Heptagonal duoexpandoprism

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 $$1:\frac{2}{1+{\tan\frac{\pi}{14}}\sqrt{3+4\cos\frac{\pi}{7}}}\approx 1:1.26060$$.