Fano Space
This is a finite space, derived from the fano configuration. This is a space where there are 7 points, and 7 lines, where every lies on three lines, and there is one line between any pair of points.
It can be replicated by using the XOR operator over three variables, not including 0. The alignment is that a, b, c = 0 if a .xor. b .xor. c = 0.
One can view it as a polytope, with seven hedra (2d surtopes), and 7 edges. As one goes higher, one gets 15, 31, ... sides of the polytope, rather like a simplex.
The connexion between 4 and 15, and the factorisation of 1111, is that the sides of the fanotopes are in binary, 1, 11, 111, 1111, 11111, etc.
An 'algebraic root' is my name for the factors that you get, when you divide something like a six-digit period into factors with one, two, three and six periods (ie one can have a six digit period of the type ababab, or abcabc. For any base, these numbers take the same form
A1 = 9, A2 = 11, A3 = 111, A6 = 91 in decimal
A1 = 7, A2 = 11, A3 = 111, A6 = 71 in octal
A1 = 1, A2 = 11, A3 = 111, A6 = 11 in binary
These numbers are easier to factorise than 999999, 777777, or 111111.
A four-figure period then has algebraic roots A1, A2, A4 = 9 * 11 * 101
The implication to geometries is that a 15-space ought substain a subspace of order 3 and prehaps 5. However, we don't see a 5-space, and therefore we don't have this.
Vector-space
It is certianly possible to have a multiplication system that uses three coordinates, such as the integer-system Z7 (the span of chords of the heptagon). However, the system has an infinite number of units, and does not map onto any known set of rotations in space.
Unlike the reals, complex, quarterons and octonions, these do not correspond to rotations and dilations of real space.
The closest one can get out of Z7, for example, is the mapping of a Z7 point onto three (different) points on a number line. A point exists for every combination of (x,y,z), and there is a particular lattice in this space (actually several), which behave as integers [like Z7, Z9, and subsets of things like Z13, Z14, Z18, Z19, Z21, &c.]
Fiddling around with this kind of stuff tells you that there must exist a matrix where the first column, in order, converges on the lengths of the chords of a polygon: for example,
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20 36 45
36 65 81
45 81 101
This matrix has a determinate of 1, and successive powers approximate the chords of the heptagon, to ever increasing decimals.