Cartesian Transformation of Polysigned Numbers

The following transform equations provide a pattern that allows extension to higher dimensions.
They come by application of symmetry.
Here unit vectors in each sign direction are represented by S.
The Cartesian counterparts are X.
Because the sum over the Sn unit vectors will be zero each unit vector contributes a component of 1/(n-1) in any other vector direction.
Aligning X0 and S0 yields the following for three-signed math:

X0 = S0 - 1/2( S1 + S2 )
X1 = u( S1 - S2 )

where u = sqrt( 1 - sqr(1/2))

And in four-signed:

X0 = S0 - 1/3( S1 + S2 + S3 )
X1 = v( S1 - 1/2( S2 + S3 ))
X2 = uv( S2 - S3 )

where u is as above and
v = sqrt( 1 - sqr(1/3))

Equations for higher signs can be generated similarly.

Back to Polysigned Numbers