Above is a graph of the unit sphere squared in four-signed math.

The rays are # (red), - (green), + (blue), and * (violet).

Each point of a unit sphere in four-signed is squared and the result graphed.

Because four-signed numbers are three-dimensional and have a well defined product such an operation is possible.

This cone exposes the magnitude behavior of the square product as a function of orientation for four-signed values.

The sphere has folded onto itself just as a square of real values folds to the positive real side. The vertex of the cone lies along a ray from the origin toward #1+1.

An animated version can also be viewed.

If the premise of polysigned numbers becomes accepted then the four-signed numbers will become analogous to complex numbers in three dimensions.

Geometrically they form a symmetrical coordinate system in three dimensions.

The basic law of cancellation is

- x + x * x # x = 0 .

The cancellation law can be viewed via the rays emanating from the center of a tetrahedron to its corners.

These are the proper directional vectors that match four-signed numbers.

If a value has equal magnitudes in each of these components the result will be a point at the origin.

This is exactly what the cancellation law states.

The use of tetrahedral directions as a coordinate system has also been developed as quadrays.

( - a + b * c # d )( - e + f * g # h )

is equivalent to+ ae * af # ag - ah

* be # bf - bg + bh

# ce - cf + cg * ch

- de + df * dg # dh .

This is simply the distribution of terms.* be # bf - bg + bh

# ce - cf + cg * ch

- de + df * dg # dh .

The resultant sign obeys a summing of the source signs in modulo four so for example:

(-1)(+3) = *3.

(+2)(+2) = #4.

(*3)(+2) = -6.

(-a)(-e) = +ae.

The four-signed numbers are the first in the series to break the law:

| A B | = | A | | B | .

They are the first of an exotic family of numbers beyond three signs.

There is a study of the behavior of deformation in the P4 product.

It is the axis of the cone graphed at the top of this page.

The axis passes through + 1 # 1 , the origin, and on

through - 1 * 1 .

Any number multiplied by any point along this axis winds up on this axis.

This axis is a natural feature of the four-signed numbers.

An axis like this exists for all even-signed number systems.

In six-signed (P6) the axis passes though (1,0,1,0,1,0), the origin, and onward through (0,1,0,1,0,1).

It is possible to get a zero result from the product of two nonzero values.

For example in P4:

( - 2 + 2 ) ( -3 * 3 ) = 0 .

This was demonstrated to me by Hero Van Jindelt and leads to the discovery of the identity axis.

Back To Polysigned Numbers