by Frodo7

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This is a pattern I have designed using simple geometric shapes: the square, and two rhombi. The term "quasi-periodic" implies this tiling pattern is somewhat between periodic tiling, where the pattern repeats itself regularly, and aperiodic tiling, where the pattern never repeats itself. (for further details see comments)

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## 7 Comments

The large size was not the point of the experiment. I did not want to break any record. It was a consequence of several limiting factors.

1. Lack of certain bricks in the palette. At first, I designed the pattern with smooth diagonal lines, not pixels. However, there are no bricks to build the rhombi or the tilted square, should they stand alone. As a result, I opted for a pixel version.

2. The pixel version is 100 percent correct geometrically: all angles and distances are correct; there is no deformation, warp, or misalignment in the pattern. The basic shapes are not too big, the sides are just a few pixels long, yet they appear as thin lines, just as I wanted. The limitation of the pixel version comes from the way we take sharp screenshots. Although there are known methods to reduce the overall size significantly, e.g by using composite bricks, or filters, at the end they all would result in fuzzy screenshots.

All in all, I would not say this experiment was a failure, or a waste of time. I enabled download and cloning for those pixel pushers who would like to play with.

Tilings or tessellations is a branch of geometry that describes the various ways to cover the plane with regular polygons without gaps or overlaps. There are simple examples of regular tilings using squares, triangles or hexagons (honeycomb), alone.

Things get more complicated, and more exiting, when we use several different shapes to tile the plane. In this particular case I used the square, one fat, and one thin rhombus. (One could argue, the square is a rhombus too, the fattest possible one.) Each of these shapes are capable of nice uniform tiling of their own. Using them together to get a nice tiling pattern is like cooking a good meal. Add a little of this and a little of that, stir, and see what happens.

Periodic Tilings - when the pattern repeats itself. Using the same basic shapes, one can make hundreds, if not thousands of different patterns, each based on a cluster of these shapes arranged carefully according to some sort of symmetry.

Aperiodic Tilings - there are tiling patterns that never repeat themselves. The famous example is the Penrose tiling.

Quasi·periodic Tiling. (quasi -from Latin; as if were, seemingly, apparently but not really) I designed a pattern, that looks as if it were aperiodic, never repeats itself, but it could be turned into a periodic pattern, at will. I don't think this is any news for mathematicians, but it might be useful for graphic designers.

While I was cooking this pattern soup, I've made several versions, by slightly changing the ratio of the ingredient elements. Adding a bit more squares resulted in a thicker soup with larger blocks or clusters of squares. In the final version I chose only to have square aggregates containing no more than two squares.

@p2pnut: You are quite right, the pattern was built by adding new bits randomly, not by copy-and-paste and reflect/rotate. In a way, it was grown to look organic. I built the original version in Adobe Illustrator, and subsequently created the FS pixel version.

You have mentioned the crystalline structure. For a long time research of aperiodic/quasiperiodic tilings were considered as pointless exercise of a small group of mathematicians. They also studied the 3D equivalents of such tilings, space filling polyhedra with aperiodic properties. The discovery of quasicrystals - crystals with quasiperiodic structure - has made a huge difference to the field. In fact, not only to geometry, but to many other disciplines, too, ranging from nanotechnology to supramolecular chemistry, and information technology using optoelectric circuits. In 2011 Daniel Shechtman won the Nobel Prize for Chemistry for his discovery.

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