About a toilet paper.
It all started with a very old question (from the times of Kepler) about the shape of tiles best suitable for laying the floor. The rectangular, square or honeycomb tiles can be placed on the floor in such a way that they do not overlap one another and there are no unsightly “gaps” between them. Tiles in the shape of regular pentagons do not work at all, because there is no way they can be properly arranged.
Most of the known designs made with these or other types of tiles are quite monotonous – every now and then the pattern is recurring. If we walked along such a pavement with our noses fixed on the floor, we would have the impression that we are coming back to the same place from time to time. Mathematicians describe such arrangements with the word “periodic”.
And is it possible to lay tiles in such a way that the pattern is never, nowhere repeated?
At the beginning, it seemed that, in order to get such an infinite variety of patterns, one must have a huge number of various shapes of tiles at one’s disposal.
In 1966, Robert Berger showed that such art is possible, provided that exactly 20 426 (yes! more than twenty thousand) pieces of various sophisticated forms of tiles are used. After several years of work, he was able to get a similar result with “only” 104 different shapes.
In 1973, Sir Roger Penrose found a solution requiring only two tiles. Two tiles and an endless variety of arrangements. Puzzles, mosaics and stained glasses of non-repeating patterns.
Sometime later, this mathematical result in an unexpected way became available to “every Tom, Dick and Harry.” The Kimberly-Clark company launched a toilet paper with the pressed Penrose’s pattern. Because it is so irregular, the “holes” of the embossing did not fell one on the other, thanks to which the roll seemed thicker and fluffier. Penrose, who had patented his design beforehand, was not delighted with this form of popularization of science. He took the company to court, won the case and… received compensation. How much did he earn on the toilet paper? He cannot talk about it, because one of the conditions for signing the settlement was a promise that he would not comment on this matter.