Fractal and multifractal geometry: scaling symmetry and statistics
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Left to right: Two examples of self‐similar fractals, one with some compositions forbidden, one nonlinear, and one random.
A driven IFS example: is the unoccupied address 323 forbidden or just unlikely?
Cartoon and increment graphs for Δt1 = 4/9 (top left), Δt1 = 1/3 (top right), Δt1 = 2/9 (bottom left), and Δt1 = 1/9 (bottom right).
Left: the sixth iterate of the process of Figure 21. Right: a randomized version of the left image.
The initiator (left), generator (middle), and first iterate (right) of the cartoon with generator vertices (24).
Top: Fractional Brownian motion simulations with α = 0.25, α = 0.5, and α = 0.75. Bottom: increments X(t + 1) − X(t) of the graphs above.
Left: the Koch curve. Center: a randomized Koch curve. Right: a randomized Sierpinski gasket.
The f(α) curves from Eq. (20), with p1 = 0.3 (left) and p1 = 0.45 (right).
Left to right: IFS with memory, random, nonlinear, and self‐affine IFS, on a grid for computing the dimension.
Left: A small DLA simulation. Right: Natural fractal dendrites, modeled by DLA.
Left: Steps in measuring the length of the Koch curve. Right: Steps in measuring the area of the Koch curve.
Top: Time series and driven IFS for the logistic map with equal‐size bins. Bottom: Equal‐weight bins.
Left: IFS driven by the DNA sequence for amylase. Right: an IFS random except that T4 never immediately follows T1.
Left: 2‐IFS generated by ℱ5. Middle: an incorrect realization of the corresponding second higher block IFS. Right: the correct implementation.
Left: IFS and vertex transition graph violating conditions (2) (and (3)). Right: violating only condition (3).
Top: four examples of IFS with memory. Bottom: Empty length 2 and 3 addresses.
Some iterations of a process generating the left‐most fractal of Figure 1.
