What is quasi self-similarity?
Quasi self-similarity of Julia sets. The fractal quality of Julia sets is formalized by the concept of quasi-self-similarity. Self-similarity occurs when a set is a dilated, isometric copy of one of its proper subsets, see e.g. the Von Koch curve or the Cantor set.
Is the Mandelbrot set self-similar?
The Mandelbrot set is highly complex. It is self-similar – that is, the set contains mini-Mandelbrot sets, each with the same shape as the whole.
What is the similarity dimension of the Koch curve?
The relation between log(L(s)) and log(s) for the Koch curve we find its fractal dimension to be 1.26. The same result obtained from D = log(N)/log(r) D = log(4)/log(3) = 1.26.
Do all fractals have self-similarity?
Simply put, a fractal is a geometric object that is similar to itself on all scales. If you zoom in on a fractal object it will look similar or exactly like the original shape. This property is called self-similarity. An example of a self-similar object is the Sierpenski triangle show below.
What is the weakest type of self-similarity?
•Statistical self-similarity— This is the weakest type of self-similarity; the fractal has numerical or statistical measures which are preserved across scales. Most reasonable definitions of “fractal” trivially imply some form of statistical self-similarity. (Fractal dimension itself is a numerical measure which is preserved across scales.)
What are the different types of self-similarity in fractals?
There are three types of self-similarity found in fractals: •Exact self-similarity— This is the strongest type of self-similarity; the fractal appears identical at different scales. Fractals defined by iterated function systems often display exact self-similarity.
What are some examples of self-similarity?
Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals.
What is a more general notion than self-similarity?
A more general notion than self-similarity is Self-affinity . The Mandelbrot set is also self-similar around Misiurewicz points . Self-similarity has important consequences for the design of computer networks, as typical network traffic has self-similar properties.