Pairwise augmented cupola

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The pairwise augmented cupolae are a family of twelve acrohedra (nonconvex polyhedra with regular faces) discovered by Jim McNeill and named by Alex Doskey. The family is notable for containing solutions to the 7-4-3, 9-4-3, and 11-4-3 acrons, which are unusual as few acrons containing heptagons, enneagons, or hendecagons have known acrohedra. They are non-self-intersecting.

Pairwise augmented cupola are created by starting with a regular n-gon for odd n, and attaching an alternating ring of (n + 1) / 2 triangles and (n - 1) / 2 squares, connected like a cupola with a gap. Every square is augmented with an additional triangle and these triangles are joined at a single point. The resulting shape, which McNeill calls a module, looks like a cupola augmented with a "n/2-gonal pyramid" but with a rhombus-shaped gap. Two modules can be joined together at their gaps to produce a valid acrohedron (ortho- and para- depending on relative orientation of the modules) or joined with a "n/2-gonal tegum" (meta-).

The complete list is as follows:

  • n = 5 (5-4-3 acrohedra)
  • n = 7 (7-4-3 acrohedra)
  • n = 9 (9-4-3 acrohedra)
  • n = 11 (11-4-3 acrohedra)

The angular defect of an n-4-3 acron is . This is 42 degrees for n = 5, 22.429 degrees for n = 7, 10 degrees for n = 9, and only 2.727 degrees for n = 11, making the 11-4-3 pairwise augmented cupolae close to flat. For n = 13, the angular defect is negative, which is geometrically unrealizable for three faces meeting at a vertex, so no pairwise augmented cupolae exist for n = 13 and above.

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