CQuats -- A New Quad Algebra

Abstract: Involving the development of new software to render 3D Mandelbrot/Julia sets. This differs from others in the field in that the algebra most 3D rendering software uses is either quaternion or hypercomplex, while complexified quaternion(cquat) algebra is neither. This results in a radically different perspective for 3D imagery of this type.

Since November 2000, with the encouragement of Godwin Vickers [1], I've been developing specialized quaternion software to augment his Quaternion Video CD. My long term projects include Fractal Zplot and Fractal ViZion, generalized fractal software. But quaternion generators require all formulas be extended to four or more dimensions and fairly connected formulas at that. Not all formulas that work in 2D can produce 3D images of any esthetic value. The new software which I call QuaSZ (Quaternion System Z) has provisions for hypercomplex, quaternion, cubic Mandelbrot and octonion fractal types, with 150 built-in formulas.

In the process of exploring all possible extensions to a generator of this type, we considered using discrete modifications of the standard quaternion math to discover new and exciting images. I recalled Fractal Ecstasy [5], and how its author reconstructed the Mandelbrot set by altering the discrete complex math of z^2+c. The extension of this to quad algebra was intriguing. Then I was reminded in a dream of the possibility of different forms of quad algebra besides quaternion or hypercomplex types. After all, why stop at two forms of quad algebra for computer graphics, when rules could be developed for any number of quad algebras.

Having played with complexified octonion algebra, as described in Charles Muses' NonDistributive Algebras [2], I wondered what shapes a complexified quaternion algebra would produce. I wanted something that was between hypercomplex and quaternion algebra, so I set to work defining a matrix that would combine features of both. Quaternion shapes tend to be composed of mainly rounded lines, and hypercomplex shapes are mainly square (see Figures 1 and 2), so I was hoping to see a blend of each behavior from the experimental algebra.

For those not familiar with the matrices of hypercomplex and quaternion algebra, here are the tables:

         i        j      k

i      -1       k      -j

j       k      -1      -i

k      -j      -i       1

        i        j       k

i      -1      k       -j

j      -k     -1       i

k      j       -i      -1

In both quaternion and hypercomplex math i^2 = -1. The hypercomplex rules provide for one real variable, two complex variables (i and j) and one variable which Charles Muses refers to as countercomplex (k). In quaternion math there is one real variable and three complex variables. In hypercomplex math, unlike quaternion math, the commutative law holds, that is reversing the order of multiplication doesn't change the product. The basics of quaternion and hypercomplex arithmetic are covered in Appendix B of Fractal Creations [7]. One other concept that mathematicians like to dwell on is the idea of a "ring". There is one ring in quaternion and hypercomplex math, "ijk". If you start anywhere in this ring and proceed to multiply three variables in a loop, backwards or forwards, you get the same number, 1 for hypercomplex, -1 for quaternion.

         i        j       k

i      -1      -k       -j

j      -k       1        i

k      -j        i       1

Note that there are two countercomplex variables here (j and k), the commutative law holds like in hypercomplex math, and the "ring" equals -1 in either direction, like quaternion math.

Multiplying two identical quad numbers together, (x+yi+zj+wk)(x+yi+zj+wk) according to the rules of my new matrix , combining terms and adding the complex constant, I derived the following discrete formula for the "complexified" quaternion set, q^2+c:

Just to get a feel for this new formula, I decided to use a fairly basic constant, -1+0i, for the initial 3D test. The extraordinary picture "Equilibrium" was the result. Being familiar with the quaternion and hypercomplex renditions of the Julia set -1+0i, it appeared that this image was a leap into hyperspace, the fractal seemed to literally expand in all directions at once. I decided to try out a Siegel disk constant, -.39054-.58679i, which Roger Bagula [3] had recently sent me. The Siegel image strongly suggested that cquats were indeed a new form of space-filling fractal. Since then Godwin Vickers has ported the cquat formula to the Persistence of Vision Ray-tracer [6], and verified that the Equilibrium image wasn't just an artifact of QuaSZ. Nearly identical images have been obtained in POV, using Pascal Massimino's [4] custom formula algorithm for 3D Mandelbrot/Julia sets.

There remains the extension of cquat math to transcendental and exponential functions. Any ideas for this are welcome. Already I've revised the built-in formulas in QuaSZ to include cquat variations where possible.

6 Persistence of Vision Ray-tracer, POV Team, Victoria, Australia, http://www.povray.org/