Tim Wegner wrote:
Does anyone know anything about these beasties? I haven't had a chance to read this in depth yet, but they appear to be 6 dimensional cousins of quaternions/hypercomplex numbers.
Could probably define trig functions for these, and get some fractals out of them (there's a Mandelbrot example in the article.) The question is, would there be more to discover that is not already present with hypercomplex and quaternion slices?
The group that Walsmith illustrates and describes as C2-3 is conventionally known as D3; the group of symmetries of an equilateral triangle (with 1 corresponding to the identity symmetry, j and k being rotations of 120 and 240 degrees, and negation being reflection about some particular axis of symmmetry). It would be possible to find things different from those in hypercomplex and quaternion slices (since D3 doesn't form a subgroup of either); I do wonder though if it's something that needs to be specially coded. For some time I've been tinkering with a "Groups for Fractint" collection of formulae; in it, Walsmith's Mandelbrot is implemented as 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 } Needless to say, that could be greatly generalised to allow different slices by different planes in 3-space, and there is a corresponding Julia, and it could be rendered as a projection of a 3D object. But of course, there's no inherent reason why we can't go on to D4, D5, etc. or any of the other groups out there, with a corresponding increase in the number of dimensions involved. I haven't thought any about extending the usual functions to these spaces.