[R.N. Boyd]:

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.

[Saul-Paul Sirag]:

The hyperdimensional fractal is contained in Mandelbrot's definition of a fractal:

"A fractal will be defined as a set for which the Hausdorf-Besicovitch 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 2-d space (the Complex 1-d 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 4-d real space to a 4-d 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/quant-ph/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!