrochoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryS2(I2(7)-2), order 14
ConvexYes
NatureTame

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:$\sqrt{2\cos\frac\pi7}$ ≈ 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:$\sqrt{2\cos\frac\pi7}$  ≈ 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:

$V=a^2b^2\xi,$

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