Cartesian Transformation of Polysign Numbers

The following transform equations are extensible to higher dimensions. These expressions are produced by application of symmetry.

Because the sum over the unit sign rays will be zero each unit vector contributes a component of 1/(n-1) in any other vector direction.
Aligning X0 with -, then X1 near + and so forth yields a standardized transformation.

For a value in P3

- a + b * c

x0 = a - 1/2( b + c )
x1 = u( b - c )
u = sqrt( 1 - sqr(1/2))

The expressions for x0 and x1 are real valued(P2) expressions on both the right hand side and the left hand side.

For four-signed numbers

- a + b * c # d

x0 = a - 1/3( b + c + d )
x1 = v( a - 1/2( b + c ))
x2 = uv( b - c )

where u is as above for P3 and

v = sqrt( 1 - sqr(1/3))

Equations for higher signs can be generated similarly. As each cartesian component is computed the remaining system is reduced by one dimension. Symmetry still applies on the remainder just as it did on the initial system.

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