Russell Walsmith wrote:
TMan { c1=real(pixel),c2=imag(pixel),c3=p1 z1=z2=z3=0: t1=z1*z1+2*z2*z3 t2=z3*z3+2*z1*z2 t3=z2*z2+2*z3*z1 z1=t1+c1,z2=t2-c2*c2,z3=t3+c3 z=z1+z2+z3 |z| < 8 }
It seems a bit challenging to define a Julia set version of the TMan; Maybe someone can see a way to go with that. I'm looking forward, eventually, to seeing these objects in full-on 3-D.
I note that by allowing c3 to be complex, TMan is in fact four-dimensional, not three. While I don't understand the development behind TMan well enough to figure out what its Julia sets would look like (historically, the Julia sets came first, and then Mandelbrot came along several decades later to summarise them) the D3_Man set I gave on 2002-05-20: D3_Man { c1=real(pixel),c2=imag(pixel),c3=0 z1=z2=z3=0: nz1=z1*z1+z2*z3+z3*z2+c1 nz2=z1*z2+z2*z1+z3*z3+c2 nz3=z1*z3+z2*z2+z3*z1+c3 z1=nz1,z2=nz2,z3=nz3 z=nz1+flip(nz2) |z-pixel| < 100 } Has a straightforward Julia: D3_Jul { z1=real(pixel),z2=imag(pixel),z3=0 c1=real(p1),c2=imag(p1),c3=real(p2): nz1=z1*z1+z2*z3+z3*z2+c1 nz2=z1*z2+z2*z1+z3*z3+c2 nz3=z1*z3+z2*z2+z3*z1+c3 z1=nz1,z2=nz2,z3=nz3 z=nz1+flip(nz2) |z-pixel| < 100 } (The critical point is still at the origin.) I don't know why I used the bailout conditions I did: I wrote these things a long time ago, and didn't comment them. A more sensible bailout condition would be |z|+sqr(nz3) < 100. (I.e., bail out if the point gets at least 10 away from the origin in 3-space). As well as fudging the constant c (e.g., replacing c2 by -c2*c2) before it is used, slices parallel to the three coordinate planes can be found by permuting the initialisers, using some number other than 0, and/or using other components in the definition of z. More useful though would be a generalisation in which an arbitrary plane in 3-space can be specified which serves as the cutting plane through the M-Set/J-Sets. If Gerald K. Dobiasovsky could work his magic (hint, hint) on these numbers, we could have a two-dimensional projection of the entire three-dimensional set. Morgan L. Owens "Meanwhile, Canberra burns."