A Seed Indeed

Like plants, fractals grow from seeds. But plants start with a small seed that gets bigger. Fractals start with a big seed that gets smaller. For example, perhaps the most famous fractal of all is the Koch snowflake. The seed of the Koch snowflake is step #2 here:

Stages of the Koch snowflake (from Fractals and the coast of Great Britain)

To create the Koch snowflake, you replace each straight line in the initial triangle with the seed:

Creating the Koch snowflake (from Wikipedia)

Animated Koch snowflake (from Wikipedia)

Now here’s another seed for another fractal:

Fractal stage #1

The seed is like a capital “I”, consisting of a line of length l sitting between two lines of length l/2 at right angles. The rule this time is: Replace the center of the longer line and the two shorter lines with ½-sized versions of the seed:

Fractal stage #2

Try and guess what the final fractal looks like when this rule is applied again and again:

Fractal stage #3

Fractal stage #4

Fractal stage #5

Fractal stage #6

Fractal stage #7

Fractal stage #8

Fractal stage #9

Fractal stage #10

I call this fractal the hourglass. And there are a lot of ways to create it. Here’s an animated version of the way shown in this post:

Hourglass fractal (animated)

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