(In which we define the uniqueness of planes)
If you cycle the procession of numbers through incremental levels, so as to simulate a gradual increase in the size of a cycle, patterns of number types and uniqueness unfold in the individual degrees of each successive band. It can be seen, early, in the short cycle that the numbers of most unique identity rotate about a centre which is 1/2 of that required to define any one unique object. Inclusive, also, in this particular weave of uniqueness through the numbers, are the first numbers present at which one can define a plane space. In simple terms we can call these the prime numbers, the first multiple numbers for any prime set, and the prime squared numbers. These numbers represent the threshold of every unique topological magnitude and they all appear in the same braid of the cycle, as an apparent “rule”.
It can also be seen early in the procession that an exception to this rule exists in the number 9. Nine is the square of 3. As a square, by this rule, it represents the point of threshold into a unique degree of plane space. Other numbers with this characteristic are found in the braid rotating about the half. But those with a factor of three are not. We define 3 as a prime number, and, in all respects the definition holds. It is it in the uniqueness of 3, understanding, that this exception is to make the rule, arises.
As you observe the procession gradually increasing you can note in the image that each degree either sorts or scatters factors across its degrees but always about poles defined by the factor 3.
Prime uniqueness holds for the number 3 in the sense that while all other prime values expand into plane spaces as variants on two dimensional surface, through the squares and the first multiples, in contrast, 3 multiples uniquely have a capacity to expand in an orthogonal direction of those planes.
I will suggest to you, now, that this relates back to my previous post on 3 being the most basic number of “things” which must necessarily exist to define any unitary “thing”. Prior to 3 our integers are not accurate referents of anything in reality, from a philosophical standpoint. Things we define using 1 or 2 as a referent are abstracted from their true identity, so what we surmise from the premise of using them this way, will mislead our results. On the other hand, if we construct our reality from the basis of three it assembles itself into these unique parts naturally and we can say that 1 and 2 (in their factor behaviour) accurately represent interdependent objects in a universe, while the polarity of three against both of these patterns equally satisfyingly looks like a platform orthogonal to both, on which they may exist.