Compactification and Pythagorean n-tuples

[richard]

Compactification (shrinkage of higher dimensions) in Unity Root Matrix
Theory (URM3) gives another method to extend any particular
Pythagorean triple to an infinite set of Pythagorean n-tuples, i.e. it
can use any Pythag triple as the root of a infinite tree of Pythag n-
tuples, choose your n.

In the 3x3 formulation of Unity Root Matrix Theory (URM3), the
'position' eigenvector Xm (~ Xminus ~ X-) is a Pythagorean triple, as
is also the acceleration eigenvector Xp (~ Xplus ~ X+)

Eigenvector equations, for nxn matrix A, eigenvalue C (big C), i.e.

A * Xm = -C * Xm

A * Xp = +C * Xp

(corrected from sci.math original post 1 hour ago)

There is a third 'zero' eigenvector X0, eigenvalue zero, i.e.

A * X0 = 0

This satisfies a hyperbolic, energy conservation equation, conserved
quantity C^2.

To retain the Minkowski-like feature of the 3x3 vectors Xp and Xm,
when extending to an arbitrary number of dimensions 'n', these two
vectors become n-dimensional, but are now Pythagorean n-tuples
instead.

In the case of Xp, this is trivially an n-element vector, all zero
except for the last three elements, which are trivially the 3x3
Pythagorean triple Xp (renamed Xp3), i.e.

Xp = (0, 0, 0, ... Xp3)

But in the case of Xm this now comprises n elements, all generally non-
zero.

The first n-3 elements of Xm, i.e. all higher dimensions n>3, are
simply a multiple of an evolution Tj (jth dimension), j=4..n
multiplied by the eigenvalue C. The last three elements are the
evolved form of the 3D vector Xm (now renamed Xm3)

A four dimensional example is

Xm = ( C*T4, -(T4)^2 * (Xp3) + (Xm3) )

where T4 = evolutionary time fourth dimension

A five dimensional example

Xm = ( C*T5, C*T4, -[(T5)^2 + (T4)^2]Xp3 + Xm3 )

where T5 = evolutionary time fifth dimension

For n, non-zero element n-tuples Xm, set all evolutionary times Tj,
j=3..n to non-zero.

For more details (links below):

3x3 Pythagorean triples Xp (Xp3), Xm (Xm3) See Appendix A (pages
51-54) for analytic and numeric examples

For the four dimensional eigenvectors see Section 8, Page 20.

For the five dimensional eigenvectors see Section 11, Page 31.

For the n-dimensional eigenvectors see Section 13, Pages 36,37.

Appendix D for 5D example


'Compactification of an n-dimensional eigenvector space over long
evolutionary timescales'

Free PDF paper

http://www.urmt.org/urmt_dimensional_compactification.pdf

or try

http://www.microscitech.com/urmt_dimensional_compactification.pdf

http://www.urmt.org


For an overview of URMT, see

http://www.urmt.org/presentation_URMT_shortform.pdf


Have a good weekend

Richard Miller
see web site for email
http://www.urmt.org

"If Pythagoras was alive today, he'd turn in his grave."