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D_q=1/(1-q)lim_(epsilon->0)(lnI(q,epsilon))/(ln(1/epsilon),) (1) where I(q,epsilon)=sum_(i=1)^Nmu_i^q, (2) epsilon is the box size, and mu_i is the natural measure. The ...

The attractor of the iterated function system given by the set of "fern functions" f_1(x,y) = [0.85 0.04; -0.04 0.85][x; y]+[0.00; 1.60] (1) f_2(x,y) = [-0.15 0.28; 0.26 ...

Determining the length of a country's coastline is not as simple as it first appears, as first considered by L. F. Richardson (1881-1953) and sometimes known as the ...

A Julia set with constant c chosen at the boundary of the Mandelbrot set (Branner 1989; Dufner et al. 1998, p. 225). The image above was computed using c=i.

Informally, self-similar objects with parameters N and s are described by a power law such as N=s^d, where d=(lnN)/(lns) is the "dimension" of the scaling law, known as the ...

Define the "information function" to be I=-sum_(i=1)^NP_i(epsilon)ln[P_i(epsilon)], (1) where P_i(epsilon) is the natural measure, or probability that element i is populated, ...

A polygon that can be dissected into n smaller copies of itself is called a rep-n-tile. The triangular polygonal spiral is also a rep-tile. The above figure shows the zeroth ...

Sea horse valley is a name sometimes given to the portion of the Mandelbrot set centered around -0.75+0.1i.

A "curve" (i.e., a continuous map of a one-dimensional interval) into a two-dimensional area (a plane-filling function) or a three-dimensional volume.

A space-filling function which maps a one-dimensional interval into a two-dimensional area. Plane-filling functions were thought to be impossible until Hilbert discovered the ...