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@@ -1,4 +1,40 @@ # MandleBrot Set - -# wikipedia link https://en.wikipedia.org/wiki/Mandelbrot_set + +# wikipedia definition + +**Formal definition** + +The Mandelbrot set is the uncountable set of values of $c$ in the complex plane +for which the orbit of the critical point $z = 0$ under iteration of the quadratic map + +$$z \mapsto z^2 + c$$ + +remains bounded. Thus, a complex number $c$ is a member of the Mandelbrot set if, +when starting with $z_0 = 0$ and applying the iteration repeatedly, the absolute +value of $z_n$ remains bounded for all $n \in \mathbb{Z}^+$. + +For example, for $c = 1$, the sequence is 0, 1, 2, 5, 26, ..., (sequence A003095 +in the OEIS) which tends to infinity, so 1 is not an element of the Mandelbrot set. +On the other hand, for $c = -1$, the sequence is 0, -1, 0, -1, 0, ..., which is bounded, +so -1 does belong to the set. + +The Mandelbrot set can also be defined as the connectedness locus of the family +of quadratic polynomials $f(z) = z^2 + c$, the subset of the space of parameters +$c$ for which the Julia set of the corresponding polynomial forms a connected set. +In the same way, the boundary of the Mandelbrot set can be defined as the bifurcation +locus of this quadratic family, the subset of parameters near which the dynamic +behavior of the polynomial (when it is iterated repeatedly) changes drastically. + +## running simulation +a pre-built binary is provided, use either + +''' +./mandelbrot_set +''' + +or + +''' +make run +''' |