September 01, 2018 at 11:16PMA delightful visualization of prime factorizations for the numbers 1 through 49.
As per the Fundamental Theorem of Arithmetic, for every natural number, there exists a unique prime factorization (in the naturals). Each and every number can be broken down into basic, irreducible components–or is itself irreducible (i.e. it cannot be factored; it is prime).
I liken this to chemistry: the primes are the periodic table, an array of atomically indivisible elements. (Of course, the elements are “divisible subatomically” (into rationals or reals), but that is outside our focus.
Following this analogy, all compound numbers are like chemical compounds, whose bonds are formed by multiplying. The properties and relationships of the compounds are determined by their structural arrangements. Insofar as the complexity of our natural world is (to some extent) attributable to these fundamental chemical relationships, so too are much of mathematics’ exotic objects and qualities the result of “numerical chemistry” amongst the primes.
As you examine the structures of the images above, I encourage you to consider the numbers of dots, “sub-groupings of dots,” and their overall symmetries. You will notice that the primes cannot be broken down into “sub-groupings” (i.e. they cannot be factored). As for the compound numbers, you will observe that their number of symmetries is related to the number of dots. The reader is encouraged to consider why this is, comparing the diagrams with the numerical details of the various prime factorizations.
Mathematics is beautiful. <3
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