These fractals are based on hypercomplex numbers, which like quaternions are a four dimensional generalization of complex numbers. It is not possible to fully generalize the complex numbers to four dimensions without sacrificing some of the algebraic properties shared by real and complex numbers. Quaternions violate the commutative law of multiplication, which says z1*z2 = z2*z1. Hypercomplex numbers fail the rule that says all non-zero elements have multiplicative inverses - that is, if z is not 0, there should be a number 1/z such that (1/z)*(z) = 1. This law holds most of the time but not all the time for hypercomplex numbers.

However hypercomplex numbers have a wonderful property for fractal purposes. Every function defined for complex numbers has a simple generalization to hypercomplex numbers. Fractint's implementation takes advantage of this by using "fn" variables - the iteration formula is:

      h(n+1) = fn(h(n)) + C.
where "fn" is the hypercomplex generalization of sin, cos, log, sqr etc. You can see 3D versions of these fractals using fractal type Julibrot. Hypercomplex numbers were brought to our attention by Clyde Davenport, author of "A Hypercomplex Calculus with Applications to Relativity", ISBN 0-9623837-0-8. See also Quaternion and Quaternion and Hypercomplex Algebra

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