Quad Algebra and Fractals |
I use the expression "quad" as short for any four-dimensional complex number system. The most common 4-D system of this type is quaternion algebra, discovered by Alexander Hamilton in the 1830's. Quaternion algebra has its own rules for addition, subtraction, division and multiplication of variables in four dimensions. Squaring and cubing quaternions is derived from those same rules. You may have seen quaternion fractals on some websites based on the simplest of quaternion formulas q^2+c. But there are also transcendental and exponential extensions in quaternion algebra that most non-mathematical people have never heard of. Mystic Fractal software features those extensions in many built-in formulas. The parser in programs like QuaSZ and Fractal Zplot also directly supports "quad" variables. Unless you are a whiz at math and can create your own 4-D formulas in Ultra Fractal or Fractint, this is the only software that makes the whole gamut of quaternion algebra accessible to everyone. We also include other quad algebras in our programs, like hypercomplex and cquat algebra (complexified quaternion) and even octonion algebra (8-dimensions!) You won't find that combination of algebras in any other fractal software. The process of designing a 3-D fractal program is very time-consuming and requires lots of research and plenty of patience to root out unforeseeable bugs. If you'd like to see some tips on how the pieces fit together to form a program, here they are. So what do slices of 4-D transcendental fractals look like? Browse our galleries, then try our software and see, folks. Terry W. Gintz |
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