# 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 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)- ortho-decatetradihedron
- meta-decatetradihedron
- para-decatetradihedron

*n*= 7 (7-4-3 acrohedra)- ortho-tetrakaidecahexadihedron
- meta-tetrakaidecahexadihedron
- para-tetrakaidecahexadihedron

*n*= 9 (9-4-3 acrohedra)- ortho-octakaidecaoctadihedron
- meta-octakaidecaoctadihedron
- para-octakaidecaoctadihedron

*n*= 11 (11-4-3 acrohedra)- ortho-icosikaididecadihedron
- meta-icosikaididecadihedron
- para-icosikaididecadihedron

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.

## External links[edit | edit source]

- Jim McNeill. 7-4-3, 9-4-3, and 11-4-3.