Tensor Calculus and Matrix Geometry, I Suggest We Un-Tesserate

Let us start with the relativistic physics concept, because it is mostly the reason why we are interested in matrix geometries today. First, we have the same concept as in quantum mechanics. We will choose a time and a place (that is in relative motion, that bit is more problematic) from which to make a measurement from, and that choice of a place to measure from will reveal a relativity in our data based on the time and place we have chosen to measure from. That is highly abstract, but it is fairly reasonable. It is also supported by loads of data.

Our next assumption is the quadratic assumption. We have altitude, latitude, longitude and time. We treat time as our tensor. The tensor will be inverse in relation to the other three-dimensions. My first question is, if all of our dimensions are relative positions, then why couldn't any of our dimensions be measured as our tensor?

So at one time four-dimensional geometries interested me a great deal, and I cobbled at them with the resources I had. I don't have any interest in four-dimensional geometries, no matter how complex - any longer. I've moved on to another geometry that is gutter-simple in comparison, and you'll see.

Now, let me boast a little about my knowledge. I hope I have this definition correct, as I have no access to anyone to correct me if I'm wrong. I formulated a - get this - partially-tesserated, hyperbolic, matrix-integral tensor. The equation looks about like this:

Eqn:. [(x + y + z) - c]^3 - [(x + y + z) - c]

I did a derivative-inverse by hand and it was interesting, but the full integral to this is way beyond my capability of solving by hand. If you're a mathematician with a specialty in the tensor and some supercomputer time, this might be worth a go. The integral itself is defiantly complex, and the number and vast size of many of the real solutions is way out in outer space.

However, my opinion is that we can drop the quadratic tensor entirely. As a mathematician, one develops hunches and intuitions. Just as an absolute discrete-finistic-reduction of mathematics, a finite number of finite sets - is incredibly attractive, so is this next piece of work I came up with. The next part to mention is that this is not as pie-in-the-sky as such a dramatic finistic reduction, and that in fact, this was just a thought that occurred to me one morning.

I did the integration in about fifteen minutes by hand before bedtime one day. Also, it is so utterly simple in contrast to what you see above. It is a three-dimensional version of that particular tesseract above. A three-dimensional tensor. Now we would imagine, we are leaving out a spatial-dimension or the time-dimension. Possibly. It may have no relationship to physics, but this integration produced an unusual result that is tickling my brain with a hunch.

So the equation before the integration looks like so:

Eqn:. [(x + y) - c]^2 - [(x + y) -c]

The integrated form looks like this, unless I mixed up a sign somewhere:

Int. Eqn:. x^2 + 2xy + y^2 + c^2 + c - 2xc - 2yc - x - y

What is so unusual about this is that I went over the factors quite a bit and I see no way to group or reduce. So there you have it. I've checked my work on the integration a couple of times, and you just multiply everything together until she goes - so I think it's right. Pretty good material - I must boast - whether it is useful for physics or simply theoretical mathematics.

Let me just plug in something fun here. When I was a kid, I used to sit and do permutations and combinatorics at the table in my home when I was bored. So I was looking at this particular combinatoric, another discrete math problem, and I found another axiomatic problem, and it has to do with algebra. Effing right, I am your demon! Wait until you see.

Eqn:. x^2 - [(x^2 - x)/2]

Now, if we do a normal factoring here, we distribute:

Fact. Eqn:. x^2 - x^2/2 - x/2

Mult: 2x^2/2 - x^2/2 - x/2

Group: (2x^2 - x^2 - x)/2

Result Eqn:. (x^2 - x)/2

Begin Neg. Proof:

1. Select 2 as x.
2. Original Eqn:. 2^2 - (2^2 -2)/2
3. Result: 3
4. Fact. Eqn:. (2^2 - 2)/2
5. Result: 1
6. x does not equal x, null proof, from the identity function, a=a.

I really want someone to check this one and give me a heads up, as I may have performed an improper operation, or I may be misunderstanding what I achieved. It confuses me a great deal. I would like to have some peers on my team. Anyway, I will return, and then we will talk about something else, and I may do a historical article this time. We shall see.