Four-Signed Numbers (P4)

Four-Signed Sphere Squared

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.

General products

The general four-signed product
( - 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.
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.

The Identity Axis

An identity vector exists naturally in four-signed space.
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