Is there a general metod to construct a fractal?

Question

I would like to construct a fractal (traditional, self-affine, and fat fractal) with a given embedding and fractal dimension, but I don't know how to do it programmatically. The shape of the fractal is negligible.

Answer 1

Wikipedia describes a generalized Cantor set with parameter $0 < \gamma < 1$ giving fractal dimension

$$0 < D = \frac{-\log(2)}{\log\left(\displaystyle\frac{1-\gamma}{2}\right)} < 1$$

Built by removing at the $m$th iteration the central interval of length $\gamma\,l_{m-1}$ from each remaining segment (of length $l_{m-1}=(1-\gamma)^{m-1}/2^{m-1}$). At $\gamma=\frac{1}{3}$ one obtains the usual Cantor set. Varying $\gamma$ between $0$ and $1$ yields any fractal dimension $0 < D < 1$

The Cartesian product of Cantor set fractals satisfies (Falconer "Fractal Geometry: Mathematical Foundations and Applications" chapter 7):

$$\dim(F \times G) = \dim(F) + \dim(G)$$

In an $n$-dimensional space one can fit a product of $n$ one-dimensional Cantor sets (each with $0 < D_i < 1$) to get any dimension $0<D<n$

The product of the middle-thirds Cantor set with itself, aka "Cantor dust":