Symmetry group P3, or 3 rotations through 120 degrees. This is quite a versatile symmetry group, and easy to accomplish. Only two lines to draw the outline of your classic tessellation. Basically, you draw any line through the edge of a hexagon, linking two of the three-way rotation points, and you will get a closed shape after 3 fold rotation. That last rotation point will need to link with the first line. Three different 120 degree rotation points around a hexagon. See the technical drawing below.

Symmetry-group-p3-explained

Three rotation points, A, B and C. The first line, red, links A to C. Any wiggly line will do and straying from the area is permitted, encouraged. The second line links point B, to anywhere on the first line. This will give you a single, repeating shape, showing up 3 times in the hexagon, shown here as the fine grey line. Now all you have to figure out is what to fit inside! The possibilities are endless and fun.

The Jewel Thief below, was created using this symmetry rule.

Symmetry-Group-p3-Jewel-Thief – © 2013 Champagne Design

Zoomed-out, the hexagon becomes quite apparent.

Symmetry-Group-p3-Jewel-Thief-zoom-out

Below is another P3 symmetry drawing that I’ve name “Rabid Rabbit”. Zoomed-out. The hexagons are quite visible in this one. Not quite as visible when you zoom-in.

Symmetry-Group-P3-Rabid-Rabbit-zoomed-out

Wicked rabbits. Garden munchers.

Symmetry-Group-P3-Rabid-Rabbit – © 2013 Champagne Design

Get out there and try it!

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.

Or this 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!

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MC Escher’s Lizards are based on this symmetry group.

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