Multi-Dimensional Matrices And The Standard Model

I've just added scale to my 3D software Argus, which involved multiplying two 4x4 matrices together. When multiplying matrices, the order matters, so a*b is not the same as b*a. I wondered how many combinations there were when matrices are in higher dimensions.

For three dimensions there are 6:
a*b*c
a*c*b
b*a*c
b*c*a
c*a*b
c*b*a

So for four dimensions, we need to multiply this up by four: the same table as above starting with a, then with b, then with c, then with d. So the formula to calculate the possible combinations is: 2*1 (=2 for 2D), 3*2*1 (=6 for 3D), 4*3*2*1 (=24 for 4D) etc. Of course this is also the number of combinations for multiple multiplications of a 2D matrix; a 2D matrix multiplied 4 times in a row, this simplifies things, although a 4D matrix could have different lengths in each dimension.

The number 6 stuck in my mind, perhaps because I'm used to even powers of two. 6 doesn't seem like a fundamental number. Also I remembered that electron shells in atoms start as 2, then 6, which I thought was interesting, and that there are 6 quarks in the standard model of particle physics.

I wondered if there were 24 particles, and that the number and variety can be explained simply by the number of dimensions in this higher-dimension matrix, and the properties explained by the order of multiplication.

It turns out that there are indeed 24 particles: 6 quarks, 6 anti-quarks, 6 leptons, 6 anti-leptons. I found this interesting partly because anti-matter can be seen as normal matter moving backwards in time, this means that, if time were a dimension in these 4D matrices, the anti-matter property of the particles could be explained by the dominance of time in the multiplication order. Also, as you can see above, the natural sorting order for similar groups of particles would be 4 groups of 6:

a*b*c*d
a*b*d*c
a*c*b*d
a*c*d*b
a*d*c*b
a*d*b*c

b*a*c*d
b*a*d*c
b*c*a*d
b*c*d*a
b*d*c*a
b*d*a*c

c*b*a*d
c*b*d*a
c*a*b*d
c*a*d*b
c*d*a*b
c*d*b*a

d*b*c*a
d*b*a*c
d*c*b*a
d*c*a*b
d*a*c*b
d*a*b*c

The mass of different particles increase with each layer, so, I thought, perhaps lower (or upper!) orders are affected by mass, as though the mass has more impact when multiplied earlier (or later).

The standard model also includes charge-carrying or other types of particles that sort of mediate or carry information about cross-interaction between particles, perhaps, I thought, akin to the multiplication itself.

Mark Sheeky, 6 September 2020