Penrose mosaic, Penrose tiles - non-periodic partition 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 (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 impossible to enlarge the hole by removing a few (finite number) of 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 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. Such a 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 own application. 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 this state of matter. And here'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.
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 decorated the local medieval architecture and noticed something familiar in 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?
And the ancientsislamic patterns
Completed the presentation
student of grade 7B, Central Education Center No. 1679
Zherder Marina.
Project managers
Sinyukova E.V. and Zherder V.M.
5klass.net
What is mosaic
Mosaic presentsa pattern
made of tiles
different forms. by them
can be paved
endless
plane without
spaces. A periodic mosaic is a mosaic,
the pattern of which is repeated through
equal intervals.
A non-periodic mosaic is a mosaic,
a pattern that can be repeated
at irregular intervals.
Mosaics in nature
There are also many examples in natureperiodic mosaic. Basically it's
crystals of solids - for example:
salt crystal
Diamond crystal
Graphite crystal
Graphene crystal
Mosaics in Escher's paintings
Mosaics are an important topic inart. Artist
M.C. Escher is famous for his
mosaics and not real
paintings.
What is a Penrose mosaic?
In 1973English
mathematician Roger
Penrose (Roger
Penrose) created
special mosaic
from geometric
figures, which
became known as the Penrose mosaic.
Polygonal mosaic slabs
The Penrose mosaic representsmosaic assembled from polygonal
tiles of two specific shapes.
Symmetry of mosaic
The resulting image looks likeas if it were some kind of “rhythmic”
ornament - a picture,
possessing
broadcast
symmetry.
Symmetry
Translational symmetry meansWhat can you choose from the pattern?
a certain piece that can be
"copy" on the plane, and then
combine these "duplicates" with each other
parallel transfer.
10. Structure of Mosaics
However, if you look closely, you cansee that there are no such in the Penrose pattern
repeating structures - it
non-periodic. But it's not a matter of
optical illusion, but the fact that the mosaic
not chaotic: she
has
rotational
fifth symmetry
order.
11. Minimum angle
This means thatimage is possible
turn on
minimum angle,
equal to 360/n degrees,
where n is the order
symmetry, in this
case n = 5.
Therefore, the angle
turn that's nothing
doesn't change, it should be
multiple of 360 / 5 = 72
degrees.
12. Unusual phenomenon
In 1984 DanShekhtman studying
studying the structure
aluminum-magnesium alloy,
discovered that on
atomic lattice
this substance
is happening
unusual for
crystals
physical phenomenon.
13. “Wrong” crystals
A sample of a substance subjected tospecial fast method
cooling, scattered the electron beam
so that on the photographic plate a
pronounced
diffraction
painting with symmetry
fifth order in
location
diffraction
maximums
(symmetry of the icosahedron).
14. Quasicrystals
Scientists have agreed onthat given
there will be an option
identify yourself
quasicrystals –
something like special
state of matter. AND
it's been a long time for him
was ready
mathematical model
- Penrose mosaic.
15.
Publication 2007In 2007, physicists Peter Lu and Paul
Steinhardt published in the magazine
Science article on mosaics
Penrose.
16. Interest in quasicrystals
It would seemunexpected here
a little: opening
quasicrystals
attracted live
interest in this
topic that led
to the appearance of a heap
publications in
scientific press.
17. Patterns in Asia
However, the highlight of the work is that itis not dedicated to modern science.
And in general - not science. Peter Lu
noticed the patterns
covering mosques
in Asia, built
back in the Middle Ages.
18.
Styles GirikhIn Islamic ornament there are two
style:
Girikh (pers.) – complex
geometric ornament,
composed of stylized
rectangular and polygonal
line shapes. In most cases
used for external
design of mosques and books in large
publication.
19. Islimi
Islimi (pers.) – a type of ornament,built on the connection of bindweed and
spirals. Embodies in stylized
or naturalistic form of the idea
continuously developing blooming
deciduous shoot. Greatest
it became widespread in clothing,
books, interior decoration of mosques,
dishes
20. Mosaics of Uzbekistan
While traveling inUzbekistan, Lou became interested in patterns
mosaics that decorated the local
medieval architecture, and noticed in
something familiar about them.
Cover of the Koran 1306-1315 and
drawing
geometric
fragments
on which it is based
pattern.
21. Mosaics from different countries
Back inHarvard, scientist became
consider
similar motives in
mosaics on the walls
medieval
buildings
Afghanistan, Iran,
Iraq and Turkey.
22. Islamic mosaics
This example is dated laterperiod - 1622 (Indian mosque).
23. Girikh schemes
Peter Lu discovered that geometricthe girikh circuits are almost identical, and
was able to identify the main elements
used in all
geometric patterns. Besides,
he found drawings of these images in
ancient manuscripts, which
ancient artists used
as a kind of cheat sheet on
wall decoration.
24. Construction order
To create these patterns we used nosimple, randomly invented contours,
and the figures that were located in
in a certain order. Ancient patterns
turned out to be accurate constructions of mosaics
Penrose!
25.
Islamic traditionsIn the Islamic tradition
there was a strict
image ban
people and animals,
therefore in the design
large buildings
gained popularity
geometric
ornament.
26. The secret of the ancient masters
Medieval mastersdid it
diverse. But what
was their secret
"strategies" - no one
knew. So here's the secret how
times it turns out to be
use
special mosaics,
who can, remaining
symmetrical,
fill the plane, not
repeating itself.
27. "focus"
Another "trick" of theseThe "trick" of the images is that,
"copying" such schemes into
various temples around
drawings, artists
would inevitably have to
allow distortions. But
violations of this
character is minimal.
This can only be explained by
that the masters are not
used drawings when
building a mosaic.
28. Tiles
For assembling weightsused tiles of five
species (ten and
pentagonal rhombuses and
"butterflies"), which are
mosaics were compiled,
adjacent to each other
without free
space between
them.
29. Symmetry of mosaics
Mosaics created from them,could have immediately
rotational and
broadcast
symmetry, that's all
rotational symmetry
fifth order (that is
were mosaics
Penrose).
30. Girihi
Fragment of the ornament of the Iranian mausoleum1304 years. On the right – reconstruction of girikhs
31. Date of appearance of mosaics
Having researched hundredsDate
appearance
photos
mosaics
medieval
Muslim
attractions,
Lou and Steinhardt were able to
date the appearance
similar trend XIII
century. Gradually this
way to get everything
great popularity and
XV century became widely
widespread.
32. Ceramic tiles
Dating approximatelycoincides with the period
technology development
decoration
palaces, mosques,
various important
glazed buildings
color
ceramic tiles
in the form of various
polygons. That
there is a ceramic one
special tiles
forms were created
specifically for girikhs.
Ceramic
tile
33. Conclusion
What Western science has discoveredbased on a huge generalization
thorny experience, eastern science
made based on intuition and feeling
beautiful. And the results are obvious: in
embodiment of the laws of geometry in
practice of Eastern thinkers
ahead of the Western ones by five centuries!
Slide 1
And ancient Islamic patterns
Penrose mosaic
The presentation was made by a student of grade 7B, Central Educational Center No. 1679, Marina Zherder. Project managers Sinyukova E.V. and Zherder V.M.
Slide 2
What is mosaic
Mosaic is a pattern assembled from tiles of different shapes. They can pave an endless plane without gaps.
Slide 3
A periodic mosaic is a mosaic whose pattern is repeated at regular intervals. A non-periodic mosaic is a mosaic whose pattern may be repeated at irregular intervals.
Slide 4
Mosaics in nature
There are also many examples of periodic mosaics in nature. These are mainly crystals of solids - for example: Salt crystal Diamond crystal Graphite crystal Graphene crystal
Slide 5
Mosaics in Escher's paintings
Mosaics are an important theme in art. The artist M.C. Escher is known for his mosaics and non-real paintings.
Slide 6
What is a Penrose mosaic?
In 1973, the English mathematician Roger Penrose created a special mosaic of geometric shapes, which became known as the Penrose mosaic.
Slide 7
Polygonal mosaic slabs
Penrose mosaic is a mosaic assembled from polygonal tiles of two specific shapes.
Slide 8
Symmetry of mosaic
The resulting image looks like it is some kind of “rhythmic” ornament - a picture with translational symmetry.
Slide 9
Symmetry
Translational 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.
Slide 10
Structure of Mosaics
However, if you look closely, you can see that the Penrose pattern does not have such repeating structures - it is non-periodic. But the point is not an optical illusion, but the fact that the mosaic is not chaotic: it has fifth-order rotational symmetry.
Slide 11
Minimum angle
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.
Slide 12
Unusual phenomenon
In 1984, Dan Shechtman, while studying the structure of an aluminum-magnesium alloy, discovered that a physical phenomenon unusual for crystals was occurring on the atomic lattice of this substance.
Slide 13
"Wrong" crystals
A sample of the substance, subjected to a special method of rapid cooling, scattered a beam of electrons so that a pronounced diffraction pattern with fifth-order symmetry in the location of diffraction maxima (icosahedral symmetry) was formed on the photographic plate.
Slide 14
Quasicrystals
Scientists agreed that this option would be called quasicrystals - something like a special state of matter. And a mathematical model for it has long been ready - the Penrose mosaic.
Slide 15
In 2007, physicists Peter Lu and Paul Steinhardt published an article in the journal Science on Penrose mosaics.
Publication 2007
Slide 16
Interest in quasicrystals
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.
Slide 17
Patterns in Asia
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.
Slide 18
There are two styles in Islamic ornament: Girikh (pers.) - a complex geometric ornament composed 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.
Styles Girikh
Slide 19
Islimi (pers.) – a type of ornament built on the combination of bindweed and spiral. Embodies in a stylized or naturalistic form the idea of a continuously developing flowering leafy shoot. It is most widespread in clothing, books, interior decoration of mosques, and dishes.
Slide 20
Mosaics of Uzbekistan
While traveling in Uzbekistan, Lou became interested in the mosaic patterns that adorned the local medieval architecture and noticed something familiar about them.
Cover of the Koran from 1306-1315 and drawing of the geometric fragments on which the pattern is based.
Slide 21
Mosaics from different countries
Returning to Harvard, the scientist began to examine similar motifs in mosaics on the walls of medieval buildings in Afghanistan, Iran, Iraq and Turkey.
Slide 23
Girikh schemes
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.
Slide 24
Construction order
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!
Slide 25
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.
Islamic traditions
Slide 26
The secret of the ancient masters
Medieval masters made it varied. 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.
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 obtained
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 explored 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...
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