Page 3 of 5 FirstFirst 12345 LastLast
Results 21 to 30 of 46

Thread: The Relativistic Unit Circle

  1. #21
    Senior Member
    Join Date
    Jun 2012
    Location
    Santa Barbara, CA
    Posts
    2,766

    Default 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).
    Last edited by BuleriaChk; 12-30-2016 at 03:18 PM.
    _______________________________________
    "Flamenco Chuck" Keyser
    The Relativistic Unit Circle 03/28/2017 07:40 AM PST
    Proof of Fermat's Last Theorem Updates 03/19/2017 8:23 PM PST
    Ignore List -The Peanut Gallery.

  2. #22
    Senior Member
    Join Date
    Jun 2012
    Location
    Santa Barbara, CA
    Posts
    2,766

    Default 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




    Last edited by BuleriaChk; 01-02-2017 at 01:46 PM.
    _______________________________________
    "Flamenco Chuck" Keyser
    The Relativistic Unit Circle 03/28/2017 07:40 AM PST
    Proof of Fermat's Last Theorem Updates 03/19/2017 8:23 PM PST
    Ignore List -The Peanut Gallery.

  3. #23
    Senior Member
    Join Date
    Jun 2012
    Location
    Santa Barbara, CA
    Posts
    2,766

    Default 3,4,5 Triangle No Accident

    Added the following comment -

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

    Three independent elements = (-, null,+)
    Four quadrants -

    (Each squared

    Think of tic-tac-toe, and why if the first player chooses the center (null), he always wins...
    Last edited by BuleriaChk; 01-02-2017 at 10:34 AM.
    _______________________________________
    "Flamenco Chuck" Keyser
    The Relativistic Unit Circle 03/28/2017 07:40 AM PST
    Proof of Fermat's Last Theorem Updates 03/19/2017 8:23 PM PST
    Ignore List -The Peanut Gallery.

  4. #24
    Senior Member
    Join Date
    Jun 2012
    Location
    Santa Barbara, CA
    Posts
    2,766

    Default 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)...
    Last edited by BuleriaChk; 01-02-2017 at 10:56 AM.
    _______________________________________
    "Flamenco Chuck" Keyser
    The Relativistic Unit Circle 03/28/2017 07:40 AM PST
    Proof of Fermat's Last Theorem Updates 03/19/2017 8:23 PM PST
    Ignore List -The Peanut Gallery.

  5. #25
    Senior Member
    Join Date
    Jun 2012
    Location
    Santa Barbara, CA
    Posts
    2,766

    Default 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.


    _______________________________________
    "Flamenco Chuck" Keyser
    The Relativistic Unit Circle 03/28/2017 07:40 AM PST
    Proof of Fermat's Last Theorem Updates 03/19/2017 8:23 PM PST
    Ignore List -The Peanut Gallery.

  6. #26
    Senior Member
    Join Date
    Jun 2012
    Location
    Santa Barbara, CA
    Posts
    2,766

    Default Re: Spin (and the Binomial Theorem)

    Added section on quaternions w.r.t. spin and quantized angular momentum
    _______________________________________
    "Flamenco Chuck" Keyser
    The Relativistic Unit Circle 03/28/2017 07:40 AM PST
    Proof of Fermat's Last Theorem Updates 03/19/2017 8:23 PM PST
    Ignore List -The Peanut Gallery.

  7. #27
    Senior Member
    Join Date
    Jun 2012
    Location
    Santa Barbara, CA
    Posts
    2,766

    Default Natural Logarithms

    Natural Logarithms

    (added to pdf)

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

























    _______________________________________
    "Flamenco Chuck" Keyser
    The Relativistic Unit Circle 03/28/2017 07:40 AM PST
    Proof of Fermat's Last Theorem Updates 03/19/2017 8:23 PM PST
    Ignore List -The Peanut Gallery.

  8. #28
    Senior Member
    Join Date
    Jun 2012
    Location
    Santa Barbara, CA
    Posts
    2,766

    Default 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..
    _______________________________________
    "Flamenco Chuck" Keyser
    The Relativistic Unit Circle 03/28/2017 07:40 AM PST
    Proof of Fermat's Last Theorem Updates 03/19/2017 8:23 PM PST
    Ignore List -The Peanut Gallery.

  9. #29
    Senior Member
    Join Date
    Jun 2012
    Location
    Santa Barbara, CA
    Posts
    2,766

    Default Relation of 2ab to 2[tex]\gamma \beta[/tex]

    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.

    Last edited by BuleriaChk; 01-04-2017 at 05:22 PM.
    _______________________________________
    "Flamenco Chuck" Keyser
    The Relativistic Unit Circle 03/28/2017 07:40 AM PST
    Proof of Fermat's Last Theorem Updates 03/19/2017 8:23 PM PST
    Ignore List -The Peanut Gallery.

  10. #30
    Moderator grapes's Avatar
    Join Date
    Nov 2010
    Location
    NC USA
    Posts
    4,006

    Default Re: Relation of 2ab to 2[tex]\gamma \beta[/tex]

    Quote Originally Posted by BuleriaChk View Post
    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

Page 3 of 5 FirstFirst 12345 LastLast

Bookmarks

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •