x̄ - >Proof of Mandelbrot and Julia set explained with visualisation and ai shapes

Let’s Break Down the Proof of the Mandelbrot and Julia Sets

Visualisation app to test the mathematical insight, fractals, Mandelbrot and Julia set.

In a more accessible way, focusing on the core ideas and intuition behind the mathematical reasoning.

Mandelbrot Set Proof Explained

The Mandelbrot set is the collection of complex numbers \( c \) where the sequence defined by \( z_{n+1} = z_n^2 + c \), starting with \( z_0 = 0 \), stays bounded (i.e., doesn’t grow to infinity).

• Intuition: Imagine starting at 0 and repeatedly squaring the result and adding \( c \). If this process keeps the numbers from blowing up, \( c \) belongs to the Mandelbrot set. The key is to figure out when this happens.
• Escape Condition: Mathematicians found that if the magnitude \( |z_n| \) ever exceeds 2, the sequence will likely escape to infinity. This is because, for large \( |z_n| \), the \( z_n^2 \) term grows much faster than \( c \), pushing the sequence outward. So, we check iterations: if \( |z_n| \leq 2 \) for all \( n \) up to a large number (say 1000), \( c \) is considered in the set.
• Why \( |z_n| \leq 2 \)?: Consider the next step: \( |z_{n+1}| = |z_n^2 + c| \). If \( |z_n| > 2 \), then \( |z_n^2| = |z_n|^2 > 4 \), and adding \( c \) (which has a finite size) won’t stop the growth. This suggests a critical radius. Rigorous analysis confirms that the filled Julia set (and thus the Mandelbrot set) is contained within the disk \( |c| \leq 2 \), though the boundary extends to this limit.
• Fractal Nature: The boundary’s complexity arises because small changes in \( c \) can switch the behavior from bounded to unbounded, creating the intricate, self-similar patterns we see.

Julia Set Proof Explained

The Julia set for a given \( c \) is the set of complex numbers \( z \) where the iteration \( z_{n+1} = z_n^2 + c \) produces chaotic behavior—neither settling to a fixed point nor escaping.

• Intuition: Start with a \( z \) value and iterate. If it escapes (e.g., \( |z_n| > 2 \)), it’s outside the filled Julia set. If it stays bounded but doesn’t settle, it’s on the Julia set’s boundary. The filled Julia set includes all points that don’t escape.
• Connectedness Link: The Julia set’s structure depends on \( c \). If \( c \) is in the Mandelbrot set, the critical point \( z = 0 \) (where the derivative \( f_c'(0) = 0 \)) has a bounded orbit, meaning the Julia set is connected (like a blob with detailed edges). If \( c \) is outside, the orbit escapes, and the Julia set becomes a disconnected “dust” of points.
• Escape Time Method: To compute this, iterate \( z_{n+1} = z_n^2 + c \) from a starting \( z \). If \( |z_n| > 2 \) within, say, 1000 steps, \( z \) escapes. The boundary (e.g., the white shape in your image) is where this behavior teeters, giving the fractal intricacy. The dimension (1.942 in your case) reflects this complexity.

Connection Between the Two

• The Mandelbrot set acts like a catalog: for each \( c \) inside it, the Julia set is connected; outside, it’s not. Your image shows a Julia set for \( c = -5 + 4i \), which lies outside the Mandelbrot set (since \( |c| = \sqrt{(-5)^2 + 4^2} = \sqrt{41} \approx 6.4 > 2 \)), explaining its disconnected, intricate boundary.
• Critical Point Role: The behavior of the critical point \( z = 0 \) under iteration determines the set’s properties. If it’s bounded, the Julia set connects; if it escapes, it fragments.

This explanation simplifies the rigorous proofs (which involve complex analysis, fixed-point theorems, and potential theory), but it captures the essence: the sets emerge from iterative dynamics, with boundaries revealing fractal beauty due to sensitivity to initial conditions.