# Thread: The Relativistic Unit Circle

1. ## Re: The Relativistic Unit Circle

In the case of Pythagorean Triples, the numbers have the relation:

, so ; that is, the numbers are on the same number line. This is also true for real numbers, where each side of the equality refer to the same single valued number (That is, ) in the Relativistic Unit Circle):

For the area of a circle:

(Note: there is no distinction between this operation and that for the area of two squares:

The actual integers the symbols represent will depend on the way the integers are counted; i.e., the number base (normally base 10 in our system).

2. ## The Axiom of Choice

I have added a section on the Axiom of Choice to The Relativistic Unit Circle.

Axiom of Choice

This shows the relation of the positive and negative axes in two dimensions as the foundation of the Pauli/Dirac matrices, and their relation to the null vector....

Edit:I just added the final comment to the pdf document:

The Null Vector

If the axes represent converging forces at the origin, then the null vector represents equal and opposing forces as the interaction” cross product.

That is, for

and
, then

, but

, so

3. ## 3,4,5 Triangle No Accident

"It is not by accident that the first Pythagorean triple is a 3,4,5 right triangle."

Three independent elements = (-, null,+)

(Each squared

Think of tic-tac-toe, and why if the first player chooses the center (null), he always wins...

4. ## Proof of Goedel's Theorem

If the real number line (system) is consistent, it is not complete, and vice versa.

E.G. The two equations

(multiply by )

and

are inconsistent (ambiguous) depending on the context (circle or rectangle)

Exercise for the student: think of other cases such as z = y/x, z=exp( x)...

5. ## Spin (and the Binomial Theorem)

Spin (and the Binomial Theorem)

If one shifts the origin to the junction of and , one can establish the pair and .

Then
,
, so .

Also,

and

so

This is the “interaction term” for the Binomial Theorem, which is easily expanded to the case for the proof of Fermat's Theorem.

This represents “spin” in the Pauli/Dirac formulation of Quantum Field Theory with a Lorentz rotation.

The terms and can then be related (at their connection) by scaling factors
and so that:

,

where represents the "separation" in the Minkowski metric in GTR.

6. ## Re: Spin (and the Binomial Theorem)

Added section on quaternions w.r.t. spin and quantized angular momentum

7. ## Natural Logarithms

Natural Logarithms

The natural logarithm can be thought of as a representation of the null vector in terms of the relativistic unit circle.

8. ## Re: Natural Logarithms

Another way of stating Fermat's theorem is the following analysis:

There is only one unique set of integers on a single number line; e.g. (a). For two sets of integers (a,b) one needs two number lines, and they must be independent. Connecting them at the origin and distinguishing them by vectors (by providing a common scaling factor) results in the relativistic unit circle, which proves the theorem (and forms the foundation of Quantum Field Theory..

In particular, it addresses the issue of re-normalization in second quantization, and is at the foundation of the Pauli/Dirac formulation and the electromagnetic field tensor, where the linear independence of + and - is actually Newton's third law ("Every action must have an equal and opposite reaction", which is represented by the null vector in the Pauli/Dirac formulations, and by the null trace in the field tensor (where the tensor is derived from the Lorentz force..

9. ## Relation of 2ab to 2$$\gamma \beta$$

Consider a triangle consisting of three lengths (a,b,c), connected to the origin or not (they can always be translated by parallel transport and rotation). The area of the triangle in the first quadrant is given by , where a is the base and b is the height.

For triangles within the unit circle, there are four quadrants, so the area of four such triangles will be , which is the equivalent area of the interaction term for for the positive interaction area of the complete circle, related to the Binomial Theorem where .

For a and b on the axes, there is no interaction area, so . (since either or is equal to 0).

For Fermat's Theorem, the proof for follows immediately.
------------------
The area of an equivalent square is , so the length of its side is . Therefore, s is not an integer, so the expression cannot be an integer (for a, b, and c integers). Therefore, cannot be an integer, and so cannot be an integer, thus validating Fermat's Theorem.

10. ## Re: Relation of 2ab to 2$$\gamma \beta$$

Originally Posted by BuleriaChk
Consider a triangle consisting of three lengths (a,b,c), connected to the origin or not (they can always be translated by parallel transport and rotation). The area of the triangle in the first quadrant is given by , where a is the base and b is the height.

For triangles within the unit circle, there are four quadrants, so the area of four such triangles will be , which is the equivalent area of the interaction term for for the positive interaction area of the complete circle, related to the Binomial Theorem where .

For a and b on the axes, there is no interaction area, so . (since either or is equal to 0).

For Fermat's Theorem, the proof for follows immediately.
------------------
The area of an equivalent square is , so the length of its side is . Therefore, s is not an integer, so the expression cannot be an integer (for a, b, and c integers). Therefore, cannot be an integer, and so cannot be an integer, thus validating Fermat's Theorem.

Here are two Pythagorean triples:

but that means

so you're wrong when you say that

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