Fractal Paper Snowflakes (Fractal LXVII)

Some time ago, I gave instructions for making your own paper fractal. I’ve decided to do another today. This time, instead of ending up with a crumpled paper ball, you’ll have a design worthy of becoming a holiday decoration.

Remember those old paper snowflakes that we would cut as children? We’re going to use the same concept today to create a fractal snowflake.

To make this fractal, you’ll need a piece of white paper, some scissors, a straight edge (a ruler, or even another folded piece of paper) and a compass. The latter two items are actually optional--you could trust your eye instead--but it helps to use a compass and straightedge to estimate lengths.

Start by folding your paper in half, and then into thirds. This is similar to what we did as children (check out Make-a-Flake if you need a refresher.) Instead of making a narrow wedge, we’re aiming to make an equilateral triangle--that is, a triangle with three equal sides. (This is where that compass might come in handy--find the spot where if you cut straight across, each length will be equal.)

Now for the scaling fractal fun. Find the outside of your triangle and, with your eye or compass, divide the edge into thirds. Now, check out the middle third: we want to turn this into the side of another equilateral triangle. Using your compass, or again your eye, find the third corner of the (smaller) triangle. Now cut that small triangle out.

At this point, if you unfold your paper, you’ll find you’ve created the first two iterations of the Koch Curve. (More precisely, it is an exterior snowflake.) But, if you keep it folded, you can repeat the last step with the fresh cut edges. Just cut triangles with sides 1/3 the original length until the triangles become frustratingly small.

With a computer, we can create Koch snowflakes that have infinite diameters--those triangles keep cutting in forever:

(A desktop sized, and for some reason better proportioned version is available by clicking the fractal.)

But for now, let’s stick to that piece of paper. To create your own ornamental Koch snowflake, cut an equilateral triangle from the tip of the wedge. Cut the same pattern of 1/3 triangles across the remaining edge. Then, create another snowflake (using the same instructions as above) on the folded triangle you cut from the tip.

When you’re all done, you’ll have two snowflakes that fit inside one another. (A little scotch tape and a broken toothpick can hold them in place nicely.) Here’s my finished snowflake:

I hope you enjoy making your own!

For a bonus fractal, look at the pile of triangles that has piled up on your desk. With these, you can create a Sierpinski triangle: Arrange each triangle upside down inside the next largest, until you create something like this:

(I used a variation of a Sierpinski pattern to create the texture of the digital snowflake above.)

All fractals created by the author using ChaosPro.