The Koch curve is constructed by iteratively replacing each line segment with four segments of 1/3 the length, forming a triangular bump. Starting with an equilateral triangle, the Koch snowflake results after infinitely many iterations.

Properties:

• Self-similar: each small section resembles the whole
• Infinite perimeter: length grows by factor 4/3 each iteration, diverging
• Finite area: bounded by the circumscribed circle
• Dimension: $d = \log 4/\log 3 \approx 1.262$

The Koch snowflake models coastlines and natural boundaries with fractal structure.

The Sierpinski gasket (or triangle) removes the central triangle from an equilateral triangle, then repeats for each remaining piece. Alternatively, it's the attractor of an IFS with three similarities, each scaling by $1/2$:

$3 \cdot (1/2)^d = 1 \quad \Rightarrow \quad d = \frac{\log 3}{\log 2} \approx 1.585$

This fractal appears in:

• Cellular automata (Rule 90)
• Pascal's triangle mod 2
• Vibration modes of triangular membranes

The gasket demonstrates that fractals arise naturally in diverse mathematical contexts.

For $f_c(z) = z^2 + c$, the filled Julia set $K_c$ consists of bounded orbits:

$K_c = \{z \in \mathbb{C} : f_c^n(z) \not\to \infty\}$

The Julia set $J_c = \partial K_c$ (boundary) is typically fractal:

• For $c = 0$: $J_0$ is the unit circle (dimension 1)
• For $c = -1$: $J_{-1}$ is a dendrite with dimension $\approx 1.27$
• For $c$ in the Mandelbrot set but near the boundary: $J_c$ has intricate fractal structure

Julia sets exhibit self-similarity under iteration of $f_c$, with dimension varying continuously with $c$. They're among the most visually striking fractals, displaying infinite complexity at all scales.

The dragon curve is the limit of folding a strip of paper in half repeatedly and unfolding at right angles. It's self-similar with two pieces, each rotated $45°$ and scaled by $1/\sqrt{2}$:

$2 \cdot (1/\sqrt{2})^d = 1 \quad \Rightarrow \quad d = 2$

Despite being a curve, its dimension is 2—it's a space-filling curve that completely covers a region. This demonstrates that topological and fractal dimensions can differ dramatically.

The Menger sponge is the 3D analog of the Sierpinski carpet: remove the central cube and face centers from a cube, then repeat. It has:

• Zero volume (measure decreases geometrically)
• Infinite surface area
• Dimension: $d = \log 20/\log 3 \approx 2.727$

The sponge models porous materials and demonstrates that fractals extend naturally to higher dimensions. Its structure appears in chemical catalysts and biological tissues.