The P4 product closely resembles P2xP3 (or RxC) when their product
is defined as follows:
Define an element in the P2xP3 space as ( r, s ).
x1 = ( r1, s1 ), x2 = ( r2, s2 ).
Having instantiated two elements in P2xP3 we arbitrarily define the arithmetic product:
x3 = x1 x2 = ( r1r2, s1s2 )
So in effect we claim this new product to simply be the independent product of its components.
This is an arbitrary definition for the purposes of comparison to the P4 product.
The above animation shows a P3 plane orthogonal to a P2 line, the darker shades being the lower system.
The system is also called a T3 tatrix.
The coloring of the axes is mnemonic.
The red axis is always the identity sign ( + in P2, * in P3, # in P4 ).
The green axis is always the - sign orientation (in P2 and up).
The blue is the + orientation (in P3 and up).
The violet is the * orientation (in P4 and up).
This coloring follows the ROYGBIV spectral mnemonic.
The sphere is multiplied by the red dot which travels the surface of the sphere.
The proposed P2xP3 product looks almost identical to the P4
However upon performing a mathematical comparison the following difference is observed:
This difference is found via a transform between the two systems.
The forward and reverse transform are verified to yield the identity matrix.
The identity axis is mapped to P2. The P3 orientation is found by squaring a random value orthogonal to the embedded P2 and mirroring it from the original. The mirrored vector is the identity sign reference of the embedded P3 plane and lies along
# 2 - 1 * 1 .
When the equivalent product definition in T3 is found this graph will be zero.
The error is surprisingly similar to its operands.
Hagen von Eitzen
has proven the isometric
isomorphism of P4 to RxC on this Usenet discussion and has
Polysign Numbers the Standard Way.
This method relies on two distance transformations but does expose that P4 and the higher dimension systems do fit into existing mathematics. Within the topic of associative algebra there are existing proofs to this effect.
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