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Real tiling with self-similar triangles

Prof. Anthony Joseph from the Department of Mathematics at the Weizmann Institute of Science talks with Dr. Yossi Alran from the Davidson Institute for Science Education about a new type of tiling - complex and amazingly beautiful

Real tiling with self-similar triangles. Prof. Anthony Joseph's lab, Weizmann Institute
Real tiling with self-similar triangles. Prof. Anthony Joseph's lab, Weizmann Institute

The tiling problem has fascinated mathematicians and artists from ancient times to the present day. Seemingly, this is a relatively simple question: is it possible to sequence an infinite plane using agreed upon rules? In ancient times, even before the Christian era, the elaborate polygons that line the plane alone were discovered: triangles, squares and hexagons. One would think that there is nothing more to innovate in this field, but mathematics always surprises us, so that every now and then a new type of tiling is discovered, usually as a result of the changing of the laws.

In the second half of the last century, two new ways to perform tiling were discovered. One: tiling where the patterns do not repeat themselves exactly, known as "non-cyclic tiling". A well-known example in this field is the flooring proposed by Roger Penrose. This tiling contributed to the discovery of quasi-periodic crystals by Prof. Dan Shechtman of the Technion, who was awarded the Nobel Prize in Chemistry for this in 2011. The second way is called "titling of self-imagination". This way was proposed by William Thurston. Recently, Prof. Anthony Joseph, from the Department of Mathematics at the Weizmann Institute of Science, proposed a new tiling: "a real tiling with self-similar triangles". Prof. Joseph gave this flooring the name "Bar Sagi flooring" - after his granddaughter who passed away at the age of 15 after an illness.

"Bar Shagiya tiling" is a general name for tiling that is based on a set of triangles derived from a perfect polygon (equal sides and equal angles). To create the group of triangles, a blocked triangle must be drawn inside an elaborate polygon with a number of (n) sides, by drawing lines between three consecutive vertices in the polygon (but not necessarily adjacent). Draw all possible blocked triangles in this way. If r,s,t are the number of sides of the polygon between three consecutive vertices, then always r+s+t=n.

It is worth paying attention to the fact that two triangles in a group will coincide if and only if their r,s,t triples are equal to the point of exchange (that is, they can be exchanged for each other). There is of course no need to count congruent triangles. Also, when r,s,t are different, the triangle is necessarily a triangle with different sides, then it is convenient to distinguish between such a triangle and its mirror image. In the figure you can see an example of the triangles derived from an elaborate octagon: four triangles in the right octagon and one more triangle in the left octagon. Two of the triangles are different sides, so we need to add their mirror images, and therefore there are a total of seven triangles in the group. For Masher (n=10), eight different triangles are derived, four of which are different sides, that is, a total of 12 triangles.

The set of triangles derived from a perfect octagon. Prof. Anthony Joseph's lab, Weizmann Institute
The set of triangles derived from a perfect octagon. Prof. Anthony Joseph's lab, Weizmann Institute

 

Now arrange the side lengths of all the derived triangles, in a group of its own according to the chosen scale so that the length of the smallest side is one unit. We will take one of the triangles from the group derived from the polygon, no matter which, and increase it to the scale of one of the lengths from the group of side lengths, no matter which. A triangle similar to the original triangle is obtained.

Prof. Joseph proved that it is possible to line up the enlarged triangle (as well as shapes consisting of the connection of the triangles), with the triangles in the original group, without there being any spaces between the connecting triangles and without the vertex of a triangle meeting the side of another triangle (this requirement is known as "real" tiling - proper ). Prof. Joseph offers a method (algorithm) that makes it possible to create such sequences given an original elaborate polygon (pentagonal or higher).

In addition, Prof. Joseph proved (with the addition of several other natural conditions) that the tiling using the triangles derived from the polygon is the only possible tiling. He also proved that tiling is possible when the ratio in which the triangle is enlarged can be any side length of any of the triangles derived from the polygon.

If you think about it, you can understand that this process can be done again, for each of the triangles that line the enlarged triangle, that is, it is possible to sequence each of the triangles in "reduced" copies from the group of triangles (reduced according to the ratio of the lengths of one of the sides of one of the triangles). In fact, it is possible to carry out the process over and over again (in several repetitions, or "iterations") and thus obtain complex "error bar tilings", astonishing in their beauty. To allow an infinite number of repetitions, using any side length we want (for the magnification ratio) and in any possible order, it is not possible to simply do the same thing over and over again. Additional constraints are necessary - and this is a complex job. These discoveries and proofs are an extension of Thurston's work. In the figure you can see an example of such a tiling created from an elaborate octagon and tiling using the group of derived triangles and three repetitions.

An example of an error bar tiling formed from an equilateral octagon in three repetitions
An example of an error bar tiling formed from an equilateral octagon in three repetitions

Prof. Joseph went on to prove that it is possible to paint "false flooring" after one repetition, using only two colors - however, in each additional repetition, at least three colors are needed. It has not yet been proven if three colors are enough for each flooring, or if there are situations where four colors are needed. In the mathematical sense, what is meant by "coloring" is "coloring maps", that is, coloring so that adjacent areas are not the same color, and in this case, coloring so that two triangles painted in the same color do not have a common side (the four-color theorem that Wolfgang Haken and Kenneth Appel proved in 1976 and according to which, no more than four colors are needed to paint a map).

An example of an "error-proof tiling" which is tiled using the set of triangles derived from an elaborated wealth in three iterations. Each repetition is painted in a different color (in order - red, blue, green), but the triangles are not painted
An example of an "error-proof tiling" which is tiled using the set of triangles derived from an elaborate wealth in three repetitions. Each repetition is painted in a different color (in order - red, blue, green), but the triangles are not painted

An example of an "error-proof tiling" which is tiled using the set of triangles derived from an elaborate wealth in three repetitions. Each repetition is painted in a different color (in order - red, blue, green), but the triangles are not painted

Prof. Joseph was fascinated not only by the beauty and mathematics of "error bar tiling", but also by the possibility of turning it into an assembly game. He cut the base triangles of such a tiling from colored paper, glued them to magnets and thus challenged his granddaughter, who was then six years old, to reassemble the tiling on the refrigerator at home. Among other things, Prof. Joseph wanted to test the feasibility of using this tiling as an aid to illustrate "plane weight diagrams", which use "Lie Algebra", his main field of research. His experiment was unsuccessful, however, the mathematical assembly methods he used led to the unexpected discovery of a method for mounting Weierstrass sections (a simplified example of which is half a line of longitude from the North Pole to the South Pole, which cuts each line of latitude at only one point and thus defines it).

This work by Prof. Joseph to create non-commutative versions of geometric algebra, led to the understanding that under certain conditions there may be similarities between classical mechanics and quantum mechanics - even though under normal conditions there are significant and tangible differences between them.

Bar Sagi, 2017-2001
Bar Sagi, 2017-2001

The inspiration for "Ritzofi Bar Sagi" also comes from the beautiful poems written by Prof. Joseph's granddaughter, Bar, which express, in part, the mental and physical suffering she experienced when she was diagnosed with blood cancer, and which were found after her death. Her poems have been collected and published and are "necessary reading material for any doctor who intends to treat cancer patients", according to oncologist Carlos Keldes from the University of Cambridge, UK.

The well-known "Theorem of the Four Colors" states that we will never need more than 4 colors to paint "false flooring".

More of the topic in Hayadan:

3 תגובות

  1. I understand that all the theorems presented in the article have been proved for Mishor. Did Prof. Joseph prove theorems also for spheres, torus, etc.?

  2. Way too many new concepts in one text.
    3 new concepts is about the maximum that is acceptable and desirable to present in the text. If you want, you can write a series of articles, each one focusing on a limited number of topics.

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