# Koch Snowflake

In the article on Koch Curves we derived the famous koch-curve, by starting with a line, then a bumpy-line, then a self-similar recursive bumpy-line. The final result looked like this:

Now we can put three of them together to make Koch's snowflake.

# Layered Koch Snowflake

Layering several of them on top of each other looks pretty cool.

## Reverse Koch Snowflake

If you're thinking what I'm thinking, then what you are thinking is "If you're thinking what I'm thinking, then what you are thinking is 'Why turn right? Why not turn left?'" Which is very clever of you. If we turn our 3 Koch curves around (by turning those right turns to left turns), they will face inward, not outward, and we'll have a reverse Koch Snowflake.

There we have an inside-outed Koch Snowflake.

## Layered Reverse Koch Curve

We can layer that as well.

## Extra Long Reverse Koch Snowflake

Oh, one more trick.

What happens if we make an extra long reverse Koch Snowflake? Instead of putting 3 reverse snowflakes together, let's put 6 of them together. We'll need a smaller angle: instead of 120 degrees between one, we'll only need 60 degrees.

(And since it's going to be a larger structure, I'll make the sides a bit smaller, 120 instead of 240)

Ah, a double length reverse Koch Snowflake is just a Koch Snowflake.

Why? Because the spaces between Koch Snowflakes are themselves Koch Snowflakes.

## These Things Pack Together With No Room To Spare!

Logo + Turtle graphics via Papert (via archive.org)

FCa