Drawing geometric whirls- with templates!

A whirl, according to Wolfram Mathworld -
Whirls are figures constructed by nesting a sequence of polygons (each having the same number of sides), each slightly smaller and rotated relative to the previous one. The vertices give the path of the  mice in the mice problem, and form logarithmic spirals.


Or it's just another really nifty thing to do with straight lines that is relaxing and fun to draw. All the images in this tutorial are approximately 800x800 pixels, or 100 dpi 8x8 inches, click on them for the full sized versions.

Triangle in a hexagon in a dodecagon

So, to draw these, you'll need a ruler, pencil, paper and a template with radial lines evenly marked. The more radial lines you have, the more lines and finer the design will be. The image above was done with lines that were 5 degrees apart. My instructions show lines 10 degrees apart. (Really, you can skip the math parts if you just want to get to drawing)

This template will do designs that have 3,4, 6, 8 and 12 sides because it has 36 radial lines and 36 is divisible by all those numbers.

So, showing using a triangle. 36/3= 12, so your initial triangle will be marked starting at the ball tips with 11 tips between angles.

Now you start doing the nested rotations. Start the next triangle one point removed from the initial points. Each triangles points are where the lines of the template and the preceding triangle intersect.

Repeat, repeat, repeat until you have enough iterations you are happy with the design!

The real problem with the above template for squares is that for a square, it will be a bit small unless you put it at a 45 degree angle. Here's a template specifically for squares.

A few whirls just to color or use for embroidery patterns.

Here's the template for the one at the top with 72 radial lines.

And all the templates in black and white so you can use them behind a sheet of paper that's transparent enough to use to trace.

I hope you enjoy this. It's a lot of fun and a bit addictive. More open designs would probably make very nice "strings" for Zentangles as well.