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Fractals & Iterative Systems — Infinite Complexity from Simple Rules - Leejohnston - 11-17-2025
Thread 10 — Fractals & Iterative Systems
Infinite Complexity from Shockingly Simple Rules
Fractals are among the most extraordinary discoveries in modern mathematics.
From simple repeated rules (called iterative systems), we get:
• infinite detail
• self-similar patterns
• chaotic boundaries
• structures found throughout nature
• shapes that are finite but infinite in perimeter
• geometry that traditional maths cannot describe
Fractals are the meeting point of:
mathematics, art, physics, biology, computing, and chaos theory.
They are everywhere in nature and in modern computing.
1. What Is a Fractal?
A fractal is a shape or pattern that has:
• self-similarity (looks similar at different scales)
• infinite detail (zoom forever, always sees more)
• fractional dimension (not 1D, not 2D, not 3D)
• generated by iteration (repeating simple rules)
Examples:
• snowflakes
• lightning
• fern leaves
• rivers
• mountain ranges
• coastlines
• blood vessels
• clouds
• turbulence
• galaxies
Fractals are nature’s handwriting.
2. Iteration — The Heart of Fractals
Many fractals come from applying a simple rule repeatedly.
For a number x:
x → f(x) → f(f(x)) → f(f(f(x))) → …
This process produces:
• stability
• oscillation
• chaos
• infinite complexity
Iteration turns simplicity into beauty.
3. The Mandelbrot Set — The King of Fractals
The Mandelbrot set is defined by:
zₙ₊₁ = zₙ² + c
Where:
• z starts at 0
• c is any complex number
If the values stay bounded, c is inside the set.
If they explode to infinity, c is outside.
When plotted, you get:
• infinite spirals
• fractal filaments
• self-similarity
• chaotic boundaries
• structures that repeat but never exactly
The edge of the Mandelbrot set is so detailed that:
it contains more information than any computer could ever store.
4. Julia Sets
Julia sets use the same iteration as the Mandelbrot set, but fix c and vary z.
They form:
• spider shapes
• lightning filaments
• snowflake patterns
• whirlpools
• spirals
Each value of c produces a completely different fractal.
5. Natural Fractals — Nature’s Geometry
Fractals appear naturally because nature uses efficient rules.
Examples:
• coastlines → fractal roughness
• trees → self-similar branching
• lungs → maximised surface area
• mountains → erosion processes
• clouds → turbulence patterns
• river systems → minimising energy
• lightning → branch optimisation
• snowflakes → symmetry + iteration
Nature is not smooth — it is fractal.
6. Fractals in Physics & Computing
Physics:
• turbulence
• chaotic orbits
• diffusion-limited aggregation
• percolation in materials
• quantum wavefunctions
• cosmic structure formation
Computer Science:
• data compression
• image generation
• procedural terrain
• computer graphics
• noise algorithms (Perlin, Simplex)
Biology:
• neural networks
• heartbeat dynamics
• DNA folding patterns
Fractals describe systems too irregular for traditional geometry.
7. Fractal Dimensions
Fractals do not have whole-number dimensions.
A coastline is not 1D.
A mountain range is not 2D.
Fractals have fractional dimension, such as:
• 1.26
• 1.58
• 2.41
This measures their “roughness.”
8. The Big Insight
Fractals show us that the universe is not smooth, clean, or simple.
It is:
• chaotic
• irregular
• infinitely detailed
• self-similar
• shaped by simple rules repeated endlessly
Fractals are the geometry of reality itself.
Written by Leejohnston & Liora — The Lumin Archive Research Division