An n-gonal cupola is defined as being an n-gon (the "n-face") attached to a 2n-gon (the "2n-face") by a row of alternating rectangles and isosceles triangles. These pages concentrate on regular cupolas where the triangles and rectangles become equilateral triangles and squares respectively. Three members of the family, the triangular, square and pentagonal cupolas exist as Johnson Solids; indeed the term 'cupola' was coined by Dr Norman Johnson in the early 1960s when he was investigating regular-faced solids and found there were no established names even for many of the obvious ones, except for pyramids and bipyramids1. The definition can be extended to cover star polyhedra, so for example the 7/3-cupola illustrated above consists of a 7/3-gon attached to a 14/3-gon.
Using n/d to describe an n/d cupola, then in the case of a regularly faced cupola, an upper limit exists of n/d<6 beyond which the distance between the n/d-gon and the 2n/d-gon becomes too great to span with triangles and squares. If n/d < 2 then the base n/d-gon can be thought of as being retrograde, cupolas such as the 4/3-cupola (which has a retrograde square connected to an 8/3-gon) then become possible. Note that n/d>6/5 is necessary for the triangles and squares to span between the two faces.
The height (H) of an n/d cupola can be shown to be:
This has a maximum value of sqrt(3)/2 if n/d = 2 (i.e. the triangular faces are upright. The height diminishes to 0 at n/d = 6 and n/d = 6/5 to explicitly set the limits described above.
In order for a cupola to exist, n
and d must be co-prime, d must also be odd.
FOOTNOTE
1. Dr Norman Johnson, private communication, December 2004.