From 78a520f77c849626ec50db9e6a379fec1a170156 Mon Sep 17 00:00:00 2001
From: Tanner Robison <Robison.tnr@gmail.com>
Date: Thu, 27 Aug 2026 12:51:56 -0700
Subject: update README

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 README.md | 40 ++++++++++++++++++++++++++++++++++++++--
 1 file changed, 38 insertions(+), 2 deletions(-)

(limited to 'README.md')

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 # 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
+'''
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