Constructible polyhedra in the light of constructiblepolygons?

Identifying Polyhedra Enabling Memorable Strategic Mapping (Part #5)

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If the challenge is one of presenting coherently the elements on a map of some kind -- some form of mind map -- the geometrical constraints in the case of a polygon are one point of departure. As noted above, there is a well-recognized understanding of what constitutes a constructible polygon -- notably because of the constraints on pattern formation by prime numbers.

The table below is of particular interest in that it covers the range of numbers up to 1,000 -- namely the range which typically includes the number of representatives in a legislative assembly. For example, seated in a hemicircle, the European Parliament numbers 705 representatives, the total being restricted to 751 by treaty, according to a system of apportionment.

Table 4: Number of sides of known constructible polygons having up to 1000 sides (bold) or odd side count (red)

Extracted from table in Wikipedia by Cmglee / CC BY-SA

An earlier exercise highlighted the challenge to governance of numbers of elements beyond 100, most evidently the tendency for numbers of parliamentary representatives to be several hundred (Dependence of viable global governance on pattern management? 2020). The cases of the European Parliament and any potential World Parliament Assembly were considered. With respect to memorable mappability that exercise noted the mathematical literature on constructible polygons in 2D, usefully summarized by that table.

Of potential interest is whether the numbers in that table are especially indicative of "constructible polyhedra", however that might be understood -- irrespective of more sophisticated mathematical approaches to the refinement of that question and detection of possible candidates. To that end a first process was simply to copy into the following table the corresponding elements from the more promising candidates in the range up to 100 (where they matched the numbers in Table 1 above). For the higher numbers, the procedure was then to extend the earlier process with the numbers in the range up to 1000.

Note that in the method for the following table no account is taken of polyhedra generated with prime numbers other than Fermat primes.

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