7-2 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

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:$\sqrt{2\cos\frac\pi7}$ ≈ 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".