The post actually gives the heptagonal fibonacci series (sevenly flat). This is a two-dimensional series, where there is a convergent region where the three values stand in the ratios of the chords of the heptagon. The thing is periodic over every prime, generally in a period of p²+p+1 (for primes mod 2,3,4,5 mod 7), or p-1 (for primes modulo 0,1,6 mod 7). You will probably recognise some of these numbers from your thesis (which deals with Z7 in a large part).
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2.246 979 603 7 5377 = 2.246 970 330
1.000 000 000 0 1.801 937 736 eg 2393 4312 = 1.000 000 000 1.801 922 273
Where B is one vertical, and A is one step horizontal, the calculation rules are the same as what holds in multiplication: A² = 1+B, AB = A=B, B² = 1+AB. For example, where 4312 is at (8,4), then 14001 = 4312 + 9689, 17459 = 7770 + 9689, and 21771 = 4312 + 17459.
Using the same relations on an grid with counters, one can show from these three relations (alone), things like A solves A³+1 = A²+2A, B solves B³+1 = 2B²+B, that (-1+AB)³ = 7 AB, and that any one of the relations can be found from the other two. This is an 'abacus' or stone-board, for calculating heptagonal numbers.
Such convergent regions exist for all polygons, the region for the 13-gon is also been found by a general process.
One can do similar things with the pentagonal numbers (by the rule that F² = F+I, which is the same rule that creates the fibonacci series), with octagonal numbers, (where one uses a two-dimensional grid, vertical by Q = root-2, and horizontal by A = sqrt(2)+1, the two rules are A = Q+I, and AQ = A + I. These by themselves generate the two octagonal series. A = alpha, the unit of the octagonal numbers.
The dodecagonal series uses Q and W, being sqrt(2) and (sqrt(6)+sqrt(2))/2. One uses the relation that QQ = I + I, and WW = WQ + I. (W = omega, the symbol i use for the shortchord of the dodecagon). Its square is the unit of the numbers of the system Z6 (span of heptagonal chords).
Once i learnt enough from these systems, and the transport across branches on the dynkin symbol, one can successfully predict the sorts of numbers that is necessary to make the chords of any polygon. A slightly less advanced trick, based on the same system, gives the sorts of numbers that make the chords of rational angles, from which one can (by looking at the cosine of an angle), reject without further consideration, that certian angles are not rational. For example, the van Oss polygon of the {3,5/2,3} gives a shortchord of 1.070466 or sqrt(3/2.61803398875). However, the only rational angles that can have a weight of sqrt(3) belong to 120 deg ie (sqrt(3)), and that this particular value is not a valid rational angle for that weight. The van Oss polygon does not close, and therefore {3,5/2,3} is infinitely dense.
The use of transport across the dynkin branch helps to calculate the sides etc to give required edges. It also helps to determine the sorts of chords necessary for any composite polygon, eg {21} or {14}.
Many of the things learnt here were then used to develop hyperbolic geometry from first principles.
The general theory exists with the general polytope. It's one of the filters that i use to tell me what works and what does not.
It should be remembered, that polytopes are the integers of geometry, and one can as much understand the nature of the containing geometry by looking on the polytopes it contains, as one does find the reals from looking at the integers of various types, that rest on the reals (and complex, and quarterions etc).