Wikipedia has an article "Pythagorean triple".
It contains a lot of useful information, but what draws my particular attention is this sentence in the section "Spinors and the modular group":
The group Γ(2) is the free group whose generators are the matrices
Consequently, every primitive Pythagorean triple can be obtained in a unique way as a product of copies of the matrices U and L.
We will call these matrices U2 and L2. I really like the matrix U2. Here is a piece from page 96 of my book Quantum Fractals.
We will be dealing with matrices. While it is a plausible hypothesis that God created the integers (see my previous post "The magic of numbers 3,4,5", it is not clear who invented matrices with their noncommutative multiplication rule. Till now nothing is known about matrices being featured in the Bible, Talmud or Babylonian tablets. The online paper "When was Matrix Multiplication invented?" informs us that:
1850 Sylvester first use of term "matrix" (matrice=pregnant animal in old french or matrix=womb in latin as it generates determinants)
1858 Cayley matrix algebra [7] but still in 3 dimensions [14]
1888 Giuseppe Peano (1858-1932) axioms of abstract vector space [12]
I will assume that the Reader is acquainted with matrix multiplication. We will be dealing here first with 2x2 matrices: two rows and two columns. Then with 3x3 matrices, then with 4x4 matrices, then ... the future will tell. Matrices will have real or complex entries, but we will will be particularly interested in matrices with integer entries. The matrix U2 above has integer entries 0,1,2. Our cat Pikabu knows this matrix pretty well.
There is a particular matrix with integer entries, we call it E. Wikipedia calls it the identity matrix and denotes it by the capital letter I:
The identity matrix is often denoted by , or simply by if the size is immaterial or can be trivially determined by the context.[1]
If A and B are square (the same number rows and columns) their matrix product (in the order written) is denoted AB. In general AB is not the same as BA. For an arbitrary matrix A and the identity matrix we have though AE=EA=A. If A,B are two matrices and AB=E, one can prove that then also BA=E. The matrix B with this property is unique and called the inverse A. Quoting Wikipedia again:
In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular, nondegenerate or (rarely used) regular), if there exists an n-by-n square matrix B such that
where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.[1] If this is the case, then the matrix B is uniquely determined by A, and is called the (multiplicative) inverse of A, denoted by A−1. Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.
Here Wikipedia is using bold letters to denote matrices. We will use normal letters. We will also need the concept of the determinant and trace. Exploiting the handy Wikipedia again:
The determinant of a 2 × 2 matrix is
The trace Tr(A) of the matrix A is simply the sum of all its diagonal elements. In the acse of the 2x2 matrix above
Tr(A ) = a+d.
There is a beautiful little paper by Roger C. Alperin "The Modular Tree of Pythagoras", p.807, we notice the following interesting sentence
After a version of this article was made available as a preprint, it was pointed out to the author that this tree structure was noticed earlier by Hall using three-by-three matrices [4]. The connection between our use of two-by-two matrices versus Hall’s three-by-three matrices is presumably a reflection of the isomorphism between the group of linear fractional transformations and a subgroup of the Lorentz group preserving the quadratic form x2+y2-z2
We will follow this path, but having in mind physics of light rays we will write the quadratic form as
x2+ z2- t2
This is for two space dimensions (x,z) and one time dimension t. When we add also y, we will be dealing with Pythagorean quadrples. After we are done with them, nothing will prevent us to add another space dimension and get interseted in Pythagorean quintuples.
Exercise: calculate the products U2 and L2 in both orders. Calculate the inverse matrices of these two
To be continued
P.S.1. 04-08-23 A conference of the Polish Physical Society is taking place right now in the city of Gdansk. At a poster session G. Koczan (who wrote here about Aristotle's physics) is presenting a paper about "Photon's position operator", the subject that I am also interested in for a long time. Below is the poster. It is in Polish, but the pictures on the poster are in an international format:
Well darn, normally those exercises are safely above my education and even when they aren't like with visualizing 4-dim spaces, I don't expect to get it exactly right at first. This I should since even engineers get taught Computational Linear Algebra as I believe the course was called... had to look up the inverse formula though I knew the determinant was needed. Do still remember matrix multiplication correctly though fairly recently I remember getting slightly confused when the matrices weren't square ones.
ReplyDeleteIt is not only you that you have problems with matrix multiplication. It seems that none of the readers can do it. I will have to rethink my ideas about what to write about. Philosophy seems to be okay but any real math, however simple, seems to be a no-no. People like to talk about Plato or Pythagoras, but they do not really want to know what Plato and Pythagoras were considering as really interesting, beyond trivialities. Nobody wants to learn. It's a pity. But that's life.
DeleteYeah matrix multiplication with integers is probably easier not to mess up than long division but lots of people probably have traumatic memories of long division. I remember the first time I did Cl(8)xCl(2)=Cl(10) in terms of showing their rows of the Pascal Triangle. Also not any more difficult than matrix multiplication with integers or long division but it was truly exciting to see the answer come out. My daughter sees me putting numbers on a scrap sheet of paper and can't believe I'm doing math again just for fun. For some reason lots of numbers paralyze most people.
DeleteI remember when the Cs said you should teach them some algebra and that was both an exciting and frustrating thought. There's no reason everybody on the forum shouldn't appreciate mathematical physics in the same way I do but doing that in a shortcut kind of way is needed since I did spend thousands of hours reading even if I didn't need to use anything more difficult than long division. College calculus for engineers was way more difficult than any actual math I tried for my interest in physics just for fun. I got A's in college calculus and I'd fail all those tests right now (too much of it was memorization). Heck Tony failed a PhD general exam because he forgot too much math while being a lawyer but that didn't prevent him from doing fun things with mathematical physics.
Maybe you could mirror posts here and on the Cass forum and like when people email you, use forum comments as coming from an unnamed reader or readers on the forum can comment here directly if they want. You could maybe start with references to sessions like 4 pluses and 2 minuses and hexagons and getting to the conformal group where you have things like XY YZ XZ spatial rotations. The X Y Z is space but it's using it like information since XY could be thought of as 110 since there's an X and a Y but no Z and for the whole conformal group it's 110000 since the conformal group has six different extended spacetime dimensions, 4 spacelike (4 pluses) and 2 timelike (2 minuses). Then go on to the rest of the conformal group with conformal map projection analogies or something then stick the 15 conformal dimensions as 6 things taken two at a time on the Pascal triangle. That's the kind of thing that originally hooked me in on Tony's site but then I'm a weird guy who jots math numbers down on scrap sheets of paper just for fun as my daughter can confirm.
The general idea of information seems to interest the forum. Would have to separate the statistical entropy kind from the exact Pascal triangle kind.
Evidently you were using some software. What is it? GAP?
ReplyDeleteI used GNU Octave (https://octave.org/).
ReplyDeleteThanks. I will learn it and use in this blog whenever opportunity arises!
ReplyDeleteConcerning scripts: in the link about scripts that you shared I do not see if there is a way to save some output to a file in a text format. Probably there is a way, but it needs a special syntax to open the file, to write to it, and to close it. Yes?
ReplyDeleteOK. Found it. It is all in the manual. Except there are very few examples in the manual.
ReplyDeleteBut I have found this book:
ReplyDeleteGNU Octave by Example: A Fast and Practical Approach to Learning GNU Octave by Ashwin Pajankar, Sharvani Chandu
and it looks just right for me!