Symmetry Group P4m

This tessellation method uses 3 reflections in a 45/45/90 degree triangle.  Fairly rigid looking tessellation as the 3 mirrors are quite apparent, and, on a zoom-out, the whole texture feels like a tartan cloth. To accomplish this tessellation, draw from the 3 points in the triangle to an arbitrary hook-up point somewhere inside the triangle, not necessarily the centre. No straying  over the mirror line is allowed though. You end up with 3 different outlines, all with bilateral symmetry. Quite similar to the previous post on symmetry group P3m1, but the angles are different here, 45/45/90 in this case.

The image below explains how three lines, from each of the triangle’s points, joined somewhere in the main area of the triangle, will create a tessellation in the P4m group. The red, the blue and the green lines joining at the yellow dot. The black lines are the mirrors. Flip the line in the mirror, each of the mirrors and repeat… ad evitam eternam. Then, just like imagining the Man in the Moon, draw a figure inside each of the three shapes.

Symmetry Group P4m-explained

Symmetry Group P4m-explained

Below, is one such design, zoomed-out… looks like a tartan cloth, or a tablecloth on the picnic table.

Symmetry-Group-p4m-zoomed-out

Symmetry-Group-p4m-zoomed-out

The image below was accomplished using P4m symmetry group, three lines create the outline of all 3 shapes, joining at the two hands and the edge of the black cape. All three figures have bilateral symmetry. Including the Peace sign on his t shirt.

Symmetry-Group-P4m-Prissy-Groovy-Cool- © 2013-Champagne-Design

Symmetry-Group-P4m-Prissy-Groovy-Cool- © 2013-Champagne-Design

Oxymora below is still another P4m tessellation.The Happy Buddhist, the Rasta Office Clerk, and the Fit Senior. The Buddhist’s fingertips, the Rasta Dude’s shoulder and the Senior’s elbow, is the point at which the three divider lines join.

Symmetry-Group-P4m-Oxymora-2013 © Champagne-Design

Symmetry-Group-P4m-Oxymora-2013 © Champagne-Design

That’s it for now, till the next symmetry group. A few more to go.

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!

10 thoughts on “Symmetry Group P4m

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