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 ).