Scientists Say: Fractal

Math allows us to create eternally unfolding patterns

The Mandelbrot set (pictured) is an example of a fractal. If you zoom in on the shape’s jagged edges, you’ll find keep finding smaller and smaller versions of the large overall shape.

Wolfgang Beyer/Wikimedia Commons (CC BY-SA 3.0)

By Katie Grace Carpenter

August 12, 2024 at 6:30 am

Fractal (noun, “FRAK-tal”)

A fractal is a geometric shape made from parts that repeat at smaller and smaller scales.

Consider the leafy fronds of a fern. If you look closely, you’ll notice that each frond appears to consist of repeating smaller fronds. This pattern is one way fractal shapes occur in nature. The structures of snowflakes follow fractal patterns. As do the buds on Romanesco broccoli.

In a fractal called the Sierpinski triangle, a triangle is divided into four smaller triangles — each of which is then divided into another four, smaller, triangles.Beojan Stanislaus/Wikimedia Commons (CC BY-SA 3.0)

If you’ve ever sketched a shape and filled it in completely with smaller versions of the same shape, then congrats! You’ve drawn a fractal. One example is the Sierpinski triangle.

To draw this simple fractal, start with an equilateral triangle. That’s a triangle with three equal sides. Then, divide the triangle’s area into four smaller equilateral triangles. (Hint: Draw one upside-down triangle inside the first.) Continue to subdivide those triangles into yet smaller ones. In theory, you could draw this shape in greater detail for eternity. But the triangles will soon become too tiny for your pen. Computers can help us visualize these patterns repeating forever.

The Sierpinski triangle is a simple kind of fractal described as self-similar. That means its repeating pattern consists of the same repeating shape. In this case, a triangle. But thanks to geometry, fractals can be even more complex.

This animation shows the complex geometry of the Mandelbrot set.

One well-known example is a geometric pattern called the Mandelbrot set. This pattern creates intricate fractals that morph, twist and spiral, making shapes that repeat and keep on repeating. These shapes and patterns underlie some computer-generated special effects seen in many movies today.

From the tiniest snowflake to the big screen, fractals are infinitely complex. These repeating formulas define geometric shapes that stretch into infinity.