The theory of orthogonal fractal fields is that these sets depend on
the dimensional self similarity of the fractal . Just how the
dimensionality affects the orthogonality relationship is not an easy
question. I have found a relationship that seems to give the desired
orthogonal sets. The procedure is more of a monitor since it everywhere
depends on the original fractal set: it is not an independent fractal.
The equations are very simple:
1) M = {{x,y},{xs,ys}}
2) determinant [M] = 1  ds ( cross product)
3) (x,y) dot product (xs,ys) = x * xs + y * ys = ds
It was finding a ds that gave a size and orientation that was
orthogonal to the original set that is the experimental element. I tried
a lot of combinations, but the following relationship gives the best
output:
4) ds = 1  1 / s: 1 <= s <= s(Max)
where s is the Moran similarity dimension for the fractal.
Since in harmonic functions, the orthogonal relations also produce the
derivatives, the solutions:
5) ys = (y * ds  x * (1  ds)) / (x^2 + y^2)
6) xs = y * ys / x + ds / x
can be called the fractal derivatives of the fractal functions. As I
said before, they are not independent functions. The (ds,1  ds) seem to
be probabilities. If so this relationship can only work when s is
greater than one.
In three dimensions the equations are:
7) M = {{x,y,z},{xs,ys,zs},{iv,jv,kv}}
8) det[M] = a * iv + b * jv + c * kv
9) sqr(a^2 + b^2 + c^2) = 1  ds
10) x * xs + y * ys + z * zs = ds
Suppose you had a 3d strange attractor whose Hausdorff dimension is
known, then this kind of relationship determines a second strange
attractor that is dependent on the first and is fractally orthogonal to
the first. In terms of signalling and coding such orthogonality can be
very useful. In the three dimensional case the nature of the root
equations will give more than one solution to the above equations in
most cases. Suppose you have tuned Chua oscillators in chaotic mode
that are being used to encode a transmission: the orthogonality
relationship would allow as many as three messages to be sent
orthogonally at the same time. To an ordinary receiver the signals would
appear to be noise. I state this idea here as a demonstration of the
usefulness of fractal field theory in practical terms. Since these
fields also have scaling orthogonality properties by their self
similarity, image compression types of coding like with wavelets may
also be a result of this mathematics.
The twin dragon results seem to suggest a new dragon that is orthogonal
to the twin dragon with an inner border that is itself a dragon. The
relationship of the twin dragon to dualistic logic and fuzzy set
analysis makes this result very interesting in theoretical terms. The
exterior border to the orthogonal set is not as deterministic as one
would expect.
The relationship of the Cantor set to the Sierpinski gasket set seems
important in fractal field orthogonality terms. It suggests some
experiments in intersections and unions of fractal sets. That we don't
know everything about fractals and how they work, doesn't mean that we
aren't developing the science in very useful directions. A new
definition of a fractal derivative that is independent of traditional
calculus methods is also suggested by this line of research.
