You can create fractals with mathematical equations and algorithms, but there are also fractals in nature. At their most basic, fractals are a visual expression of a repeating pattern or formula that starts out simple and gets progressively more complex.

Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Learn more about fractals, chaos theory, and how they relate to nature and systems.

Learn what fractals are, how they arise from self-similar patterns and iterated rules, and how to measure their dimensions. Explore fractals in geometry, algebra, and nature with interactive diagrams and exercises.

Fractals are distinct from the simple figures of classical, or Euclidean, geometry—the square, the circle, the sphere, and so forth. They are capable of describing many irregularly shaped objects or spatially nonuniform phenomena in nature such as coastlines and mountain ranges.

Fractal is a pattern that never ends. It elaborates mathematical constructs that exhibit self-similarity, meaning they display similar patterns or structures when zoomed in or out. This self-similarity occurs at many levels of magnification, revealing elaborate details that are replicated at every scale.

Fractals are mathematical sets, usually obtained through recursion, that exhibit interesting dimensional properties. We’ll explore what that sentence means through the rest of this section. For now, we can begin with the idea of self-similarity, a characteristic of most fractals.

fractal is a geometric shape that has a fractional dimension. Many famous fractals are self-similar, which means that they consist of smaller copies of themselves. Fractals contain patterns at every level of magnification, and they can be created by repeating a procedure or iterating an equation infinitely many times.

Fractals are infinitely complex patterns that are self-similar across different scales. Learn how fractals are common in nature, how they are made by simple iterative processes, and how they help us understand the world on many levels.

A fractal is a self-similar object or quantity that has the same "type" of structures on all scales. Learn about the fractal dimension, the coastline paradox, and the prototypical fractals such as the Mandelbrot set and the Koch snowflake.