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
bipyramids^{1}. 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.