Symmetrical mosaic. Algorithm for constructing Penrose mosaics – models and quasicrystals. Mosaics from different countries

Algorithm for constructing Penrose mosaics – models and quasicrystals

Student
Vladimir State University named after

A. G. and, Pedagogical Institute,
Faculty of Physics and Mathematics, Vladimir, Russia
Email:*****@***com

Quasicrystals are a relatively recently discovered type of solid, intermediate between crystals and amorphous solids. Their occurrence is associated with substances experimentally discovered in 1982 that give a diffraction pattern with functional Bragg peaks and symmetry incompatible with the translation lattice. For their discovery, Israeli physicist and chemist Dan Shechtman received the Nobel Prize in 2011.

Non-periodic point systems with long-range order are usually used as mathematical models of quasicrystals. Such mathematical quasicrystals, unlike physical ones, can be defined in any dimension.

A two-dimensional model of a quasicrystal is the Penrose mosaic, which was studied by mathematicians even before the discovery of quasicrystals. The Penrose mosaic is not a periodic partition, since it does not transform into itself by any parallel transfers - translations. However, there is a strict order in it, determined by the algorithm for constructing this partition.

There are many approaches to defining mathematical quasicrystals. The most well-known approach is based on projecting lattices from higher-dimensional spaces into lower-dimensional spaces, which is called “model sets”. When applied to Penrose tiling, this approach is called the Baaki method.

This method is most convenient for studying and analyzing the diffraction pattern of quasicrystals both from a theoretical point of view and from the point of view of computer algorithms. Based on this analysis, subsequent conclusions can be drawn about the properties of quasicrystals.

To analyze the properties of the Penrose mosaic, we wrote a computer program using the Baaki algorithm, according to which the window is determined https://pandia.ru/text/79/142/images/image002_56.gif" width="51 height=24" height="24 ">.gif" width="104" height="24">, where .

Sets https://pandia.ru/text/79/142/images/image007_19.gif" width="61" height="24">, , , , , where is the golden ratio. Then the projections of the points onto the model set will be as follows : and where https://pandia.ru/text/79/142/images/image016_12.gif" width="23" height="20">..gif" width="121" height="23">. The vertices are connected by an edge when the distance between them is 1. Thus, a Penrose mosaic is constructed using the above algorithm.

We discovered that Baaki's method is not entirely accurate and the resulting partition is not exactly a Penrose partition, since “extra” vertices and edges of the partition appear. It turned out that this construction is correct up to the vertices and boundaries of pentagons.

Using a computer experiment, it was possible to refine the Baaki method, resulting in a Penrose mosaic (Fig. 1):

Fig.1 Penrose mosaic obtained using a modification of the Baaki algorithm

The method described above for constructing a Penrose tiling is called weak parametrization of the Penrose tiling.

There is another construction method - strong parameterization of the partition vertices, where you can obtain the parameters of neighboring vertices using the parameter of a given vertex. The entire set of parameters is divided into polygons, in each of which the first local environment of the point is uniquely defined, as well as a star consisting of vectors connecting the point with neighboring points.

In 1973, the English mathematician Roger Penrose created a special mosaic of geometric shapes, which became known as the Penrose mosaic.
Penrose mosaic is a pattern assembled from polygonal tiles of two specific shapes (slightly different rhombuses). They can pave an endless plane without gaps.

Penrose mosaic according to its creator.
It is assembled from two types of rhombuses,
one with an angle of 72 degrees, the other with an angle of 36 degrees.
The picture turns out to be symmetrical, but not periodic.

The resulting image looks like it is some kind of “rhythmic” ornament - a picture with translational symmetry. This type of symmetry means that you can select a specific piece in a pattern that can be “copied” on a plane, and then combine these “duplicates” with each other by parallel transfer (in other words, without rotation and without enlargement).

However, if you look closely, you can see that the Penrose pattern does not have such repeating structures - it is aperiodic. But the point is not an optical illusion, but the fact that the mosaic is not chaotic: it has fifth-order rotational symmetry.

This means that the image can be rotated by a minimum angle equal to 360 / n degrees, where n is the order of symmetry, in this case n = 5. Therefore, the rotation angle, which does not change anything, must be a multiple of 360 / 5 = 72 degrees.

For about a decade, Penrose's invention was considered nothing more than a cute mathematical abstraction. However, in 1984, Dan Shechtman, a professor at the Israel Institute of Technology (Technion), while studying the structure of an aluminum-magnesium alloy, discovered that diffraction occurs on the atomic lattice of this substance.

Previous ideas that existed in solid state physics excluded this possibility: the structure of the diffraction pattern has fifth-order symmetry. Its parts cannot be combined by parallel transfer, which means that it is not a crystal at all. But diffraction is characteristic of a crystal lattice! Scientists agreed that this option would be called quasicrystals - something like a special state of matter. Well, the beauty of the discovery is that a mathematical model for it had long been ready - the Penrose mosaic.

And quite recently it became clear that this mathematical construction is much older than one could imagine. In 2007, Peter J. Lu, a physicist from Harvard University, along with another physicist, Paul J. Steinhardt, but from Princeton University, published an article in Science on mosaics Penrose. It would seem that there is little unexpected here: the discovery of quasicrystals attracted keen interest in this topic, which led to the appearance of a bunch of publications in the scientific press.

However, the highlight of the work is that it is not dedicated to modern science. And in general - not science. Peter Lu drew attention to the patterns covering mosques in Asia, built in the Middle Ages. These easily recognizable designs are made from mosaic tiles. They are called girihi (from the Arabic word for "knot") and are a geometric design characteristic of Islamic art and consisting of polygonal shapes.

An example of a tile layout shown in a 15th-century Arabic manuscript.
The researchers used colors to highlight repeating areas.
All geometric patterns are built on the basis of these five elements.
medieval Arab masters. Repeating elements
do not necessarily coincide with tile boundaries.

There are two styles in Islamic ornament: geometric - girikh, and floral - islimi.
Girikh(pers.) - a complex geometric pattern made up of lines stylized into rectangular and polygonal shapes. In most cases, it is used for the external decoration of mosques and books in large publications.
Islimi(pers.) – a type of ornament built on the combination of bindweed and spiral. Embodies in stylized or naturalistic form the idea of ​​an ever-evolving flowering foliage shoot and includes an endless variety of options. It is most widespread in clothing, books, interior decoration of mosques, and dishes.

Cover of the Koran of 1306-1315 and drawing of geometric fragments,
on which the pattern is based. This and the following examples do not match
Penrose lattices, but have fifth-order rotational symmetry

Before Peter Lu's discovery, it was believed that ancient architects created giriha patterns using a ruler and compass (if not by inspiration). However, a couple of years ago, while traveling in Uzbekistan, Lou became interested in the mosaic patterns that adorned the local medieval architecture and noticed something familiar about them. Returning to Harvard, the scientist began to examine similar motifs in mosaics on the walls of medieval buildings in Afghanistan, Iran, Iraq and Turkey.

This example is dated to a later period - 1622 (Indian mosque).
Looking at it and the drawing of its structure, one cannot help but admire the hard work
researchers. And, of course, the masters themselves.

Peter Lu discovered that the geometric patterns of girikhs were almost identical and was able to identify the basic elements used in all geometric designs. In addition, he found drawings of these images in ancient manuscripts, which ancient artists used as a kind of cheat sheet for decorating walls.
To create these patterns, they used not simple, randomly invented contours, but figures that were arranged in a certain order. The ancient patterns turned out to be exact constructions of Penrose mosaics!

These images highlight the same areas,
although these are photographs from a variety of mosques

In the Islamic tradition, there was a strict ban on the depiction of people and animals, so geometric patterns became very popular in the design of buildings. Medieval masters somehow managed to make it diverse. But no one knew what the secret of their “strategy” was. So, the secret turns out to be in the use of special mosaics that can, while remaining symmetrical, fill the plane without repeating itself.

Another “trick” of these images is that, by “copying” such schemes in various temples according to drawings, artists would inevitably have to allow distortions. But violations of this nature are minimal. This can only be explained by the fact that there was no point in large-scale drawings: the main thing was the principle by which to build the picture.

To assemble girikhs, five types of tiles were used (ten- and pentagonal rhombuses and “butterflies”), which were assembled in a mosaic adjacent to each other without free space between them. Mosaics created from them could have either rotational and translational symmetry at once, or only fifth-order rotational symmetry (that is, they were Penrose mosaics).

Fragment of the ornament of the Iranian mausoleum of 1304. On the right – reconstruction of girikhs

After examining hundreds of photographs of medieval Muslim sites, Lu and Steinhardt were able to date the trend to the 13th century. Gradually this method gained increasing popularity and by the 15th century it became widespread. The dating roughly coincides with the period of development of the technique of decorating palaces, mosques, and various important buildings with glazed colored ceramic tiles in the shape of various polygons. That is, ceramic tiles of special shapes were created specifically for girikhs.

Researchers considered the sanctuary of Imam Darb-i in the Iranian city of Isfahan, dating back to 1453, to be an example of an almost ideal quasicrystalline structure.

Portal of the shrine of Imam Darb-i in Isfahan (Iran).
Here two systems of girikhs are superimposed on each other.

Column from the courtyard of a mosque in Turkey (circa 1200)
and the walls of a madrasah in Iran (1219). These are early works
and they use only two structural elements found by Lu

Now it remains to find answers to a number of mysteries in the history of Girikh and the Penrose mosaics. How and why did ancient mathematicians discover quasicrystalline structures? Did medieval Arabs give mosaics any meaning other than artistic? Why was such an interesting mathematical concept forgotten for half a millennium? And the most interesting thing is what other modern discoveries are new, which in fact are well-forgotten old?

Penrose mosaic, Penrose tiles - non-periodic division of the plane, aperiodic regular structures, tiling of the plane with two types of rhombuses - with angles of 72° and 108° (“thick rhombuses”) and 36° and 144° (“thin rhombuses”), such (the proportions are subject to “Golden ratio”) that any two adjacent (that is, having a common side) rhombuses do not form a parallelogram together.Named after Roger Penrose, who was interested in the problem of “tessellation,” that is, filling a plane with figures of the same shape without gaps or overlaps.

All such tilings are non-periodic and locally isomorphic to each other (that is, any finite fragment of one Penrose tiling occurs in any other). “Self-similarity” - you can combine adjacent mosaic tiles in such a way that you again get a Penrose mosaic.

Several segments can be drawn on each of the two tiles so that when laying out the mosaic, the ends of these segments are aligned and several families of parallel straight lines (Amman stripes) are formed on the plane.

The distances between adjacent parallel lines take exactly two different values ​​(and for each family of parallel lines the sequence of these values ​​is self-similar).

Penrose tilings, which have holes, cover the entire plane except for a figure of finite area. It is not possible to enlarge the hole by removing a few (finite number) tiles and then completely pave the uncovered part.

The problem is solved by tiling with figures that create a periodically repeating pattern, but Penrose wanted to find just such a figure that, when tiled on a plane, would not create repeating patterns. It was believed that there were no tiles from which only non-periodic mosaics could be built. Penrose selected many tiles of various shapes, in the end there were only 2 of them, having the “golden ratio”, which underlies all harmonious relationships. These are diamond-shaped figures with angles of 108° and 72°. Later, the figures were simplified to a simple rhombus shape (36° and 144°), based on the principle of the “golden triangle”.

The resulting patterns have a quasicrystalline shape that has 5th order axial symmetry. The mosaic structure is related to the Fibonacci sequence.
( Wikipedia)

Penrose mosaic. The white dot marks the center of 5th order rotational symmetry: a rotation around it by 72° transforms the mosaic into itself.

Chains and mosaics (magazine Science and Life, 2005 No. 10)

Let us first consider the following idealized model. Let the particles in an equilibrium state be located along the transport axis z and form a linear chain with a variable period, changing according to the law of geometric progression:

аn = a1·Dn-1,

where a1 is the initial period between particles, n is the serial number of the period, n = 1, 2, …, D = (1 + √5)/2 = 1.6180339… is the number of the golden proportion.

The constructed chain of particles serves as an example of a one-dimensional quasicrystal with long-range symmetry order. The structure is absolutely ordered, there is a systematic pattern in the arrangement of particles on the axis - their coordinates are determined by one law. At the same time, there is no repeatability - the periods between particles are different and increase all the time. Therefore, the resulting one-dimensional structure does not have translational symmetry, and this is caused not by the chaotic arrangement of particles (as in amorphous structures), but by the irrational ratio of two adjacent periods (D is an irrational number).

A logical continuation of the considered one-dimensional structure of a quasicrystal is a two-dimensional structure, which can be described by the method of constructing non-periodic mosaics (patterns) consisting of two different elements, two elementary cells. This mosaic was developed in 1974 by a theoretical physicist from Oxford University. R. Penrose. He found a mosaic of two rhombuses with equal sides. The internal angles of a narrow rhombus are 36° and 144°, and of a wide rhombus - 72° and 108°.

The angles of these rhombuses are related to the golden ratio, which is expressed algebraically by the equation x2 - x - 1 = 0 or the equation y2 + y - 1 = 0. The roots of these quadratic equations can be written in trigonometric form:

x1 = 2cos36°, x2 = 2cos108°,
y1 = 2cos72°, y2 = cos144°.

This unconventional form of representing the roots of equations shows that these rhombuses can be called narrow and wide golden rhombuses.

In the Penrose mosaic, the plane is covered with golden rhombuses without gaps or overlaps, and it can be infinitely extended in length and width. But to build an infinite mosaic, certain rules must be followed, which differ significantly from the monotonous repetition of identical elementary cells that make up a crystal. If the rule for adjusting golden diamonds is violated, then after some time the growth of the mosaic will stop, as irremovable inconsistencies will appear.

In Penrose's infinite mosaic, golden rhombuses are arranged without strict periodicity. However, the ratio of the number of wide golden diamonds to the number of narrow golden diamonds is exactly equal to the golden number D = (1 + √5)/2= = 1.6180339…. Since the number D is irrational, in such a mosaic it is impossible to select an elementary cell with an integer number of rhombuses of each type, the translation of which could obtain the entire mosaic.

The Penrose mosaic also has its own special charm as an object of entertaining mathematics. Without going into all aspects of this issue, we note that even the first step - building a mosaic - is quite interesting, as it requires attention, patience and a certain intelligence. And you can show a lot of creativity and imagination if you make the mosaic multi-colored. Coloring, which immediately turns into a game, can be done in numerous original ways, variations of which are presented in the pictures (below). The white dot marks the center of the mosaic, a rotation around which by 72° turns it into itself.

Penrose mosaic is a great example of how a beautiful construction, located at the intersection of various disciplines, necessarily finds its application. If the nodal points are replaced by atoms, the Penrose mosaic becomes a good analogue of a two-dimensional quasicrystal, since it has many properties characteristic of this state of matter. And that's why.

Firstly, the construction of the mosaic is implemented according to a certain algorithm, as a result of which it turns out to be not a random, but an ordered structure. Any finite part of it occurs countless times throughout the mosaic.

Secondly, in the mosaic one can distinguish many regular decagons that have exactly the same orientations. They create a long-range orientational order, called quasiperiodic. This means that there is an interaction between distant mosaic structures that coordinates the location and relative orientation of the diamonds in a very specific, albeit ambiguous, way.

Thirdly, if you sequentially paint over all rhombuses with sides parallel to any selected direction, they will form a series of broken lines. Along these broken lines, you can draw straight parallel lines spaced from each other at approximately the same distance. Thanks to this property, we can talk about some translational symmetry in the Penrose mosaic.

Fourth, sequentially shaded diamonds form five families of similar parallel lines intersecting at angles that are multiples of 72°. The directions of these broken lines correspond to the directions of the sides of a regular pentagon. Therefore, the Penrose mosaic has, to some extent, rotational symmetry of the 5th order and in this sense is similar to a quasicrystal.

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In the February 2007 issue of Science magazine, an article by American scientists Peter Lu and Paul Steinhardt on medieval Islamic architecture appeared, which immediately became a scientific sensation. According to the authors of the article, the mosaic patterns decorating the walls of medieval mausoleums, mosques and palaces were made using mathematical laws discovered by European scientists only in the 70s of the twentieth century. From here, it clearly follows that medieval architects were several centuries ahead of their European colleagues.

This discovery, like many things in modern science, happened completely by accident. In 2005, Harvard University graduate student Peter Lu came to Uzbekistan as a tourist. Admiring the wall decor of the Abdullakhan mausoleum in Bukhara, he saw in it an analogue of the complex geometric structures that he had once studied at the university. The bizarre forms of patterns on numerous Samarkand ornaments only confirmed the correctness of his guess. Upon returning home, he told his thesis supervisor, Princeton University professor Paul Steinhardt, about his discovery.

A thorough study of the structure of wall paintings and ornamentation of medieval Muslim architectural monuments in Uzbekistan, Afghanistan, Iran, Iraq, Turkey and India confirmed the correctness of Peter Lu's guess and became the subject of the sensational article mentioned above.

In order to understand the meaning of the discovery of Peter Lu and Paul Steinhadt, one should become familiar with such concepts as the parquet problem, quasicrystalline structure, golden number, etc. Therefore, let's begin the presentation in order.

The parquet problem and Penrose structures

In mathematics, the problem of completely filling a plane with polygons without gaps or overlaps is called parquets. Even the ancient Greeks knew that this problem was easily solved by covering the plane with regular triangles, squares and hexagons.

At the same time, regular pentagons cannot serve as elementary elements of parquet, since they cannot be fitted tightly to each other on a plane without gaps. The same can be said about seven-, eight-, nine-, ten-, etc. squares. Gradually, ways were invented to fill the plane with regular polygons of different types and sizes. For example, this is how you can fill a plane by combining quadrilaterals and octagons of different sizes:

A much more complex development of this problem was the condition that the structure of the parquet, composed of several types of polygons and completely covering the plane, would not be quite “regular” or “almost” periodic. For a long time it was believed that this problem had no solution. However, in the 60s of the last century it was finally solved, but this required a set of thousands of polygons of various types. Step by step, the number of species was reduced, and finally, in the mid-70s, Oxford University professor Roger Penrose solved the problem using only two types of diamonds. Below is shown a variant of quasiperiodic (i.e. almost periodic) filling the plane with rhombuses with acute angles of 72 and 36°. They are also called “thick” and “thin” diamonds.

To obtain a non-periodic pattern when arranging diamonds, you should adhere to some non-trivial rules for their combination. It turned out that this seemingly simple structure has very interesting properties. For example, if we take the ratio of the number of thin rhombuses to the number of thick ones, then it always turns out to be equal to the so-called “golden ratio” 1.618... Since this number is “not exact”, and as mathematicians say, irrational, the structure turns out not to be periodic, but almost periodic. Moreover, this number determines the relationship between the segments inside the decagons that form a five-pointed star - a pentagram, which is considered a geometric figure with ideal proportions. Note that the highlighted decagons have the same orientation, which coordinates and defines the arrangement of the diamonds that make up the Penrose tiling. It is amazing that this purely geometric construction turned out to be the most suitable mathematical model for describing the quasicrystals discovered in 1984.

What are quasicrystals

We included this section in our article in order to tell another interesting story about how a mathematical construction, which was the fruit of the pure imagination of scientists, unexpectedly found important practical application.

All substances in nature can be divided into two types: amorphous, in which there is no regularity in the mutual arrangement of atoms, and crystalline, characterized by their strictly ordered arrangement. From the laws of crystallography it follows that for crystals only symmetry axes of the first, second, third, fourth and sixth orders are possible, i.e. By analogy with parquet, crystals with fifth-order symmetry cannot exist in nature. This circumstance was strictly proven on the basis of the mathematical theory of groups in multidimensional spaces. But nature, as always, turned out to be much more inventive, and in 1984 the work of Shekhtman’s group was published, which reported the discovery of an aluminum-manganese alloy with fifth-order rotational symmetry. Subsequently, many similar alloys with hitherto unknown properties were synthesized. These alloys were called quasicrystals, and are now considered to be intermediate between amorphous and crystalline forms of matter.

It was thanks to this discovery that Penrose's geometric construction, which turned out to be the most suitable tool for modeling the structure of quasicrystals, gained great popularity and was further developed. And that is why it is included in university courses. At present, a three-dimensional generalization of the Penrose mosaic has already been obtained, composed of thin and thick rhombohedrons - hexagonal figures, each face of which is a rhombus.

What geometry underlies medieval mosaics

After analyzing about 3,700 mosaic tiles, Lu and Steinhardt came to the conclusion that at the turn of the 13th century, the technology of decorating mausoleums, mosques and other buildings with periodic mosaics made up of a set of five polygons, namely, a decagon, a hexagon, and a bow tie, had spread throughout Muslim countries. (terminology of the authors of the article), pentagon and rhombus. This was essentially a solution to the parquet problem described above using a set of five “Muslim” polygons. Patterns made up of such polygons are called “girikh” (from Persian - knot).

Please note that the faces of all polygons have the same dimensions, which allows them to be joined on any side. In addition, each polygon tile has decorative lines, but they are drawn according to strict geometric rules: any two pattern lines converge in the middle of each side at angles of 72 or 108°, i.e. multiples of 36°. This ensures that the pattern remains consistent as you move from one tile to another.

To build such a mosaic, it was enough to have a compass and a ruler at your disposal. By the way, before the discovery of American scientists, it was believed that medieval masters, when creating the decoration of buildings, used only the simplest tools such as a ruler and compass. It has now become obvious that this is not entirely true.

The 15th century marks the most creative period of the flowering of science and culture in the countries ruled by the Timurids. It was at this time that a qualitative leap occurred in the art of ornament. This is confirmed by the fact that numerous studied monuments such as the mausoleum of Darb-e-Imam in Iran, the tomb of Haj Abdullah Ansari in Herat and others belong to the Timurid era.

The combination of the girih mosaic, which had become traditional by this time, and the geometric figures “arrow” and “kite” (again in the terminology of Lu and Steinhardt) made it possible to create

non-periodic patterns reminiscent of Penrose mosaics. It follows that they may have been using more sophisticated tools by this time, but it is clear that there was a conceptual leap in decorative techniques in the 15th century!

In subsequent interviews after the publication of the article, Lu and Steinhardt noted that they could not say to what extent the medieval architects themselves understood the details of their discovery, but that they see it as an analogue of Penrose’s structures. And they are absolutely sure that what they discovered cannot be just some random coincidence.

Lyrical digression

It is done. I managed to understand the intricacies of geometric patterns that give unique beauty to the creations of our ancestors, and I hope to some extent satisfy the curiosity of our compatriots. Of course, some kind of dissatisfaction remains, because I, too, have admired the beauty and elegance of Samarkand ornaments hundreds of times. Why did this thought never occur to me? To justify myself, I can only say that when the quasiperiodic Penrose structure was included in university courses, I was already working on my PhD thesis in my narrow specialty. And Peter Lu is only 28 years old, and he has already gone through the Penrose structures at the university. Of course, knowing and recognizing the manifestation of some pattern in a completely unexpected place are completely different things, but in order to do this, you must at least know that such a law exists.

But this is not what this digression is about. It took me two days, or rather two sleepless nights, to understand the essence of the article in Science magazine, but the reasons why I did not do this earlier have, it seems to me, a deep philosophical meaning. When I read about the article by Lu and Steinhardt on the Internet, I immediately called my colleague, an expert in the field of geometry. He immediately understood what was going on, but upset me by telling me that I had caught him before leaving for the airport. Having learned that he was returning from a foreign business trip only after three months, I asked him to at least recommend me some book in which I could read about Penrose structures. He told me the book and added that this is very complex mathematics and it is unlikely that it will be possible to quickly understand everything, much less explain it popularly to ordinary people. When I leafed through the book recommended to me, stuffed with such concepts as multidimensional invariant spaces, factor space of conjugate irrational space, my enthusiasm quickly faded.

After the report of the Jahon news agency, the interest of our scientific, and not only the scientific community in this issue began to grow like an avalanche. Among the learned men of the Academy of Sciences and the National University, of course, there were specialists who understand complex issues of Lie algebras, group theory, multidimensional symmetries, etc. But they were all unanimous in their opinion that it was impossible to explain these things popularly. The other day a trivial thought suddenly struck me: Wait. But how did medieval architects come up with this, because they did not have the most powerful apparatus of modern mathematics? This time I decided to try to understand this not through the complex mathematical apparatus of the Penrose quasiperiodic structure, which turned out to be a dark forest for me, but to follow the path of medieval architects. First, I downloaded the original article by Lu and Steinhardt from the Internet. Their method amazed me. To explain the essence of their discovery, they also took exactly this path, i.e. using the conceptual apparatus of medieval architects, and operating with such simple things as “girikh” mosaic, “arrow” tiles, “kite”, etc.

The philosophical point of all this is that in order to understand the laws of nature (and perhaps society) it is not necessary for everyone to follow the same path. Human thinking is also multidimensional. There is an eastern approach, and there is a western approach. And each of them has the right to exist, and in a particular case may unexpectedly turn out to be more effective than the opposite. This is what happened in this case: what Western science managed to discover on the basis of a huge generalization of thorny experience, Eastern science did on the basis of intuition and a sense of beauty. And the results are obvious: in the practical implementation of the laws of geometry into practice, Eastern thinkers were five centuries ahead of Western ones!

Shukhrat Egamberdiev.
Astronomical Institute of the Academy of Sciences of the Republic of Uzbekistan.

The full text of the article with color illustrations can be found in the next (the article was written in 2008. EU) issue of the magazine “Fan va turmush” - “Science and Life of Uzbekistan”.

Project participants

Nikiforov Kirill, 8th grade student

Rudneva Oksana, 8th grade student

Poturaeva Ksenia, 8th grade student

Research topic

Penrose mosaic

Problematic question

What is a Penrose mosaic?

Research hypothesis

There is a non-periodic tessellation of the plane

Objectives of the study

Get acquainted with the Penrose mosaic and find out why it is called the “golden” mosaic

Results

Penrose mosaic

Plane tiling is covering the entire plane with non-overlapping shapes. In mathematics, the problem of completely filling a plane with polygons without gaps or overlaps is called parquets or mosaics. Probably, interest in paving first arose in connection with the construction of mosaics, ornaments and other patterns. Even the ancient Greeks knew that this problem was easily solved by covering the plane with regular triangles, squares and hexagons.

This tiling of the plane is called periodic. Later we learned how to perform tiling using a combination of several regular polygons.

A more difficult task was the creation of not quite “correct” or “almost” periodic parquet. For a long time it was believed that this problem had no solution. However, in the 60s of the last century it was finally solved, but this required a set of thousands of polygons of various types. Step by step, the number of species was reduced, and finally, in the mid-1970s, Oxford University Professor Roger Penrose, an outstanding scientist of our time, actively working in various fields of mathematics and physics, solved the problem using only two types of rhombuses.

Roger Penrose

We investigated a method for constructing such a mosaic, which is now called the Penrose mosaic. To do this, draw diagonals in a regular pentagon (pentagon). We get a new pentagon and two types of isosceles triangles, which are called “golden”. The ratio of the hip to the base in such triangles is equal to the “golden” proportion. The angles in the triangles are 36°, 72° and 72° in one and 108°, 36° and 36° in the other. Let's connect two identical triangles and get “golden” rhombuses. The scientist used them in the construction of parquet, and the parquet itself was called “golden”.

Penrose mosaic

Penrose mosaic has the following properties:

1. the ratio of the number of thin rhombuses to the number of thick ones is always equal to the so-called “golden” number 1.618...