The four-signed numbers (P4) fail
to conserve magnitude when their product is taken.

Yet the
mathematical operations of summation and multiplication are
clean.

The distributive, associative, and commutative laws apply
just as they do for the real numbers.

The following images should
help explain the product behavior.

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

In the following animated image
the unit sphere graphed on the right is

multiplied by the
traveling red point on the right.

The result is graphed on the
left.

As the red dot travels the spherical surface a complete
image of the four-signed product is surveyed.

The small colored
circles on the sphere indicate the original sign vector
positions.

In
the following animation the sources are reoriented to the identity
axis.

The
product behavior of the four-signed numbers is complicated.

The
principles by which they behave are general and exist in any whole
dimension.

As one travels downward in dimension the same
principles yield the complex numbers and the real numbers.

A
comparison has been made between P4
and R x C ( P2 x P3 ).