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.
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,
must be even, d must be odd. This is because n odd cannot
appear in Occurrence IV and d even cannot appear in
I, II, III, IV or V if n is also even.
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.
I. n/d co-prime: n-face in an n/d
(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)
II. n/d co-prime: Retrograde n-face in an n/d*
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)
III. n/d co-prime: Faceted 2n-face in n/2d semicupola
gyrobicupola if n is even) with faces n/2d and n/d
(Example: 4/3 sesquicupola with 4/3(x2) and 8/3 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|
|pentagon (5/1)||5/1||5/4||5/2||n odd||5/1ssq||compound|
|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|
|7/3-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|
|enneagon (9/1)||n/d 6||n/d* < 6/5||9/2||n odd||2n/2d 6||2n/2d* < 6/5|
|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|
to Uniform Polyhedra
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