The Diaxial Plane and the Cantor Set

By Roger Bagula
Compiled and graphics by Paul Bourke
21 Sept 1998

My triaxial tritorus is a very unusual surface. Only from directly above does it seem to fit our ideas of how a surface should look. The idea of dividing a plane into three redundant axes and then, doing a dot product to give a space covering parametric was an idea of worth. In the bath tub I was soaking and reflection on this when I thought of how the Canter Set is very like two axes on a line:

1) x = x / 3
2) x = x / 3 + 2 / 3

Switching randomly between these equations chops up a line. Suppose we made redundant axes on a line like those on a plane:

3) x1 = cos(t)
4) x2 = cos(t + 4 * Pi / 3)

where i leave out the 2*Pi/3 axis as the middle third. I call this concept the diaxial line.

The diaxial plane is the result of a dot product of two diaxial lines. The same kind of parametric as in the triaxial case results:

5) x = cos(t) * cos(p)
6) y = cos(t + 4 * Pi / 3) * cos(p + 4 * Pi / 3)

That is the diaxial plane. One of the angles behaves as the radius does in circle plane coordinates( polar coordinates). It is the diaxial ditorus that sparks interest form this approach:

7) x = cos(t) * (1 + sin(t))
8) y = cos(t + 4 * Pi / 3) * (1 + sin(t + 4 * Pi / 3))

The result covers a four cornered region of the plane. For you who must have a fractal:

9) x =cos(s * t) * (r^s + sin(s * p)) + a
10) y = cos(s * t + 4 * Pi / 3) * (r^s + sin(s * p + 4 * pi / 3)) + b
11) t = angle(x,y)
12) p = 2 * Pi * log(r) / log(2)

Surprisingly s = 0.5 works very well in this to give a Mandelbrot iteration.

Surprisingly the diaxial approach can be applied to the Cantor set equations. I tried variations on:

13) x = x * y / 9 + a(i)
14) y = (x / 3 + 2 / 3) * (y / 3 + 2 / 3) +b(i)

and I had to slow it down by:

15) z = sqr(2) * z

to avoid overflows. By mixing it with raw complex Cantor sets I got a very interesting fractal that with Paul Bourke's box dimension calculation program gave approximately:

16) d(diaxial diCantor) = log(4)

Since this is a four part nonlinear IFS, the Moran equation doesn't give us a way to get the similarity dimension so Bourke's program is very useful. I wish I had made use of it before. The output is:

 s  log(1/s) N(s) log(N(s))
 -  -------- ---- ---------
 1  0        597  6.39192
 2 -0.693147 223  5.40717
 4 -1.38629   88  4.47734
 5 -1.60944   70  4.2485
 6 -1.79176   52  3.95124
 9 -2.19722   31  3.43399
13 -2.56495   21  3.04452
16 -2.77259   13  2.56495
20 -2.99573   13  2.56495
23 -3.13549   10  2.30259
27 -3.29584    7  1.94591
54 -3.98898    4  1.38629

So we have used the redundant axis concept in a new and more primitive context and even used it to develop new fractals. The idea of angular symmetry in coordinate geometry being as important as as distance symmetry seems to be new and foreign to most people. The best interpretation of coordinates based on group theory is that of angular symmetry and not radial distance symmetry. The idea of using over determination at a lower level to give space filling determination at a higher level seems to work well as a generalization. I have also found that it allows the making of surfaces in higher dimensional geometry as well as paths. The multiaxial approach has yielded polygonal functions as well as strange new surfaces. As a fundamentally new approach to geometry visualization and symmetry, the multiaxial approach is a mathematical advance in our understanding.