The Koch Snowflake is a fractal named after the Swedish mathematician Helge von Koch (1870-1924). It’s simple to make, attractive to see:
And here’s how you making it, starting with an equilateral triangle:
Make the Flake #0
Make the Flake #1: Erect a 1/9-sized equilateral triangle on middle of each side
Make the Flake #2: Then a 1/81-sized equilateral triangle on the middle of each straight line created by #1
Make the Flake #3: And so on.
Make the Flake #4
Make the Flake #5
Make the Flake #6
Make the Flake #0-6 (animated at ezGif)
In the end, the Koch snowflake has an infinitely long perimeter around a finite area (see discussion at Wikipedia). The Koch anti-snowflake or un-flake also combines an infinitely long perimeter and finite area:
Koch un-Flake
You make the un-flake like this:
Make the Un-Flake #0
Make the Un-Flake #1
Make the Un-Flake #2
Make the Un-Flake #3
Make the Un-Flake #4
Make the Un-Flake #5
Make the Un-Flake #6
Make the Un-Flake #0-6 (animated at ezGif)
And you can combine the Koch snowflake and the Koch anti-snowflake like this:
Make the Flake+Un-Flake #0
Make the Flake+Un-Flake #1
Make the Flake+Un-Flake #2
Make the Flake+Un-Flake #3
Make the Flake+Un-Flake #4
Make the Flake+Un-Flake #5
Make the Flake+Un-Flake #6
Make the Flake+Un-Flake #0-6 (animated at ezGif)
Here’s another variation, what you might call the Koch-Cross flake:
Koch-Cross Flake #0
Koch-Cross Flake #1
Koch-Cross Flake #2
Koch-Cross Flake #3
Koch-Cross Flake #4
Koch-Cross Flake #5
Koch-Cross Flake #6
Koch-Cross Flake #0-6 (animated at ezGif)
And the Koch-Cross un-flake:
Koch-Cross un-Flake
And combined Koch-Cross Flake+un-Flake:
Combined Koch-Cross Flake+un-Flake
Here’s an animation of the combined Koch-Cross Flake+un-Flake:
Combined Koch-Cross Flake+un-Flake (animated at ezGif)
And the combined Koch-Cross Flake+un-Flake created on the sides of a square:
Combined Koch-Cross Flake+un-Flake on square
And a final variant of the infinitely many on offer:
Variant Koch snowflake stage #1
Variant Koch snowflake stage #6
Un-Flake of variant Koch snowflake
Combined variant Koch Flake+un-Flake
