Symmetry Group P6m — Perfect Snowflakes | Tessellations, images in regular division of the plane

It’s been a year last week, since I started drawing with the KaleidoPaint iPad app. I was avoiding this P6m symmetry group because of the tight space for drawing within and the abundance of mirrors. A spoke of 12 mirrors at 30 degrees — a spoke of 6 and one of 4 — how do you ever fit anything in there? Very challenging. Kept putting it off. Till now. It snowed.

This is the perfect app to draw snowflakes. This one took like 30 seconds to draw. Not a true tessellation in my definition, but still cool ( no pun intended).

Symmetry Group P6m — Perfect for drawing Snowflakes

The sketch below illustrates how you can achieve a true tessellation, one where all space is occupied with the figures. There is no negative space, i.e., no background visible between the various elements (like the dark blue behind the snowflakes). Everything interlocks perfectly. Not easy with this system. To get a true tessellation, draw 3 lines, from each of the triangle’s corners (A, B, and C), to an arbitrary point somewhere inside the triangle (the green dot). You end up with 3 figures that tessellate the plane, perfectly.

Symmetry Group P6m — 3 mirrors around a 30-60-90 degree triangle

The above 3 lines were elaborated into the tessellation below. In the best of worlds we would all grow our own food (the gardener), bike and not drive vehicles (the mountain biker), and our ecology would be in balance (the frog).

Symmetry Group P6m — In the Best of Worlds

Lots of software out there to help you accomplish this type of design.

An iPad app is available, which is what I have used here to create these images: KaleidoPaint by Jeff Weeks.

There is also a java-based program “Escher Web Sketch” at the Ecole Polytechnique de Lausanne. Make sure Java is enabled and not blocked by your security software.

Also, another screen-based software by Anselm Levskaya Escher Sketch v2.

Or a pair of scissors and a piece of cardboard works quite well. That’s how I learned.

Comments are always welcome!

2015-11-19 I’ve since undertaken the study of Islamic Geometric Patterns, Eric Broug’s wonderful teaching book. Most of these beautiful patterns are based on this symmetry system. Once you start drawing following these strict rules, it is amazing the results. I am an absolute beginner at this! Most of them below have construction lines in a faded grey with the resultant final lines in a broader, brighter stroke. More later.