Cupola OR Semicupola

For each n/d (within the bounds of 6/5<n/d<6) one of three conditions applies:

1. n/d are co-prime, d odd: A cupola exists with bases n/d and 2n/d

2. n/d are co-prime, d even: A semicupola exists with bases n/d and 2n/d which is equivalent to 2 x n/(d/2), the latter face being faceted.

3. n/d are not coprime: No elementary cupola or semicupola exists, but compound cupolas or semicupolas may.

So for any n/d there may be a cupola or a semicupola but not both.

Table of cupolas and semicupolas

Cupolas in red, semicupolas in blue.

The list is complete for cupolas for n<=10 and for semicupolas with n<=12. The term ssq is used to denote a sesquicupola. 
5/2 5/3
  7/2 7/3
7/4 7/5
  9/2   9/4        
  11/2   11/4   11/6    11/8


There is more than one way in which a particular n/d-gon can be represented in these families. For the purpose of this analysis the starting assumption is made that the the n/d-gon itself is not retrograde. This means n/d>2. Empirically, it appears the maximum number of possible appearances a particular n/d-gon can make (including as a model for a faceted face, and including sesquicupolas) is 6. For an n/d-gon to appear in all six cases, n must be even, d must be odd. This is because n odd cannot appear in Occurrence IV and d even cannot appear in Occurrence I, II, III, IV or V if n is also even. Discounting retrograde polygons, of all those with up to 10 edges only the square, the 8/3-gon and the 10/3-gon appear in all six occurrences. Examples show the appearance of an 8/3-gon in each way.

The term d* is used below to denote (n-d) so a n/d*-gon is a retrograde n/d-gon.

Occurrence I. n/d co-prime: n-face in an n/d cupola (if d odd) with faces n/d & 2n/d, or semicupola (if d even) with faces n/d and n/(d/2) (Example:
                  8/3 cupola with 8/3 and 16/3 Faces)

Occurrence II. n/d co-prime: Retrograde n-face in an n/d* cupola (if d* odd) with faces n/d* & 2n/d* or semicupola (if d* even) with faces n/d* and
                    n/(d/2)  (Example: 8/5 cupola with 8/5 and 16/5 faces)

Occurrence III. n/d co-prime: Faceted 2n-face in n/2d semicupola (or gyrobicupola if n is even) with faces n/2d and n/d
                    (Example: 4/3 sesquicupola with 4/3(x2) and 8/3 faces).

The retrograde equivalent the n/2d* cupola cannot exist. The argument is as follows. As the n/d-gon is not retrograde then n/d>2. Therefore n/d*<2 and n/2d*<1. This is below the 6/5 lower limit (The example here would have been the 8/10 semicupola) Occurrence IV. n/d co-prime, n even: 2n-face in a (n/2)/d cupola with faces (n/2)/d & n/d (Example: 4/3 cupola with 4/3 and 8/3 Faces) Note that the retrograde equivalent, the (n/2)/d* cupola cannot exist. The argument is similar to the Occurrence III retrograde above. As the n/d-gon is not retrograde then n/d2. Therefore n/d*<2 and (n/2)/d*<1. This is below the 6/5 lower limit (The example here would have been the 4/5 cupola) Occurrence V. n/d co-prime:In a compound of two n/d-gons in the n-face of a sesquicupola with Faces n/d(x2) & 2n/d (Example: 8/3 sesquicupola with 8/3(x2) and 16/3 faces)

Occurrence VI. n/d* co-prime:In a compound of two n/d*-gons in the retrograde n-face of a sesquicupola with Faces n/d*(x2) and 2n/d* (Example: 8/5 sesquicupola with 8/5(x2) and 16/5 faces)

This appears to exhaust the possibilities.

The following table shows the six Occurrences by column and the n/d-gon by row to show the appearance of each n/d-gon up to n=10 in a number of cupolas/semicupolas. Modelled cupolas/semicupolas are linked. Where the cupola/semicupola name is given but not linked, it exists but has not been modelled. If it does not exist a reason is given in red as to why not.
  I: n/d cupola /semicupola II: n/d* cupola /semicupola III: n/2d semicupola/ sesquicupola IV: (n/2)/d cupola V: n/d sesquicupola VI: n/d* sesquicupola
Polygon appears as: n-face n-face Retrograde Faceted 2n-face 2n-face Compound n-face  Compound n-face Retrograde
n/d requirement: n/d co-prime n/d co-prime n/d co-prime n/d co-prime, n even n/d co-prime n/d* co-prime
digon (2/1) 2/1 2/1 n/2d < 6/5 (n/2)/d < 6/5 2/1ssq 2/1ssq
triangle (3/1) 3/1 3/2 3/2 n odd 3/1ssq compound
square (4/1) 4/1 4/3 2/1ssq 2/1 4/1ssq 4/3ssq
pentagon (5/1) 5/1 5/4 5/2 n odd 5/1ssq compound
5/2-gon 5/2 5/3 5/4 n odd compound 5/3ssq
hexagon (6/1) n/d = 6 n/d* = 6/5 3/1ssq 3/1 2n/2d = 6 2n/2d* = 6/5
heptagon (7/1) n/d 6 n/d* < 6/5 7/2 n odd 2n/2d 6 2n/2d* < 6/5
7/2-gon 7/2 7/5 7/4 n odd compound 7/5ssq
7/3-gon 7/3 7/4 n/2d < 6/5 n odd 7/3ssq compound
octagon (8/1) n/d 6 n/d* < 6/5 4/1ssq 4/1 2n/2d 6 2n/2d* < 6/5
8/3-gon 8/3 8/5 4/3ssq 4/3 8/3ssq 8/5ssq
enneagon (9/1) n/d 6 n/d* < 6/5 9/2 n odd 2n/2d 6 2n/2d* < 6/5
9/2-gon 9/2 9/7 9/4 n odd compound 9/7ssq 
9/4-gon 9/4 9/5 n/2d < 6/5 n odd compound 9/5ssq
decagon (10/1) n/d 6 n/d* < 6/5 5/1ssq 5/1 2n/2d 6 2n/2d* < 6/5
10/3-gon 10/3 10/7 5/3ssq 5/3 10/3ssq 10/7ssq

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