First, let's start with the property of fractals we observed in the Romanesco cauliflower.

Property: Self-Similarity is the property that zooming into an object produces a never-ending repeating pattern.



Another example of self-similarity in nature are the repeating patterns of crystallizing water and snowflakes.

How do we describe these self-similar patterns and how do we generate self-similar shapes mathematically that are reproducible at any magnification? We have seen fractal patterns in snowflakes, so let's start by generating a self-similar pattern resembling a snowflake.

Koch Snowflake Starting with an equilateral triangle, create an equilateral triangle using the middle third of each side as a base, and then remove the base of the triangle. Now, repeat this process for each line segment in the resulting figure. Here are the first few iterations: Continuing this process gives the Koch snowflake in the limit. Here is a close-up of the border after multiple iterations: Since zooming into the Koch snowflake gives a curve that is a copy of itself at a smaller scale (called the Koch curve), the Koch snowflake displays self-similarity. If the equilateral triangle we start with has side length 1, then notice that by replacing each line segment by 4 4 4 segments of one third the length, we multiply the length by 4 3 \frac{4}{3} 34​ at each step. This shows that after n n n steps, the length of the perimeter is 3 ⋅ ( 4 3 ) n 3 \cdot \left( \frac{4}{3} \right)^n 3⋅(34​)n, so the Koch star has infinite perimeter if measured as a 1-dimensional curve. However, as we will see later, this arises because the Koch snowflake should be thought of as having more than 1 dimension and trying to measure a shape in the wrong dimension gives a meaningless answer. This is similar to trying to measure the amount of a very thin thread needed to cover a 2-dimensional square. We would need an infinitely long thread since we are trying to measure a 2-dimensional object with a one-dimensional curve.

A B C D E What is the area enclosed by a Koch snowflake starting from an equilateral triangle with side length 1? A. 1

B. 1 2 \frac{1}{2} 21​

C. 2 3 5 \frac{2\sqrt{3}}{5} 523 ​​

D. 2 3 4 2 \frac{\sqrt{3}}{4} 243 ​​

E. Area is infinite

The Koch snowflake shows that even though fractals are complex, they can be generated by repeatedly applying simple rules. We can think of the starting triangle of the Koch snowflake as the initiator and the step of replacing each line by a peak as the generator. If we instead start with a line segment as initiator and use the following generator, we obtain a different pattern.

These examples demonstrate the following properties of fractals.

Fractals have detail at arbitrarily small scales and display irregularity that cannot be described by traditional geometrical language.

In other words, fractals are objects which, at any magnification, will never “smooth out” to look like Euclidean space.