I’ve already given you a foretaste of this P4 symmetry group with the cougar tessellation, and story, a few weeks ago. A thrilling event, possibly once in a lifetime. If you haven’t read it yet, do. But now for the technical side of this P4 group, with a few more examples.
Workie boots are heavy. Steel toe and sole. Ever try running in a pair of those?
Symmetry groups P2, P3, P4, and P6, have evolved into my favourite symmetry groups. Possibly because the constructions have no mirrors in them. There is fluidity and freedom in these symmetry groups. You are encouraged to stray outside the mesh limits. Fish-eye lens exaggerations usually occur in my case. (Did you know you could get a clip-on fish-eye lens for your iPhone? Oh, so cool.) In the case of symmetry group P4, there are two different 4-way rotation points, one inside the box and one outside the box, and four identical 2-way rotation points on all corners of the box (mesh). The image below has these points highlighted. I’ve drawn in a faint red line to outline the box/mesh.
Point A, the 4-way rotation is linked to one of the 2-way rotation points (C) with a line, the red one. The second 4-way rotation point B, links to anywhere on the first line. These two wavy straying lines will give to a single repeating shape. As in all tessellations, the trick is to tweak the outline, then fill the region with a recognizable figure. Any shape can also be subdivided into any number of figures, if two creatures become apparent to you within your shape, go for it!
The sketch below, has the same structure. Four identical 2-way rotation points, at the rump, and two different 4-way rotation points; one at the ear and one at the front hoof.
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!