My favourite book on fractals is Measure, Topology, and Fractal Geometry by Edgar. A short book and not very well known. It has a great many exercises all very suitable at undergrad. level but it requires a good mathematical background in basic analysis and topology.
Apart from focusing on the geometrical aspects it also gives an excellent overview on different fractal dimensions definitions. But if I remember correctly this book does not cover fractals in the complex plane (Julia sets)
However, if you just want an overview on fractals I would start with Chaos and Fractals New Frontiers of Science by Peitgen, Jurgens & Saupe. It does not really give definitions but it guides the reader with simple examples or numerical experiments to the interesting properties of fractals.
A good understanding of Julia sets unfortunately requires a good understanding of Riemann surfaces, Hyperbolic geometry and Complex Analysis. The only book I know which also covers the necessary background is the Dynamics in one complex variable by (Fields medal winner) John Milnor. This is quite a tough book I read it when I was an undergrad. but I only fully understood it when I was a grad. student. But the material is great it covers Julia sets both from an analytical and topological view. It contains a collections of amazing proofs e.g. which Julia sets are smooth, when is a (polynomial) Julia set connected and a great many fun and insightful exercises.
I am actually not sure what level you are looking for but let me order the books based on difficulty (lowest easiest). Also perhaps you might be interested in reading Falconer's Fractal Geometry book. I only read some chapters many years ago, unfortunately the author introduces the Hausdorff dimension somewhere in the beginning which is perhaps one of the most difficult dimension definitions but after that it gets more easier.
Chaos and Fractals New Frontiers of Science by Peitgen. Jurgens &
Saupe (No real proofs, but contains numerics and gives excellent intuition. Also the authors always put references in their informal statements to books/articles where you can find the details)
Fractal Geometry by Falconer (only the beginning is difficult, but
contains mathematical rigour and simple proofs.)
Measure, Topology, and Fractal Geometry (nice exercises, requires
good background in basic analysis and topology. Note it also contains a section on basic topology/metric spaces)
Dynamics in one complex variable by John Milnor (Only Julia sets,
very difficult, but also very awesome)
Since the op mentioned the Mandelbrot set, I decided to focus on the Cauliflower for the Julia from the c=1/4 Mandelbrot, and see what comparisons I could find with the Weiestrass function. First, lets define the Julia as the boundary of the set of points that escape when iterating $$z \mapsto z^2 + \frac{1}{4}$$
This mapping has a fixed point of z=0.5. On the real axis, anything bigger than 0.5 or smaller than -0.5 goes to infinity. At the imaginary axis, $z=\pm \frac{i}{2}\sqrt{3}$ is the boundary point, since $z^2+0.25=-0.5$. Then in the complex plane, we have this Cauliflower for the boundary of the Julia set for c=0.25. Next, we generate the Boettcher function for the boundary of the Cauliflower, which maps the Cauliflower to the unit circle, via a Taylor series, which has a fractal boundary at the unit circle radius of convergence, and we compare it to the Weiestrass function.
I will use this form for the Weiestrass function
$$\sum_{n=0}^{\infty} 2^{-n}\cos(2^nx) = \Re(\sum_{n=0}^{\infty} 2^{-n}\exp(i2^nx))$$
$$z=\exp(ix), \; \text{Weiestrass_circle}=\sum_{n=0}^{\infty}2^{-n}z^{2^n}=z+\frac{z^2}{2}+\frac{z^4}{4}+\frac{z^{8}}{8}+\frac{z^{16}}{16}+\frac{z^{32}}{32}+\frac{z^{64}}{64}...$$
The similarity I found involves wrapping the Weiestrass function around the unit circle as a Taylor series, where the Weiestrass function is the real part of the unit circle function. By comparison, the Cauliflower Julia set unit circle function is the reciprocal of the following series; see external ray on wikipedia for some background. Later I can add the derivation of the formal Boettcher series, if the op is interested.
$$\frac{1}{\Phi(z)}=z +\frac{1}{8}z^3 +\frac{11}{128}z^5 +\frac{29}{1024}z^7 +\frac{1619}{32768}z^9 +\frac{5039}{262144}z^{11}+\frac{75391}{4194304}z^{13}...$$
$$\Phi(\pm 1)=\pm \frac{1}{2}, \Phi(\pm i) =\pm \frac{i}{2}\sqrt{3}, \;\; \Phi(z^2)=\Phi(z)^2+\frac{1}{4}$$
Here is a graph of the Weiestrass "Taylor series" function, wrapped around the unit circle, followed by Julia set Cauliflower, reciprocal of the Taylor series, also wrapped around the unit circle. Both functions have a similar kind of fractal similarity. Both functions can be represented as a Taylor series which converges inside the unit circle, and on the boundary, but not outside the unit circle. Both functions are continuous on the unit circle boundary. Both functions are nowhere differentiable at the unit circle boundary, and cannot be extended outside the unit circle boundary.
Weiestrass Taylor series, where red is real (cosine part of original definition) and green is imag
Julia unit circle, reciprocal of the Boettcher Taylor series above, red is real, green is imag
Best Answer
You can only prove selfsimilarity for fractal sets $K$ that are defined mathematically, e.g., the Cantor set, the Sierpinsky carpet, Koch's curve. In these cases $K$ is not only similar to a part of itself, but $K$ is in fact the union of (more or less disjoint) similar copies of itself: $$K=\bigcup_{i=1}^m f_i(K)\ ,$$ whereby the $f_i$ are similarities. In the above examples this is evident by inspection. All these examples go under the heading Iterated Function System. See the literature quoted in the linked Wikipedia entry, notably Barnsley and Falconer.
If you require only that $K$ is similar to a part of itself (as in your quote from Wikipedia) then there are examples which you might not consider "fractal", e.g., the logarithmic spiral $$r=e^\phi\qquad(-\infty<\phi\leq0)\ .$$