Pairwise augmented cupola
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 743, 943, and 1143 acrons, which are unusual as few acrons containing heptagons, enneagons, or hendecagons have known acrohedra. They are nonselfintersecting.
Pairwise augmented cupola are created by starting with a regular ngon 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/2gonal pyramid" but with a rhombusshaped 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/2gonal tegum" (meta).
The complete list is as follows:
 n = 5 (543 acrohedra)

orthodecatetradihedron

metadecatetradihedron
(Prize substitute) 
paradecatetradihedron
 n = 7 (743 acrohedra)

orthotetrakaidecahexadihedron

metatetrakaidecahexadihedron

paratetrakaidecahexadihedron
 n = 9 (943 acrohedra)

orthooctakaidecaoctadihedron

metaoctakaidecaoctadihedron

paraoctakaidecaoctadihedron
 n = 11 (1143 acrohedra)

orthoicosikaididecadihedron

metaicosikaididecadihedron

paraicosikaididecadihedron
The angular defect of an n43 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 1143 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.
External links[edit  edit source]
 Jim McNeill. 743, 943, and 1143.