by Robert Neil Boyd
from
Rialian Website
An interesting new idea has occurred to
me while browsing Coxeter's "Regular Polytopes".
That is, as is stated in the subject line, the fractal dimension of
a hyperdimensional space, or object. If no one has thought of this
before, I'll be surprised, but for me, the hyperdimensional fractal
is an entirely new concept.
This could be the missing piece that might allow for realistic
rendering of 3D landscapes. Hyperdimensional fractals appear to have
many other interesting properties as well, and may find some
interesting applications in the physics, particularly at the
interface between 3D and 4D hypervolumes, where the physics already
appear to be interesting.
[SaulPaul Sirag]:
The hyperdimensional fractal is contained in Mandelbrot's definition
of a fractal:
"A fractal will be defined as a set for which the
HausdorfBesicovitch dimension strictly exceeds the topological
dimension."
(p. 15 of Fractals, Form, Chance, and Dimension, by B.B.
Mandelbrot, 1977)
The Mandelbrot set is a fractal line whose (Hausdorf) dimension is
greater than 1. It is embedded in a 2d space (the Complex 1d space
C). It is generated by iteration of the map (from C to C):
z > z^2  a (where z and
a are complex numbers).
An example of a hyperspace fractal is generated by iteration of the
Henon map from C^2 to C^2 (i.e. from a 4d real space to a 4d real
space):
[x, y] > {x^2 + c  ay, x], assuming
a does not equal zero.
See The Henon Mapping in the Complex Domain by John H. Hubbard (pp.
101 111 of Chaotic Dynamics and Fractals, edited by Michael F.
Barnsley and Stephen G. Demko, 1986).
Iterations of polynomials in [x, y, z,...] would yield higher
dimensional fractals.
[Arkadiusz Jadczyk]:
Indeed.
Attractor sets of chaotic dynamical systems are usually "hyperdimensional"
fractals or multifractals.
Hyperdimensional fractals will typically arise also in quantum
systems coupled to classical systems so as to model "simultaneous
measurement" of several noncommuting observables.
See for instance:
http://xxx.lanl.gov/abs/quantph/9909085
and
http://www.cassiopaea.org/quantum_future/chaos.htm
[R. M. Kiehn]:
Sirag and Ark have answered your questions about MD fractals.
Now consider that which follows in reference to minimal surfaces,
which are useful to the study of wakes and persistent phenomena in
otherwise dissipative media (such as the Falaco Solitons). S. Lie
proved that all holomorphic functions generate minimal surfaces, as
a complex curve in 4D. Now consider the Mandelbrot generator in
terms of the quadratic polynomial and its iterates.
Then at each
step of iteration a new minimal surfaces is generated. In the
iteration limit the Mandelbrot set is produced. It would appear that
the fractal so generated is still a minimal surface!
