Fractals & Self-Similarity Reference Cheat Sheet

Key Facts

• A self-similar fractal is made of $N$ smaller copies of itself, each scaled by a factor of $\frac{1}{r}$.
• For an exactly self-similar fractal, the similarity dimension is $D = \frac{\log N}{\log r}$.
• A line segment has dimension $D = 1$, a filled square has dimension $D = 2$, and many fractals have non-integer dimensions such as $1 < D < 2$.
• The Sierpiński triangle has $N = 3$ copies scaled by $\frac{1}{2}$, so $D = \frac{\log 3}{\log 2} \approx 1.585$.
• The Koch curve has $N = 4$ copies scaled by $\frac{1}{3}$, so $D = \frac{\log 4}{\log 3} \approx 1.262$.
• In the Koch snowflake, the perimeter grows without bound while the enclosed area approaches a finite limit.
• An iterated function system uses repeated transformations such as scaling, rotation, translation, and reflection to generate a fractal.
• The Mandelbrot set is defined by iterating $z_{n+1} = z_n^2 + c$ starting at $z_0 = 0$ and checking whether the sequence stays bounded.

Vocabulary

Fractal

A fractal is a pattern or shape with detail that repeats across different scales.

Self-similarity

Self-similarity means a shape contains smaller parts that resemble the whole shape.

Scale factor

A scale factor is the ratio that compares the size of a smaller copy to the original, such as $\frac{1}{r}$.

Fractal dimension

Fractal dimension is a number that measures how a fractal's detail changes as magnification changes.

Iteration

Iteration is the process of repeating the same rule or function again and again.

Mandelbrot set

The Mandelbrot set is the set of complex numbers $c$ for which $z_{n+1} = z_n^2 + c$ remains bounded when $z_0 = 0$.