number crystals
statigrafix complexity notes
explanations for the masses series [e4m]
information visualization [viz]

god created the natural numbers
the rest is man's

leopold kronecker 1823-1891

in our time we have witnessed
at least two displays of hostility against new mathematical ideas
the first was directed against lattice theory, and its virulence all but succeded in wiping lattice theory
off the mathematical map. the second, still going on, is directed against the theory of categories. grothendieck did much
to show the simplifying power of categories in mathematics. cats have broadened our view all the way
to the solution of the weil conjecture. today, after the advent of braided categories and
quantum groups, cats are beginning to look downright concrete, and the last
remaining anticat reactionaries are beginning to look
downright pathetic

gian-carlo rota 1932-1999
combinatorial species and tree-like structures, 1998


the first 1000 natural numbers arranged as a spiral following jan boeyens in his 2008 chemistry from first principles. natural primes - highlighted in red - are localized to 8 of the 24 spiral arms. this is not a deep observation, but follows from basic facts such that there are no even primes - some arms contain only even numbers. however, the distribution of primes is a deep topic. for example, there are infinitely many pairs of primes that are 2-neighbors, and also arbitrarily large gaps between some neighboring primes. why? where does this complexity - "halfway" between order and chaos, emerge from the most basic of processes: counting?

mathematics expands on an endless process of generalization. what are the natural numbers? one could follow an algebraic characterization: the natural numbers is the initial object in the cat "semiring". in the cat "topos", which itself is a generalization of topological spaces, "natural numbers objects" consist of an object N and two arrows "0" and "+1", which represent the initial "counting unit" and addition of that unit, respectively, along with a diagram [below] displaying how these arrows force a unique arrow !:N-->A to make the diagram commute. thus numbers are constructed recursively as a discrete-time dynamical system [keep adding +1 to the numbers you already constructed, beginning with the initial number. in the cat "set", the arrow 0:1-->N is the number 0] addition, multiplication, exponentiation, and other operations can be defined on natural number objects

natural numbers objects are infinite: the subcat of finite sets has no natural numbers object.


next, reutilizing boeyen's 24-periodicity for convenience, we visualize the divisor lattices of natural numbers [mathematica code *]. the dual operations in divisor lattices are of course greatest-common-denominator and least-common-multiple. for clarity, we don't display the lattices unlabeled by the divisors, to focus on the quasi-periodicity of divisor structure. again, the q arises: how does the irregularity emerge from the regular 1,2,3,4... construction of these numbers?

- technically, what are shown are the hasse diagrams of the lattices, ie "transitive covers" of the latter [it would be senseless to show the lattices themselves - too many edges]. a lattice is a special type of order which is a special type of directed graph.

- divisor lattices of prime numbers look like a single upright stick, red points represent the number 1 @bottom and the prime itself at the top, joined by a single gray line [arrow pointing up really]. the lattice is 1-dimensional.

- prime powers, eg 8=2^3, looks like an upright 3-stick figure. 9=3^2 is a 2-stick figure. these lattices are also 1-dimensional.

- lattices that look like a single diamond, eg 6=2*3, are 2-dimensional: 2 primes generate these numbers. but also, the lattice of 24=2*2*2*3 is also 2-dim, but it looks like 3-diamonds stuck together.

- 30=2*3*5 yields the smallest 3-dim lattice, which looks like a wire-frame cube. 36=2*2*3*3 is 2-dim not 3-dim and is 4 diamonds stuck together in a larger diamond

- whether a lattice leans left, looks upright, or leans right is related to whether the number's smaller prime generators contribute more powers than the larger prime generators. eg 18=2*3*3 leans right b/c 18 is built from more 3's than 2's.

- although 24 numbers are rendered on each blue line, artificial random "jittering" is added to their rightful horizontal position, to avoid sticks or diamonds of taller lattices from overlapping completely. [jittering is very useful in data viz to break up such degeneracies]

- manfred schroeder [the inventor of mp3 codec] notes in number theory in science and communication 2006, that even many 25-digit #'s have remarkably compact lattices, generated by an average of only 5 primes. the 100,000-101,000 viz shows the growing range of complexity between the simplest [primes] and the higher-dim exemplars.

- screen real estate and memory constraints prevent higher res versions, but ideally such viz ought to be dynamic, like google maps, zoomable so that at the highest res even high-dim lattice points can be labeled by their consituent primes and powers

divisor lattices of the numbers 1-1000

divisor lattices of the numbers 10,000-11,000

divisor lattices of the numbers 100,000-101,000

the final viz shows a small subset of the infinitely many gaussian primes which are scattered with the obvious 4-fold symmetry among the gaussian integers, ie, the integer-lattice-points of the complex plane.

- gaussian primes of squared magnitude less than 1000 are highlighted in the central region. complex 0 is, of course, at the center of the cross.

- blue circles along the positive real axis mark the natural primes less than 100. some are empty. the viz illustrates that although all real gaussian primes are natural primes, the converse is not true. eg 2 factors as (1+i)*(1-i). it is akin to having a more powerful microscope.

- 4 homework, visualize the intersection of gaussian and eisenstein primes that are based on the 3rd roots of unity.. . truly man's creation


alan calvitti phd
head of research