The Penrose Mosaic, or how Central Asian architects anticipated the discovery of European scientists by five centuries. Miscellaneous. Penrose Mosaic and Ancient Islamic Patterns Symmetrical Mosaic

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

Several segments can be drawn on each of the two tiles so that when the mosaic is laid out, the ends of these segments will overlap 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 mosaics, which have holes, cover the entire plane, except for the finite area figure. It is impossible to enlarge the hole by removing a few (finite) tiles, after which it is impossible to pave the uncovered part completely.

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 tiling a plane, would not create repeating patterns. It was believed that there are no such tiles from which only non-periodic mosaics would be built. Penrose handpicked many tiles of various shapes, as a result, there were only 2 of them, having the "golden ratio", which is the basis of all harmonious relationships. These are rhomboid shapes with angles of 108 ° and 72 °. Later, the figures were simplified to the form of a simple rhombus (36 ° and 144 °), based on the principle of the "golden triangle".

The resulting patterns have a quasicrystalline form, which has axial symmetry 5th order. The mosaic structure is associated with the Fibonacci sequence.
( Wikipedia)

Penrose Mosaic. The white point marks the center of rotational symmetry of the 5th order: a 72 ° rotation around it transfers the mosaic into itself.

Chains and mosaics (journal Science and Life, 2005 # 10)

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

аn = a1 Dn-1,

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

The constructed chain of particles serves as an example of a one-dimensional quasicrystal with a long-range symmetry order. The structure is absolutely ordered, there is a systematic arrangement of particles on the axis - their coordinates are determined by one law. At the same time, there is no recurrence - the periods between particles are different and increase all the time. Therefore, the obtained 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 unit cells. This mosaic was developed in 1974 by a theoretical physicist from the University of Oxford R. Penrose. He found a mosaic of two rhombuses with equal sides. The inner angles of the narrow rhombus are 36 ° and 144 °, and the wide rhombus is 72 ° and 108 °.

The angles of these rhombuses are associated with the golden ratio, which is algebraically expressed by the equation x2 - x - 1 = 0 or by 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 representation of 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 and overlaps, and it can be infinitely spread in length and width. But to build an infinite mosaic, certain rules must be observed, which are significantly different from the monotonous repetition of the same unit cells that make up the crystal. If the rule of fitting the golden rhombuses is violated, then after a while the growth of the mosaic will stop, as unrecoverable inconsistencies will appear.

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

The Penrose mosaic 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, since it requires attention, patience and a certain ingenuity. And a lot of invention and imagination can be shown if you make the mosaic multi-colored. The coloring, which turns immediately into a game, can be performed with numerous original ways, the variants of which are presented in the figures (below). The white dot marks the center of the mosaic, a 72 ° rotation around which brings it into itself.

The Penrose Mosaic is a great example of how beautiful construction, located at the intersection of various disciplines, is sure to be used. If the nodal points are replaced by atoms, the Penrose mosaic will become a good analogue of a two-dimensional quasicrystal, since it has many properties characteristic of such a state of matter. And that's why.

First, 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 in the entire mosaic countless times.

Secondly, many regular decagons with exactly the same orientations can be distinguished in the mosaic. They create a long-range orientational order called quasiperiodic. This means that there is interaction between distant mosaic structures that reconciles the location and relative orientation of the diamonds in a well-defined, albeit ambiguous, way.

Thirdly, if you consistently paint over all the rhombuses with sides parallel to any chosen direction, then they form a series of broken lines. Along these broken lines, straight parallel lines can be drawn, spaced from each other at approximately the same distance. Due to this property, we can talk about some translational symmetry in the Penrose tiling.

Fourth, sequentially filled rhombuses 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.

A shame! The people of the Middle Ages surpassed modern scientists. We thought that advanced mathematics and crystallography were our achievements. It turns out that nothing of the kind - all this was already half a thousand years ago. In addition, modern science seems to have been surpassed not by the best mathematicians, but by simple artists. Well, maybe not very simple ... But still!

No, well, in fact - modern mathematicians are engaged in sheer nonsense! Then the paper is folded 12 times, then the Lorenz equations are crocheted, then the balls are twisted into donuts. In general, of the serious people, only Perelman and Okunkov remained - all hope is on them ...

But it is interesting that people made mathematical achievements in antiquity, sometimes not at all attaching special importance to them. It is also interesting that scientists are repeating the same "old" discoveries today, without suspecting at all that they are inventing something that has existed without their guesses for more than one century.

For example, the English mathematician Roger Penrose came up with such a thing in 1973 - a special mosaic of geometric shapes. It became, accordingly, the Penrose mosaic. What is so specific about it?

The Penrose Mosaic in the version of 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 from it turns out to be symmetrical, but not periodic (illustration from the site en.wikipedia.org).

The Penrose Mosaic is a pattern made up of polygonal tiles of two specific shapes (slightly different rhombuses). They can pave an infinite plane without gaps.

The resulting image looks like it is a kind of "rhythmic" ornament - a picture with translational symmetry. This type of symmetry means that in the pattern you can select a certain piece 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 there are no such repetitive structures in the Penrose pattern - it is aperiodic. But the point is not optical illusion, but the fact that the mosaic is not chaotic: it has a rotational symmetry of the fifth order.

Examples of quasi-steels are the alloy AlMnPd and Al 60 Li 30 Cu 10 (illustration by Paul J. Steinhardt).

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

For about a decade, Penrose's invention was considered little more than a cute mathematical abstraction. However, in 1984, Dan Shechtman, a professor at the Israeli Institute of Technology (Technion), studying the structure of an aluminum-magnesium alloy, discovered that diffraction occurs on the atomic lattice of this substance.

Previous concepts in solid-state physics excluded such a possibility: the structure of the diffraction pattern has a 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 just for the crystal lattice!

How to be here? The question is not easy, so scientists agreed that this option will be called quasicrystals - something like a special state of matter.

Shown here is one of the tile patterns shown in a 15th century Arabic manuscript. The researchers highlighted the repeating areas with colors. All the geometric patterns of medieval Arab masters studied by Lou and Steinhardt are based on these five elements. As you can see, the repeating elements do not necessarily line up with the borders of the tiles (illustration by Peter J. Lu).

Well, the whole beauty of the discovery, you guessed it, is that a mathematical model has long been ready for it. And, as you probably understood, this is a Penrose mosaic. But this one is not at all ten years old, but much more. This became known only in our days, at the dawn of the XXI century, and this model turned out to be much older than one could imagine.

In 2007, Peter J. Lu, a Harvard University physicist, teamed up with another physicist, Paul J. Steinhardt, but from Princeton University, to publish an article in Science on mosaics. Penrose (Lou should be known to regular readers of "Membranes" - we have already talked about his discoveries of diamond cutting of ancient axes and the most complex old machines). It would seem that there is not much unexpected here: the discovery of quasicrystals attracted a keen interest in this topic, which led to the appearance of a heap of publications in the scientific press.

However, the highlight of the work is that it is far from being devoted to modern science... And in general - not a science.

"Quasicrystalline" patterns have found their place not only in architecture. Here you can see the cover of the Koran from 1306-1315 and the drawing of the geometric fragments on which the pattern is based. This and the following examples do not correspond to Penrose lattices, but have rotational symmetry of the fifth order (illustration by Peter J. Lu).

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 pattern that is characteristic of Islamic art and consists of polygonal shapes.

For a long time it was believed that these patterns were created using a ruler and compasses. However, a couple of years ago, while traveling in Uzbekistan, Lu became interested in the mosaic patterns that adorned local medieval architecture and noticed something familiar in them.

Returning to Harvard, the scientist began to consider similar motifs in the mosaics on the walls of medieval buildings in Afghanistan, Iran, Iraq and Turkey.

He found these patterns to be nearly the same and was able to highlight the basic elements of the girihs 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.

But all this, it turns out, is not so important. To create these patterns, they used not simple, randomly invented contours, but figures that were arranged in a certain order. And this is not particularly surprising.

What is really interesting is that, having forgotten about such schemes, people met with them again later. Yes, yes, ancient patterns are nothing more than what, centuries later, will be called Penrose lattices and will be found in the structure of quasicrystals!

The same areas are highlighted in these images, although they are photographs from a wide variety of mosques (illustration by Peter J. Lu).

In the Islamic tradition, there was a strict prohibition on the image of people and animals, therefore, geometric patterns have become very popular in the design of buildings. Medieval craftsmen managed to somehow make it different. But what was the secret of their "strategy" - no one knew. So, the secret turns out to be in the use of special mosaics, which can, while remaining symmetrical, fill the plane without repeating.

Another "trick" of these images is that, while "copying" such schemes in various churches according to drawings, the artists would inevitably have to admit distortion. But violations of this nature are minimal. This is explained only by the fact that there was no sense in the large-scale drawings: the main thing is the principle by which to build a picture.

For the assembly of girichs, tiles of five types (ten- and pentagonal rhombuses and "butterflies") were used, which were made up in the mosaic adjoining each other without free space between them. Mosaics created from them could have both rotational and translational symmetry at once, and only rotational symmetry of the fifth order (that is, they were Penrose mosaics).

Fragment of the ornament of the Iranian mausoleum of 1304. On the right is the reconstruction of the girichs (illustration by Peter J. Lu).

After examining hundreds of photographs of medieval Muslim landmarks, Lou and Steinhardt were able to date a similar trend to the 13th century. Gradually, this method became more and more popular and by the 15th century it had become widespread.

The 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 quasi-crystalline structure.

This discovery impressed many people. American Association for the Advancement of Science (

Project participants

Nikiforov Kirill, grade 8 student

Rudneva Oksana, grade 8 student

Poturaeva Ksenia, grade 8 student

Research topic

Penrose Mosaic

Problematic question

What is Penrose Mosaic?

Research hypothesis

There is a non-periodic tiling of the plane

Research Objectives

Get to know the Penrose mosaic and find out why it is called the "golden" mosaic

Results

Penrose Mosaic

A plane tiling is the covering of an entire plane with non-overlapping shapes. In mathematics, the problem of solid filling a plane with polygons without gaps and overlaps is called parquets or mosaics. Probably, the first interest in paving 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.

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

A more difficult task was to create a 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 still solved, but this required a set of thousands of polygons different 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 thigh to the base in such triangles is equal to the "golden" ratio. 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 design of the parquet, and the parquet itself was called "golden".

Penrose Mosaic

Penrose mosaic has the 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 ...

And the ancients
islamic patterns
The presentation was made by
student of grade 7B, Central Education Center No. 1679
Gerder Marina.
Project leaders
Sinyukova E.V. and Zherder V.M.
5klass.net

What is mosaic

Mosaic presents
a pattern,
assembled from tiles
different shapes. By them
can be paved
endless
plane without
spaces.

Periodic mosaic is a mosaic,
the drawing of which is repeated through
equal intervals.
Non-periodic mosaic is a mosaic,
which pattern can be repeated
at irregular intervals.

Mosaics in nature

There are also many examples in nature.
periodic mosaic. Mainly
crystals of solids - for example:
Salt crystal
Diamond crystal
Graphite crystal
Graphene crystal

Mosaics in Escher's paintings

Mosaics are an important topic in
art. Artist
M.K. Escher is known for his
mosaics and not real
paintings.

What is Penrose Mosaic?

In 1973
English
mathematician Roger
Penrose (Roger
Penrose) created
special mosaic
from geometric
figures, which and
became known as the Penrose mosaic.

Polygonal Mosaic Slabs

The Penrose Mosaic is
mosaic assembled from polygonal
tiles of two specific shapes.

Mosaic symmetry

The resulting image looks like
as if it is some kind of "rhythmic"
ornament - a picture,
possessing
translational
symmetry.

Symmetry

Translational symmetry means
what in the pattern you can choose
a certain piece that you can
"copy" on the plane and then
combine these "duplicates" with each other
parallel transfer.

10. Structure of Mosaics

However, if you look closely, you can
see that there are no such
repeating structures - it
non-periodic. But the point is not at all about
optical illusion, but that the mosaic
not chaotic: she
possesses
rotational
symmetry of the fifth
order.

11. Minimum angle

It means that
image can
turn on
minimum angle,
equal to 360 / n degrees,
where n is the order
symmetry, in this
case n = 5.
Therefore, the angle
turning that nothing
does not change, should be
multiple of 360/5 = 72
degrees.

12. Unusual phenomenon

In 1984 Dan
Shechtman doing
study of the structure
aluminum-magnesium alloy,
found that on
atomic lattice
of this substance
going on
unusual for
crystals
physical phenomenon.

13. "Wrong" crystals

A sample of a substance subjected to
special method of rapid
cooling, scattered the electron beam
so that the photographic plate formed
pronounced
diffractive
picture with symmetry
fifth order in
location
diffractive
highs
(symmetry of the icosahedron).

14. Quasicrystals

Scientists have agreed on
that given
option would be
name
quasicrystals -
something special
state of matter. AND
for him for a long time
was ready
mathematical model
- Penrose mosaic.

15.

Publication 2007
In 2007, physicists Peter Lou and Paul
Steinhardt published in the magazine
Science article on mosaics
Penrose.

16. Interest in quasicrystals

Seemingly,
unexpected here
a little: opening
quasicrystals
attracted a lively
interest in this
topic that led
to the appearance of the heap
publications in
scientific press.

17. Patterns in Asia

However, the highlight of the work is that it
is not devoted to modern science.
And in general - not a science. Peter Lou
drew attention to the patterns,
covering mosques
in Asia, built
back in the Middle Ages.

18.

Styles. Girih
In Islamic ornament, there are two
style:
Girikh (pers.) - difficult
geometric ornament,
composed of stylized in
rectangular and polygonal
line shapes. In most cases
used for external
decoration of mosques and books in large
edition.

19. Islimi

Islimi (pers.) - a type of ornament,
built on the junction of the bindweed and
spirals. Embodies in stylized
or a naturalistic form of an idea
continuously developing blooming
deciduous shoot. The greatest
he got spread in clothes,
books, interior decoration of mosques,
dishes.

20. Mosaics of Uzbekistan

While traveling in
Uzbekistan, Lu became interested in patterns
mosaics that adorned the local
medieval architecture, and noticed in
them something familiar.
Cover of the Quran 13061315 and
drawing
geometric
fragments,
on which the
pattern.

21. Mosaics from different countries

Back in
Harvard, scientist became
consider
similar motives in
mosaics on the walls
medieval
buildings
Afghanistan, Iran,
Iraq and Turkey.

22. Islamic Mosaics

This sample is dated at a later date.
period - 1622 (Indian mosque).

23. Schemes of girihs

Peter Lou discovered that geometric
the girih schemes are practically the same, and
was able to highlight the main elements,
used in all
geometric ornaments. Moreover,
he found drawings of these images in
ancient manuscripts, which
ancient artists enjoyed
as a kind of cheat sheet for
wall decoration.

24. Build order

To create these patterns, no
simple, randomly invented contours,
and the figures that were located in
a certain order. Ancient patterns
turned out to be exact constructions of mosaics
Penrose!

25.

Islamic traditions
In Islamic tradition
there was a strict
ban on image
people and animals,
therefore in the design
buildings large
gained popularity
geometric
ornament.

26. The secret of the ancient masters

Medieval craftsmen
did it
varied. But what
was their secret
"strategies" - nobody
knew. So, the secret is how
once it turns out
using
special mosaics,
who can stay
symmetrical,
fill the plane, not
repeating.

27. "focus"

Another "trick" of these
The "focus" of images is that,
by "copying" such schemes into
various temples in
drawings, artists
would inevitably have to
allow distortion. But
violation of this
character are minimal.
This is explained only by the fact
that the masters are not
used drawings when
building a mosaic.

28. Tiles

For assembling girihs
used tiles from five
species (ten- and
pentagonal rhombuses and
"butterflies"), which in
mosaics were compiled,
adjacent to each other
without free
spaces between
them.

29. Symmetry of mosaics

Mosaics made from them,
could be possessed as soon as possible
rotational and
translational
symmetry, and only
rotational symmetry
fifth order (i.e.
were mosaics
Penrose).

30. Girihi

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

31. Date of appearance of mosaics

Examining hundreds
date
appearances
photos
mosaics
medieval
Muslim
attractions,
Lou and Steinhardt were able to
date the appearance
a similar trend XIII
century. Gradually this
the way acquired everything
great popularity and to
XV century became widespread
common.

32. Ceramic tiles

Dating approximately
coincides with the period
development of technology
decorating
palaces, mosques,
various important
buildings glazed
color
ceramic tiles
in the form of various
polygons. That
have ceramic
special tiles
forms created
specifically for the girikhs.
Ceramic
tile

33. Conclusion

What western science has been able to discover
based on huge generalization
thorny experience, oriental science
made on the basis of intuition and feeling
beautiful. And the results are obvious: in
the embodiment of the laws of geometry in
practice oriental thinkers
ahead of the West by five centuries!

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

The Penrose Mosaic in the version of its creator.
It is assembled from two types of rhombuses,
one at 72 degrees, the other at 36 degrees.
The picture is symmetrical, but not periodic.

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

However, if you look closely, you can see that there are no such repetitive structures in the Penrose pattern - it is aperiodic. But the point is not optical illusion, but the fact that the mosaic is not chaotic: it has a rotational symmetry of the fifth order.

This means that the image can be rotated by a minimum angle of 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 little more than a cute mathematical abstraction. However, in 1984, Dan Shechtman, a professor at the Israeli Institute of Technology (Technion), studying the structure of an aluminum-magnesium alloy, discovered that diffraction occurs on the atomic lattice of this substance.

Previous concepts in solid-state physics excluded such a possibility: the structure of the diffraction pattern has a 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 just for the crystal lattice! Scientists have agreed that this option will be called quasicrystals - something like a special state of matter. Well, the whole beauty of the discovery is that a mathematical model was already ready for it - the Penrose mosaic.

And quite recently it became clear that this mathematical construction is much more years old than one could imagine. In 2007, Peter J. Lu, a Harvard University physicist, teamed up with another physicist, Paul J. Steinhardt, but from Princeton University, to publish an article in Science on mosaics. Penrose. It would seem that there is not much unexpected here: the discovery of quasicrystals attracted a keen interest in this topic, which led to the appearance of a heap of publications in the scientific press.

However, the highlight of the work is that it is far from being devoted to modern science. And in general - not a science. Peter Lu drew attention to the patterns covering the 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 pattern that is characteristic of Islamic art and consists of polygonal shapes.

Tile pattern shown in a 15th century Arabic manuscript.
The researchers highlighted the repeating areas with colors.
All geometric patterns are built on the basis of these five elements.
medieval Arab masters. Repeating elements
do not necessarily match the borders of the tiles.

In Islamic ornament, two styles are distinguished: geometric - girih, and floral - islimi.
Girih(pers.) - a complex geometric ornament made up of lines stylized into rectangular and polygonal figures. In most cases it is used for external design mosques and books in a large edition.
Islimi(pers.) - a type of ornament built on the connection of a bindweed and a spiral. Embodies in stylized or naturalistic form the idea of ​​an ever-growing, flowering leafy shoot and includes an infinite variety of options. It is most widely used in clothing, books, interior decoration of mosques, and dishes.

Cover of the Koran 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 rotational symmetry of the fifth order

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

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

Peter Lou discovered that the geometric patterns of the girichs were practically the same and was able to highlight 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. Ancient patterns turned out to be accurate constructions of Penrose mosaics!

In these pictures, the same areas are highlighted,
although these are photographs from a variety of mosques

In the Islamic tradition, there was a strict prohibition on the image of people and animals, therefore, geometric patterns have become very popular in the design of buildings. Medieval craftsmen managed to somehow make it different. But what was the secret of their "strategy" - no one knew. So, the secret turns out to be in the use of special mosaics, which can, while remaining symmetrical, fill the plane without repeating.

Another "trick" of these images is that, while "copying" such schemes in various churches according to drawings, the artists would inevitably have to admit distortion. But violations of this nature are minimal. This is explained only by the fact that there was no sense in the large-scale drawings: the main thing is the principle by which to build a picture.

To assemble girichs, tiles of five types (ten- and pentagonal rhombuses and "butterflies") were used, which were made up in the mosaic adjoining each other with no free space between them. Mosaics created from them could have both rotational and translational symmetry at once, and only rotational symmetry of the fifth order (that is, they were Penrose mosaics).

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

After examining hundreds of photographs of medieval Muslim landmarks, Lou and Steinhardt were able to date a similar trend to the 13th century. Gradually, this method became more and more popular and by the 15th century it had become widespread. The dating roughly coincides with the period of development of the technique of decorating palaces, mosques, various important buildings with glazed colored ceramic tiles in the form of various polygons. That is, ceramic tiles of special shapes were created specifically for the girichs.

The 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 quasi-crystalline structure.

Portal of the sanctuary of Imam Darb-i in Isfahan (Iran).
Here, two systems of girihs are superimposed on each other at once.

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

Now it remains to find answers to a number of mysteries in the history of the girih and Penrose mosaics. How and why did the ancient mathematicians discover quasicrystalline structures? Did the 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 - what other modern discoveries are new, which in fact is a well-forgotten old?