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.

Thus:

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
product:

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
produced Understanding
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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Polysigned Numbers