Burnett, David wrote:
This got me thinking (a rare occurance and not without its dangers). Our fractals are mostly based on the micro-interactions of the real and imaginary components of a complex number. Has anyone explored using very imaginary numbers (ie n^4 = -1) or where a very complex number has the form ax + biy + cjz (where j^4 = -1). Or further extensions into any power of 2 = -1. probably been done (hasn't it all) - but you never know........ too hot here for anything but such meandering musings.
This is what Russ Walsmith has been investigating re: his triternions. Also available from within Fractint are quaternions and hypercomplex numbers (quaternions evolved as a result of finding numbers that relate to rotations in three-dimensional space in the same way that complex numbers relate to rotations in two-dimensional space - discovered by William Rowan Hamilton on October 16, 1843; while hypercomplex numbers are kind of "complex complex" number - they're of the form a+jb, where j^2=-1 and a and b are both of the form x+iy, where i^2=-1.) z^4 = -1 has the solution z = cos(pi/4) + i*sin(pi/4) = 1/sqrt(2) + i*1/sqrt(2). That's like taking the point at (1,0) and rotating it one-eighth of the way around the origin anticlockwise. Squaring that transformation doubles the degree of rotation to one-quarter (taking the point (1,0) to the point (0,1)). Squaring it again means doubling the rotation again, to a half circle (from (1,0) to (-1,0). To break out of complex numbers, one has to go beyond addition, subtraction, multiplication, division, and exponentiation (the jargon for this is to say that the "field of complex numbers is algebraically closed"). There are all sorts of directions this can go in - the subject is called algebra. Morgan L. Owens "Ditch the metaphysical baggage and go for complete independence."