Aperiodic tiling, and how we become interested

From the Wikipedia:

A given set of tiles, in the Euclidean plane or some other geometric setting, admits a tiling if non-overlapping copies of the tiles in the set can be fitted together to cover the entire space.

A given set of tiles might admit periodic tilings, tilings that remain invariant after being shifted by a translation. (For example, a lattice of square tiles is periodic.) It is not difficult to design a set of tiles that admits non-periodic tilings as well (For example, randomly arranged tilings using a 2×2 square and 2×1 rectangle will typically be non-periodic.)

An aperiodic set of tiles (as in the picture above), however, admits only non-periodic tilings, an altogether more subtle phenomenon.An ordering is non periodic if it lacks translational symmetry, which means that a shifted copy will never match exactly with its original.

The Penrose tiles are an aperiodic set of tiles, since they admit only non-periodic tilings of the plane.


Any of the infinitely many tilings by the Penrose tiles is non-periodic.


I find this kind of patterns both fascinating. When we look at one locally, there is always an impression of regularity, of some pattern that repeat itself and thus easy to grasp. But, when the eye tries to cover a larger portion of the pattern, there is always a disturbance, a kind of a surprise, for we find that in fact it only seems to repeat itself but never really does. It looks as if part of the pattern can let us grasp the whole of it, but soon we discover how eluding it really is.

I have this thought that aperiodic tiling is how things become interesting to us. When we experience something new, we try to 'tile' it with all our known concepts and memorized experiences. If we succeed, we soon enough categorize the new experience, archive it, and lose interest. If we fail, we most often feel disconnected to the experience and soon enough we tend to discard it altogether.

But if we succeed to 'tile' the new experience with our previous preconceptions, yet we get this peculiar and subtle arrangement of aperiodic patterns, then, something else happens. We become interested. We gain enough grasp of the subject matter, yet its edges are always eluding and demand more variations. The theme is clearly there, yet... not quite. We become interested.

For a long time, I was thinking about how we use physical space as a metaphor for our mental spaces, and how geometrical tiling is, in a manner of speaking, analogous to the way we organize our experiences. It is fascinating that certain sets of tiles would admit only aperiodic tiling. Here is a thought: certain sets of concepts would admit only interesting organization of mental spaces and experiences. How about creating such a set?

Life is interesting. :-)

Mon, Dec 31, 2007  Permanent link
Categories: Mental spaces, Geometry