Geometry

Self-Similarity

Big Picture

A geometric shape is said to be self-similar when it is exactly similar to part of itself. Self-similarity results in a lot of repetitive and interesting shapes.

Key Terms

Self-Similarity: When one part of an object can be enlarged (or shrunk) to look like the whole object.

Iteration: A step in a sequence of repetition.

Self-Similarity

There are several examples of self-similarity: Sierpinski triangle, fractals, and the Cantor set. Self-similar objects can be formed through an iterative process. The first iteration is typically called the Start Level or Stage 0.

Sierpinski Triangle

The Sierpinski triangle iterates an equilateral triangle by connecting the midpoints of the sides. The central triangle is shaded.

Sierpinski Triangle

All the triangles inside of the big triangle are similar to each other and to the big triangle, so they’re all self-similar.

Fractals

A fractal is a self-similar object that is repeated at successively smaller scales. Also called a Koch snowflake, the fractal below is also made up of equilateral triangles.

Fractals

Cantor Set

The Cantor set is created by dividing a segment into thirds and erasing the middle third.

Cantor Set

Notes