# Heptagonal duoexpandoprism

Heptagonal duoexpandoprism
Rank4
TypeIsogonal
Notation
Bowers style acronymHedep
Coxeter diagramxo7xx ox7xx&#zy
Elements
Cells49 tetragonal disphenoids, 98 wedges, 49 rectangular trapezoprisms, 14+14 heptagonal prisms
Faces196 isosceles triangles, 196 isosceles trapezoids, 98+98 rectangles, 28 heptagons
Edges98+98+196+196
Vertices196
Vertex figureMirror-symmetric triangular antiprism
Measures (based on two heptagonal-tetradecagonal duoprisms of edge length 1)
Edge lengthsEdges of duoprisms (98+98+196): 1
Lacing edges (196): ${\displaystyle {\frac {\sqrt {2}}{2\sin {\frac {\pi }{7}}}}\approx 1.62971}$
Circumradius${\displaystyle {\sqrt {\frac {3+2\cos({\frac {\pi }{7}})}{2-2\sin({\frac {3\pi }{14}})}}}}$
Central density1
Related polytopes
ArmyHedep
RegimentHedep
DualHeptagonal duoexpandotegum
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryI2(7)≀S2, order 392
ConvexYes
NatureTame

The heptagonal duoexpandoprism or hedep is a convex isogonal polychoron and the sixth member of the duoexpandoprism family. It consists of 28 heptagonal prisms of two kinds, 49 rectangular trapezoprisms, 98 wedges, and 49 tetragonal disphenoids. 2 heptagonal prisms, 1 tetragonal disphenoid, 3 wedges, and 2 rectangular trapezoprisms join at each vertex. It can be obtained as the convex hull of two orthogonal heptagonal-tetradecagonal duoprisms, or more generally heptagonal-diheptagonal duoprisms, and a subset of its variations can be constructed by expanding the cells of the heptagonal duoprism outward. However, it cannot be made uniform.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is ${\displaystyle {\frac {2}{1+{\tan {\frac {\pi }{14}}}{\sqrt {3+4\cos {\frac {\pi }{7}}}}}}\approx 1.26060}$.