Most simple of all symmetry systems. A translation of your design both vertically and horizontally. That’s it! The shape is up to you as long as it repeats in those two directions, and is built around a four sided figure, square, rectangle, lozenge. No mirrors, no rotations, no glides — simply a translation. Straying outside the basic parallelogram is encouraged.

The image below illustrates two simple lines translated both up/down and left/right. Again, as in all tessellations, it up to you to figure out what to place inside the shape. Just like imagining sheep in cloud shapes. Tweak the edges, repeatedly and something might emerge.

Symmetry group p1 = one translation horizontal, one vertical, anything goes

Any shape works! As you can see from the tessellation below. Follow the cat’s tail and see how far you get. A most unusual stamp!

Symmetry Group p1-cat-with-tail-into-the-next-galaxy © 2013 Champagne Design

Sometimes the shapes can be easy to fill, sometimes they can be challenging. The most simple shapes have slight modifications from the basic repeating squares, rectangles, parallelograms. The tessellation below started off fairly easily, the brown puppy repeated by translation.

Symmetry Group P1 – A zoom-out shows the repeating parallelogram © 2013 Champagne Design

The challenge was to fill the rest of the unit cell with something adequate to the topic. Three more puppies. That last white one was a stretch. The tunes by Three Dog Night come to mind, and morph into Four Dog Day.

Symmetry Group P1 – Four Dog Day © 2013 Champagne Design

For a simpler, but still quite pleasing tessellation, the Crashing Waves below are a simple modification to a parallelogram (highlighted). Feels more like wallpaper, wrapping paper than an actual tessellation, but it illustrates the concept well.

Symmetry Group P1 – Crashing Waves – © 2013 Champagne Design

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. (I used to have a basic Mac-based software, way back in prehistory, 1995, by the same name. It worked on the cutesy first Mac)

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

Comments are always welcome!