As my enlightenment is my own, perhaps scrutiny will expose faults in the supposition that follows.
An adequate accumulation of contextual datum can give rise to patterns, the comprehension of which offers understanding. It is however the data points themselves, their spatial and temporal juncture with lines of inquiry, that are fundamental. Whether these are literal or figurative in occurrence, answers can only arrive as a consequence of intersection.
~~~
I begin this thought experiment with the premise that multiplication is orthogonal.
Consider a three-dimensional unit in an enumerable form, of which you have many. A sufficient collection is thus capable of a space filling orientation in three perpendicular directions.
The quantities one, two, and three are incapable of this feat as it is four that is first to possess such attribute, although it has no arrangement equally divisible in each direction.
Therefore, an “irregular quantity” could be defined as any quantity that, while capable of contiguously populating a lattice arrangement in three orthogonal directions, is not equally divisible in each (see Fig. 2. of original draft below).
Eight then, is the first quantity that enjoys both the space filling attribute and the characteristic of being equally divisible in each direction.
As such, a “regular quantity” could be defined as any quantity that can fulfill a regular lattice arrangement in three orthogonal directions and is thereby equally divisible in each.
Put another way, “regular quantities” are the product of three natural numbers, all greater than one.
~~~
An intuitive extension of the aforementioned resides in the semi-abstractness of two dimensions where the tiling of a plane in perpendicular directions is possible. In this scenario, the quantities of one and two lack any ability to fill in two orthogonal directions. Three is the first quantity with the filling attribute yet it does not have the equal divisibility characteristic.
Applying the prior definitions in this two-dimensional construct reveals that not only are composite quantities like four, six, eight, nine, etc. plane filling, but they can be arranged with equal divisibility in both directions (see Fig. 1. of original draft below).
Using this spatial definition for composites, is primality implicit of the remainder? Generally, yes. Specifically… not so fast. Foregoing the debate on the primality of one, it is the quantity of two that presents itself as unreconcilable in this context. Two’s absence of the filling attribute, as with one, two, and three in three dimensions, brings pause.
If we assume that for one dimension (where all quantities are equally divisible) the first linear filling quantity in one direction is two, and that for three dimensions, the first space filling quantity in three directions is four, it will seem logical that for two dimensions, it’s first plane filling quantity is three. (Note that extending this beyond three dimensions seems suggestive of the equal divisibility characteristic becoming less and less likely.)
Obviously, the equal divisibility characteristic, sanctioning the exclusion of two from primality, is less than a palatable notion. In the pure abstract of mathematics, the idea runs up hard against the Fundamental Theorem of Arithmetic and afoul of Riemann’s conclusive affirmation of primes.
Yet if such characteristic were applicable, two would be neither prime nor composite in this two-dimensional construct. It is the seeming absurdness of just this, that begs the question: What then of the quantities one, two, and three in three dimensions? They are neither space filling nor equally divisible by definition. What then, are they?
This question could be rendered moot by withholding consideration of the space filling attribute, relying solely on the divisibility characteristic to classify them as “irregular quantities”? But this, seems quite the dodge. And perhaps reconciliation simply resides in the forest… behind all those trees.
~~~
As a sidebar to this thought experiment, it has caused me to consider the divisibility of a dimension. It would seem to divide something of nth dimension… requires a “knife” of n-1.
Stephen A. Wait
8th of February, 2024
