8-2 step prism

8-2 step prism
(OFF file)
Rank	4
Type	Isogonal
Elements
Cells	8+8 phyllic disphenoids, 4 rhombic disphenoids
Faces	16+16 scalene triangles, 8 isosceles triangles
Edges	4+8+8+8
Vertices	8
Vertex figure	Ridge-triakis notch
Measures (circumradius ${\sqrt {2}}$, based on a uniform duoprism)
Edge lengths	6-valence (8): ${\sqrt {4-{\sqrt {2}}}}\approx 1.60804$
	4-valence (4): 2
	4-valence (8): ${\sqrt {4+{\sqrt {2}}}}\approx 2.32685$
	3-valence (8): ${\sqrt {6}}\approx 2.44949$
Central density	1
Related polytopes
Dual	8-2 gyrochoron
Abstract & topological properties
Flag count	480
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(8)-2), order 16
Flag orbits	30
Convex	Yes
Nature	Tame

The 8-2 step prism is a convex isogonal polychoron and a step prism. It has 4 rhombic disphenoids and 16 phyllic disphenoids of two kinds as cells, with a total of 10 (2 rhombic and 8 phyllic disphenoids) joining at each vertex. It is one of 3 isogonal polychora with 8 vertices, and the only one not to be uniform (the other 2 are the hexadecachoron and tetrahedral prism).

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {\sqrt {6+2{\sqrt {2}}}}{2}}$ ≈ 1:1.48563.

Vertex coordinates

Coordinates for the vertices of an 8-2 step prism inscribed in an octagonal duoprism with base lengths a and b are given by:

• (a*sin(πk/4), a*cos(πk/4), b*sin(πk/2), b*cos(πk/2)),

where k is an integer from 0 to 7. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:${\frac {\sqrt {4+2{\sqrt {2}}}}{2}}$ ≈ 1:1.30656.

Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora:

• Phyllic disphenoid (8): 8-2 step prism
• Scalene triangle (8): 8-2 step prism
• Scalene triangle (16): 16-2 step prism
• Edge (8): 8-2 step prism

External links

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