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Finnish Artist Makes Progress in Geometry

When I met Sir Roger Penrose at the centenary conference of Alan Turing in Manchester in 2012 I told him that Penrose tiling is now featuring as a tiling of Keskuskatu, a central street in Helsinki, he was upset. He said “They should always ask me before they use it!” I was a little dumbfounded. “Can you show a picture?” he continued. I googled it with my smart phone and showed him:

He stared at it for 30 seconds and said “It seems alright.” Sir Roger Penrose is a physicist, mathematician and a philosopher of mind. His ideas are often controversial. He believes that Gödel’s incompleteness implies that the human mind cannot be simulated by a Turing machine which in turn would imply that the human mind radically depends on quantum mechanics which in turn… requires revision according to Penrose, who doesn’t believe in the Schrödinger’s cat. Despite being controversial, he has written numerous books which are excellent at popularizing mathematics, physics and philosophy of mind thereby attracting numerous students to these areas whether they agree with him or not.

[EDIT: I have now heard that Penrose was asked for the permission about Keskuskatu and he gave one. Apparently he either didn’t remember that or he didn’t realize it was the same thing.]

One of the things Penrose is very famous for is Penrose tiling. It is a non-periodic way of tiling an (infinite) plane with two pieces. One of the interesting properties of Penrose tiling is that it has a five-fold rotational symmetry around some points. This can intuitively be seen from this picture which is one version of Penrose tiling:

Penrose Tiling (public domain)

Penrose Tiling (public domain)

Penrose tiling has the following properties:

  • It is non-periodic — no translational symmetry (you can’t move it without rotating so that it looks exactly the same).
  • It is uniformly recurrent: any finite patch can be found infinitely often across the tiling, moreover at a distance D from any point of the plane where D depends on the patch.
  • It has a five-fold rotational symmetry as described above, i.e. there are points around which the whole thing can be rotated by 360/5=72 degrees and look exactly the same afterwards.
  • It is a substitution tiling: the same patterns occur at larger and larger scales. Thus, the tiling can be obtained through “inflation” (or “deflation”). You get an intuitive idea of what substitution means from these illustrations:
Substitution -- frames from here, CC4.0

Substitution — frames from here, CC4.0

Illustration from here, page 7.

Illustration from here, page 7.

In the above list of properties, replace “five” by any natural number “n” and you get a mathematical geometry problem: can you find a tiling of the plane which satisfies all these properties which has an n-fold rotational symmetry? Well, for n=4, for example, just cover the plane with squares and for n=6 with hexagons, that was easy! But for higher n the problem has apparently remained open up until a Finnish artist Markus Rissanen doodled a solution (after doodling for some years though) and eventually described his solution together with a mathematician Jarkko Kari from the University of Turku in this paper. They call Rissanen’s invension roses, because they “greatly resemble a flower with its petals” and the construction generalizes the Penrose tiling in the above sense. One of the differences to Penrose tiling is that Rissanen’s tiling can also be periodic (but doesn’t have to be). For example one can see that the plane can be tiled periodically with the above rhombus-like substitution shape.

It is more frequent that artists adopt geometrical ideas from mathematics. In this case the pattern has been reversed. An artistic intuition was behind a solution to a mathematical problem! Here are some illustrations from their paper which gives an idea of how they did it, but for more precise proofs, just look into the paper!roses1

roses2 roses3 roses4

Reference:

Kari, Jarkko, and Markus Rissanen. “Sub Rosa, a system of quasiperiodic rhombic substitution tilings with n-fold rotational symmetry.” Discrete & Computational Geometry 55.4 (2016): 972-996.

 

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