# 7-2 step prism

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 ${\displaystyle {\sqrt {2}}}$, based on a uniform duoprism)
Edge lengths5-valence (7): ${\displaystyle 2\sin {\frac {\pi }{7}}{\sqrt {3+2\cos {\frac {2\pi }{7}}}}\approx 1.78831}$
4-valence (7): ${\displaystyle 2{\sqrt {\sin ^{2}{\frac {\pi }{7}}+\cos ^{2}{\frac {\pi }{14}}}}\approx 2.13423}$
3-valence (7): ${\displaystyle {\sqrt {4+2\cos {\frac {\pi }{7}}+2\sin {\frac {\pi }{14}}}}\approx 2.49940}$
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 ${\displaystyle 1:{\sqrt {2\cos {\frac {\pi }{7}}}}\approx 1:1.34236}$.

## 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:

• ${\displaystyle (a\sin(2\pi k/7),a\cos(2\pi k/7),b\sin(4\pi k/7),b\cos(4\pi k/7))}$,

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 ${\displaystyle 1:{\sqrt {2\cos {\frac {\pi }{7}}}}\approx 1:1.34236}$.

## Measures

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

${\displaystyle V=a^{2}b^{2}\xi ,}$
where ξ  ≈ 1.55622 is the largest real root of ${\displaystyle 884736x^{3}-1613472x-823543}$, equivalent to ${\displaystyle 49/(192\cos(2\pi /7)-96)-49/96}$.

## 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