Tuesday Special - Tetractys and Lattice Infinity

 

9 + 16 = 25

3² + 4² = 5².

a² +b² = c².

Real magic. The Pythagoras Theorem. Except that it does not belong to Pythagoras. Pythagoras was a mystic, not a mathematician.

“There is not a single mathematical sentence that can be attributed with certainty to Pythagoras as an individual. The image of Pythagoras as a mathematician and scientist is a construction of later times. In contrast, the belief in the transmigration of souls and the religious character of the Pythagorean community is consistently and early attested. The historical Pythagoras appears not as a scientist, but as the founder of a way of life.”
— Walter Burkert, Lore and Science in Ancient Pythagoreanism, trans. Edwin L. Minar Jr., Harvard University Press, 1972, p. 113

"Pythagoras appears as a figure of the shaman type, especially in his claim to possess a soul of a special kind and in his ability to recall its previous incarnations. He was thought to have a miraculous, semi-divine origin; he performed healings and miracles; he was able to travel great distances in a moment, to be in two places at once, and to communicate with the dead. These are traits which correspond to shamanistic ideas and practices found in Central Asia and the Near East."
— Walter Burkert, Lore and Science in Ancient Pythagoreanism, trans. Edwin L. Minar, Jr. (Harvard University Press, 1972), p. 298

Tetractys

But, it is said that the Pythagorean Brotherhood worshipped the Tetractys: 1- the Monad, 2 - Duality, 3 - the Triad (beginning of form), 4 - the Tetrad (the World).

Back to (3,4,5). Pythagorean triples ware know to Babylonians ~1800 BCE. They knew (3, 4, 5), (5, 12, 13), (7, 24, 25), and wrote it in cuneiform. They knew even larger triples, such as (119,120,169). But in the Pythagorean Brotherhood there were initiation rites and levels of access. I think that the members of its inner circle knew that beyond the popular Triad a²+b²=c², at a deeper level, there is the well balanced Tetrad:

  a² + b² =c² + d², (a,b,c,d integers).            (1)

Of course tetrads contain triads, when one of the numbers is zero. So, those who know Tetrads, know also Triads, but not the other way. Of course we are interested in primitive solutions, that is in cases where a,b,c,d have no common divisor.  I could not find anything about tetrads in Babylonia, but I found them on math.stackexchange: Diophantine equation a² + b² =c² + d². The complete solution can be found in the textbook L.J. Mordell, "Diophantine Equations", Academic Press 1969, on p. 15.

Well, it is not explicitly complete there, it is somewhat sketchy, but here it is (I skip the proof).

Proposition 1. Every primitive solution of  (1) is of the form

a = (mp+nq)/2,
b = (np-mq)/2,
c = (mp-nq)/2,
d = (mq+np)/2,

where m,n,p,q are integers. Conversely, for any integers m,n,p,q such that a,b,c,d are integers, the formula above provides a solution of  a² + b² =c² + d².

What it has to do with the compactified Minkowski space?

Recall the quadratic form Q from The infinity ab initio:

Q(X) = (X¹)² + (X²)² + (X³)² -(X⁴)²+ (X⁵)² - (X⁶)²,

and the equation of the null cone N:

(X¹)² + (X²)² + (X³)² - (X⁴)² + (X⁵)² - (X⁶)² = 0.

Skip two dimensions X² and X³:

(X¹)²  - (X⁴)²+ (X⁵)² - (X⁶)² = 0.

Rewrite it as

(X¹)² + (X⁵)²  = (X⁴)² + (X⁶)² .            (2)

Then set X¹ = a, X⁵ = b, X⁴ = c, X⁶ = d, and you get (1). In (2) we are interested in the projective cone PN. Each tetrad a,b,c,d satisfying (1) determines a unique point (a',b',c',d') = λ(a,b,c,d)  on PN for which a^'2+b'² = c'²+d'² = 1. We get a point on the torus S¹⨉S¹. That is the image of our compactified Minkowski space with suppressed two space-like dimensions. By assuming the X components are integers we assume that ℝ^4,2 is a regular discrete lattice.

And so I did. Using Proposition 1 I generated about 1 mln of tetrads (a,b,c,d) and plotted them on a torus. Here is the result:

627169 + 627169 + 4446 points

There is some pattern there that can be seen. I do not understand where this pattern comes from, except, perhaps, of two circles around the torus. We will meet these circles soon. They represent  infinity.

P.S. 02-07-25 19:05 I have updated the torus. Points corresponding to M₊ ∪ M_- ∪ M_∞ are in Blue, Black and Red respectively. The infinity (Red) is hard to see as it is a set of measure zero on the torus. But it is just the line separating  M₊ and M_-.  The view above is from the side of M_-.  From the side of it would look similar, but mostly blue.