This symmetry group is one of my favourites, as it uses glides as well as rotations in its construction. No mirrors here, so the tessellated figures do not need to be bilaterally symmetric constructs. Glides are one more trick to tickle your visual perception.
Accomplishing the division of the surface using this symmetry method consists in linking one of the two-fold rotation points (R1) to one of the corners (red line 1) From that same corner, draw a line to the other adjacent corner (green line 2). Only one two-fold rotation point is left (R2). Connect R2 to anywhere, using any curve. Now stare at it till your eyes cross (just kidding) and you might see a figure emerge. Imagination. Try it. Delete it. Try again till you can edit the line to form a figure.
Dad was an Usher for a summer job, when he was a youngster. A Posh Usher, uniform and all. Cool job, because you get to see all the movies. Over and over… well maybe not. The green dots and red dots indicate the two different 180 degree rotation points, same as R1 and R2 in the above. The dashed line indicates the glide-reflection line, which can be viewed as moving the figure in translation with at the same time a mirror applied to it.
The sketch below uses the same symmetry group; Involuntary Suicide… where the guy has the barrel of his weapon in the other guy’s mouth, which happens to be himself! Can you locate the two rotation points?
One rotation point at the tip of his t shirt at the belly, the other rotation point at his calf, right or left depending which clone you’re looking at.
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!