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
(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	7+7 phyllic disphenoids
Faces	14 scalene triangles, 7+7 isosceles triangles
Edges	7+7+7
Vertices	7
Vertex figure	Bilaterally-symmetric bi-apiculated tetrahedron
Measures (circumradius $\sqrt2$, based on a uniform duoprism)
Edge lengths	5-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 density	1
Related polytopes
Dual	7-2 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(7)-2), order 14
Convex	Yes
Nature	Tame

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

External links

• Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".