Periodic - aperiodic (13th of September 2010)

Toffee-cubes on a tiled plane. But there is, of course, much more to this. For a start, cubes can also be discerned in a supporting plane, though it is obviously a two-dimensional structure. This is a relatively well known illusion which I used in my book >> "Problem of the observer" to make a sort of introduction to my thoughts on Ernst Mach's understanding of possibilities and mission of science. Instead of explaining it further, I offer you >> WHOLE CHAPTER XIV OF THE BOOK "PROBLEM OF THE OBSERVER" with an illustration of periodic tiling of plane a-la-Escher. This should serve nicely for explaining this post.

And now a bit concerning the tiling of the plane with rhombi. An astute observer will not miss the fact that rhombi are two equilateral triangles joined along one side. Furthermore, the tiling can in a way be considered as periodic since it is a tiling by hexagons that show internal variations. Here is one clearer image of tiling (below) where this fact can be perhaps more easily discerned.

A basic variation in the hexagons in the way they are divided in three rhombi - there are two different ways to do this, and in the tilings shown here, these two ways appear randomly, each one with a probability of 50 %. This introduces aperiodicity in the tiling.

Besides different geometric divisions of hexagons in rhombi, there is also randomness in the coloring of rhombi where three different colors, each of them appearing with equal probability (1/3). This introduces additional aperiodicity in tiling.

This interesting mathematics is visually stimulating and it is quite easy to see a lot of densely packed cubes in the two-dimensional structure. But don't be easily fooled. It is a problem of the observer...

>> CHAPTER XIV FROM THE BOOK "PROBLEM OF THE OBSERVER".

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Last updated on 13th of September 2010.