Purl Origami Tessellation

Patterns in the Tessellation

You can see many things in any tessellation - here I'm seeing a close mimic of the purl knitting stitch, and also an Easter bunny.

I'm also seeing parallel mirror symmetry lines with a string of twists between them, a sequence of triangle-rhombus-triangle that I use in many tessellations, and an unusual arrangement of rhombi and triangles.

Which aspect I choose to emphasize varies with the lighting, the level of zoom, and who I'm talking to.

Details

As we look closely at this tessellation, we can see that there's the zigzagging rhombi and triangles, and then there's the rhombi that go between these zigazgs.

While there are 2-fold symmetric positions within the zigzags, the rhombi between are not rotationally symmetric at all - on one end they are tucked under a pair of triangles and on the other end they just meet the tips of the triangles.

This small piece of asymmetry removes any possibility of rotational symmetry from the whole pattern.

Instead, we have mirror symmetry lines lying against the ends of our triangle-rhombus-triangle strips providing all the global symmetry in the pattern.

Weave-like

Since all of the twists are on the same side of the paper, the back side appears to have multiple pieces that were woven or pieced together.

This is a common property of simple tessellations - and a nice compensation for these patterns taking up more grid space per repetition!

Often, tessellations with twists all on one side have been folded by multiple folders already - at least in the common tilings using standard twists.

But, standard tilings revolve around rotational symmetry and this tiling definitely does not, so I'm confident that Purl is a new design.

Further Explorations

Purl is the simplest tessellation in its tiling (barring spacing changes), with all twists as closed as possible and all on the same side of the paper.

Its basic unit is composed of two rhombi and two triangles, with no rotational symmetry constraints on the choice of each rhombus and each triangle.

In theory, you could choose each twist independently to be either closed or open, front or back.

You'll still want to preserve the mirror symmetry, so alternating properties of columns is probably not an option unless two columns are fused and then mirrored.

I invite you to play with this tiling and find new tessellations - then tag me (@gatheringfolds) on Instagram or Facebook to show me what you've made!

You can buy Purl's crease pattern as a reference for your own explorations - the important thing is that all of the pleats are in the same orientations between corresponding twists.