# Fractal universe

Fractal geometry is a branch of mathematics that focuses on the description of geometric objects that have properties of self-similarity, that is, the property of having a similar structure to itself at any level of magnification. Fractal geometry differs from traditional geometry, which focuses on perfectly regular and defined geometric objects, such as triangles, squares, and circles.

The Mandelbrot set is a famous fractal that is generated by a simple mathematical algorithm. It is named after the French mathematician Benoit Mandelbrot, who is considered the father of fractal geometry and who introduced the term “fractal” in 1975. The Mandelbrot set is defined as the set of complex numbers c for which the function f(z) = z^2 + c, with z starting at 0, does not diverge when iterated. The function is iterated repeatedly, and for each point in the complex plane, the sequence of iterates is calculated. A point c is in the Mandelbrot set if the absolute value of the function is always less than or equal to 2. When the Mandelbrot set is graphed, it produces a highly detailed and infinitely complex pattern that is symmetrical about the origin. The boundary of the Mandelbrot set is a fractal, meaning that it has a infinite level of detail and self-similarity. The set also exhibits a kind of “chaos” since small changes in the initial value of c can greatly change the behavior of the sequence of iterates. The Mandelbrot set has many interesting properties, such as the presence of “miniature” copies of the whole set, called “Mandelbrot islands” and “Mandelbrot dust” which are smaller copies of the set that appear as you zoom into the set. The Mandelbrot set is not only a fascinating mathematical object, but it also has many practical applications. For example, it is used in the study of dynamical systems and chaos theory, as well as in computer graphics and art. The Mandelbrot set has been used to generate a wide variety of images, from abstract patterns to realistic landscapes.

Here’s a list of famous fractals.

1. The Mandelbrot set
2. The Julia set
3. The Sierpinski triangle
4. The Koch snowflake
5. The Dragon curve
6. The Cantor set
7. The Peano curve
8. The Lorenz attractor
9. The Barnsley fern
10. The Harter-Heighway dragon