From: 3blue1brown
The Mandelbrot set is an iconic shape and a “poster child of math” that arises from the field of holomorphic dynamics [00:00:04]. This field studies what happens when complex functions are repeatedly applied [00:01:07].
Holomorphic Dynamics: The Core Concept
Holomorphic dynamics involves functions with complex number inputs and outputs that are also differentiable [00:00:37]. Geometrically, being differentiable in this context means that when zoomed in near a point, the function’s behavior looks approximately like scaling and rotating (multiplying by a complex constant) [00:00:45]. This includes common functions like polynomials, exponentials, and trigonometric functions [00:01:01].
The “dynamics” refers to the process of repeatedly applying one of these functions: evaluating on an input, then evaluating the same function on the output, and so on [00:01:07]. The resulting sequence of points can exhibit various behaviors:
- Cycles: The pattern of points gets trapped in a cycle [00:01:26].
- Limiting points: The sequence approaches a specific point, including “the point at infinity” [00:01:30].
- Chaotic behavior: No apparent pattern [00:01:45].
When these behaviors are visualized, they often result in intricate fractal patterns [00:01:50]. An example of this is Newton’s fractal, which arises from iterating a rational function [00:02:03].
Defining the Mandelbrot Set
The most popular example of a rational function studied in this context is z² + c, where c is a constant [00:03:47]. The Mandelbrot set is constructed by considering c as a changeable parameter, while always starting the iterative process with an initial value of z = 0 [00:04:09].
The iterative process is:
- Start with
z₀ = 0. z₁ = z₀² + c = c[00:04:22]z₂ = z₁² + c = c² + c[00:04:27]- And so on,
z_n+1 = z_n² + c[00:04:42].
The set is then visualized by coloring points in the complex plane based on the behavior of this iteration for different values of c:
- Points where the process remains bounded (does not “blow up”) are colored black [00:05:07]. (If the value gets as big as 2, it will blow up to infinity [00:05:01].)
- Divergent values are assigned a gradient of colors based on how quickly they rush off to infinity [00:05:11].
This results in the iconic image of the Mandelbrot set [00:05:22]. It functions as a “parameter space” because each pixel corresponds to a unique function determined by the parameter c [00:06:34].
Interpreting the Mandelbrot Set’s Features
The different sections of the Mandelbrot set represent specific dynamic behaviors:
- The largest cardioid section (main body) includes values of
cwhere the process eventually converges to some limit [00:05:52]. This corresponds to when at least one of the function’s fixed points is attracting [00:12:24]. - The large circle on the left indicates values where the process gets trapped in a cycle between two values [00:05:58].
- Smaller circles, like the top and bottom ones, show values where the process gets trapped in cycles of three values, and so on [00:06:05].
Relation to Julia Sets
While the Mandelbrot set is a parameter space (tweak c, fix z₀), Julia sets are formed by fixing c and letting the pixels represent different initial seed values (z₀) [00:06:47]. These images, often called Julia fractals, also color pixels black if the process remains bounded, and use gradients for divergent values [00:07:10].
It’s important to note that the term “Julia set” refers specifically to the boundaries of these regions, not the interior black regions [00:07:56].
Stability and Attracting Cycles
A key concept in understanding these dynamics is “stability” [00:09:48].
- A fixed point (
f(z) = z) is attracting if nearby points tend to get drawn towards it [00:09:52]. - It’s repelling if nearby points are pushed away [00:09:57].
This stability can be computed using the function’s derivative at the fixed point [00:10:00]:
- If the absolute value of the derivative is less than 1, it’s an attracting fixed point [00:11:22].
- If the absolute value is greater than 1, it’s a repelling fixed point [00:11:34].
Cycles can also be attracting or repelling. An attracting cycle means a neighborhood of points around a value from that cycle tends to get pulled in [00:15:04]. For the Mandelbrot set, the fact that we only use z = 0 as a seed value is sufficient because of a theorem by Fatou, which states that if a rational map has an attracting cycle, at least one of the values where the derivative of the iterated function equals zero must fall into that cycle [00:17:48]. This means if a stable cycle exists, the z=0 seed value will find it [00:18:35].
The Mandelbrot Set’s Universal Appearance
The appearance of the Mandelbrot set is not exclusive to the z² + c function [00:20:58]. It can be found in parameter spaces for other dynamic systems, for instance, when visualizing which cubic polynomials have attracting cycles for Newton’s method [00:20:43]. This suggests that the Mandelbrot set relates to a more general and universal property of parameter spaces in such iterative processes [00:21:03].
Why Fractals? The Link Between Chaos and Complexity
The intricate fractal boundaries in these diagrams are a result of the inherent chaotic behavior of points in the Julia set [00:23:35].
- Fatou Set: Points that eventually fall into some stable, predictable pattern are part of the Fatou set [00:23:29]. A small disc around a point in the Fatou set will eventually shrink as it falls into stable behavior [00:23:06].
- Julia Set: This is everything else; the rough boundaries between colored regions [00:23:41]. Points in the Julia set tend to behave chaotically, meaning their nearby neighbors will eventually fall into qualitatively different behaviors [00:23:42].
A key result known as the “stuff-goes-everywhere principle” (a corollary to Montel’s theorem) states that if you draw a small disc around a point on the Julia set, it tends to expand over time and eventually hits almost every point on the complex plane [00:23:49]. This principle implies that if there are three or more attracting behaviors, the Julia set cannot be smooth; it must be complicated and fractal-like [00:24:02]. This highlights a deep, logical link between chaos and the generation of fractals [00:26:45].