On the World
Trinity Numbers Before and After The Matrix
NEVATHIR
April 11, 2017
If we google the phrase, Trinity numbers, top results have nothing to do with number theory or computer sciences. While there are prime numbers, Catalan numbers, Fibonacci numbers, no sensible established number collections resemble the idea of Trinity.
The idea of Trinity is of course Christian and there is a trihedron representation known as Shield of the Trinity. Recently, popular culture united the Christian idea with computers under The Matrix. Should the union remain a fantasy or is there a number collection that fits?
As a teenager learning Euclidean geometry in 1990s, before The Matrix was filmed and popularized, I did a little research and found a trihedron collection that shares similar properties with Shield of the Trinity. Here we restrict ourselves to trihedron collections under $R^{3}$ with special orthogonal group as symmetry group that differentiates a generic trihedron from its reflection.
A trihedron has six edges that form straight line segments in $R^{3}$. Each edge has four neighbor edges and a opposite edge. If we take Shield of the Trinity and make each edge connected to God length $j$, others length $k$ that satisfies triangle inequality, it's easy to prove that sums of squares for each pair of opposite edges are equal to $j^2+k^2$ and opposite edges are orthogonal.
However, it's a boring trihedron/number collection. Calling it Trinity numbers is definitely a joke. Thus we work a little harder and establish the relationship that for every trihedron in $R^{3}$, sums of squares for each pair of opposite edges are equal if and only if opposite edges are orthogonal. The trihedron/number collection with integer orthogonal opposite edges is much more mysterious and perhaps may be justifiably called Trinity numbers.
Trinity numbers defined above are computable, but so far there is no exact closed form, which makes Trinity numbers much like prime numbers, rather than Catalan numbers. Trinity numbers are indeed related to prime number distribution via Fermat's theorem on sums of two squares and sum of two squares theorem, with a caveat that triangle inequality pushes Trinity numbers beyond simple combinatorial combination from prime numbers.
To get a little hands-on experience, thanks to sum of two squares theorem and fast modern computers, Trinity numbers can be efficiently enumerated with integer factorization. Estimates for Trinity number distribution are much less developed than those for prime numbers, though. It's unclear how to place Trinity numbers within number theory or computer sciences.
Can relationships between Trinity numbers be found? How are they related to prime number properties? Are there useful approximations to Trinity number distribution, much like prime numbers? Due to time and priority restrictions, I can not provide a satisfactory exposition for Trinity numbers yet and would be deeply grateful for anyone that does. If there is any progress regarding Trinity numbers, please send a link via email nevathir@glacier-studio.com and let's see if our mental power succeeds or fails before the mystery of numbers.