My first ever tessellation was accomplished using this P4g symmetry group.

My understanding occurred during a class on crystallography, works with the same principles. What a blast that was. Hooked since then.

The basic concept is a 4-way rotation inside a square, built of mirrors. One line from the rotation point to the perimeter corner will give you four halves of an identical figure.

First ever tessellation sketch using symmetry group p4g

If two lines are drawn from the rotation point, to anywhere on the perimeter, two figures will appear, such as in MCE’s Angels and Devils, below.

MC Escher’s Angels and Demons, are built using symmetry group P4g

Quick identification method for symmetry group P4g:

1- mirror symmetry down the body,

2- figures are head to head and feet to feet, hooked at the arms/elbows/hands/wings…

3- 90 degree rotations at the arms… always end up looking the same, in my case anyhow

Symmetry Group P4g, Look Ma, I Have Two Watches

P4g used to be my favourite symmetry group, before software came out. I’ve since expanded my repertoire and now prefer the total absence of mirrors in designs — really enjoying P6 and P3.

The image below illustrates the two lines to the perimeter concept, Two Fitness Styles, one dude posing his bodybuilding skills, the other doing his yoga thing. Note the area sharing for the heads! I seem to be doing a lot of that.

Symmetry Group P4g, Two Fitness Styles

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!