Unusual mosaics by R. Penrose

This fall, our newspaper has already twice (October 15 and November 12) told about the scientific achievements of the Honorary Doctor of Igor Sikorsky Kyiv Polytechnic Institute, 2020 Nobel Laureate in Physics Roger Penrose. First about those for whom the prize was awarded, then about the "impossible objects of Penrose", which gave some impetus to the emergence of a new direction in art - impossibilism.  Today we are talking about "Penrose mosaics", which could also be considered impossible if they did not exist.

It is known that the plane can be divided into identical squares, triangles, hexagons, rhombuses, which form a pattern characterized not only by rotational but also translational (mosaic elements are periodically repeated) symmetry. The division of a plane into figures of several types, say, octagons and squares, can have the same property. In the middle of the twentieth century, mathematicians began to look for answers to the following questions: is it possible to create a non-periodic mosaic, and what can be the minimum number of types of geometric shapes that form it?

In the early 1970s, Roger Penrose, a professor of mathematics at Oxford University, solved this problem by creating non-periodic mosaics, which were named after him - "non-periodic mosaics by Penrose.”

In 1978, in his article "Pentaplexity. A Class of Non-Periodic Tilings of the Plane"  he described how he searched and found these mosaics. First, Penrose pointed out that a right pentagon can be cut into six smaller pentagons and five triangles (Fig. 1). He noted that the six pentagons connected in this way are part of the known dodecahedron scan. Then he began to think about how to pave the plane with these pentagons. He started with the options shown in Fig. 2 and looked for shapes that can fill in the gaps between the pentagons. As a result, he created his first non-periodic mosaic (Fig. 3), which consists of six elements: a pentagonal star, a rhombus, a "paper ship" and three types of pentagons (marked in different colours). Next, he began to study the resulting mosaic and look for ways to combine any two adjacent figures into one to reduce the number of different elements. In this way he first obtained a non-periodic mosaic of five elements, then of four, and then, cutting the resulting figures into parts and combining the parts into new figures, Penrose built a mosaic of two quadrilaterals, which, according to J. Conway, are called "kite" and "dart" (fig. 4 і 5). It is interesting to note that the ratio of the length of the larger sides of these quadrilaterals to the length of the smaller sides is equal to the golden ratio (1.6180339…). Penrose then determined that several different non-periodic mosaics could be formed from these elements (two of which are shown in the figure). Analyzing these mosaics, Penrose proved that on an infinite plane the ratio of the number of "kites" to the number of "darts" is equal to the golden ratio. And since this number is irrational, it means that the mosaic of such elements is non-periodic. By the way, the mentioned article by Penrose is freely available on the Internet: https://web.ma.utexas.edu/users/radin/Pentaplexity.pdf .

Then Roger Penrose came to the conclusion that a non-periodic mosaic can be formed from two rhombuses: "thick", which has angles of 72 and 108 degrees, and "thin" with angles of 36 and 144 degrees (Fig. 6). These, as well as previous "non-periodic mosaics of Penrose", have an interesting feature: they are symmetrical relatively to the axis of the fifth order. Prior to their creation, mosaics with only axes of symmetry of the third, fourth, and sixth orders were known, because it is impossible to pave a plane with regular pentagons.

It is interesting to note that the four elements which form the "non-periodic mosaics of Penrose" can be easily obtained by drawing a regular pentagon and performing simple geometric constructions in it (Fig. 7).

Having created non-periodic mosaics, Penrose did not send an article about them to scientific journals. He understood that his mosaics could have a variety of applications and be of commercial value. So I decided to first patent them in the UK, US and Japan, and then publish articles. Thus, he has, in particular, US patent №4133152 "Set of tiles for covering a surface". It describes tiles having the shape of "kites", "darts", "thick" and "thin" diamonds, as well as tiles obtained from them by converting the straight sides of the original tiles into curves. The most interesting example of such tiles listed in the patent is now known as "Penrose chickens". The patent also describes games that use similar tiles.

Penrose's non-periodic mosaics have now become quite common. "Penrose tiles" pave floors, sidewalks, decorate walls, release their sets as puzzles. This is no accident. Mathematicians know that the hidden beauty of mathematical results is felt by people who have not studied mathematics. And in Penrose's mosaics both hidden and visible beauty are combined.