A sensation in the world of mathematics. A new type of pentagons has been discovered that cover the plane without breaks and without overlaps.
This is only the 15th type of such pentagons and the first to be discovered in the last 30 years.
The plane is covered with triangles and quadrangles of any shape, but with pentagons everything is much more complicated and interesting. Regular pentagons cannot cover a plane, but some irregular pentagons can. The search for such figures has been one of the most interesting mathematical problems for a hundred years. The quest began in 1918, when mathematician Karl Reinhard discovered the first five suitable figures.
For a long time it was believed that Reinhard had calculated all possible formulas and that no more such pentagons existed, but in 1968 the mathematician R.B. Kershner found three more, and Richard James in 1975 brought their number to nine . That same year, 50-year-old American housewife and math enthusiast Marjorie Rice developed her own notation method and, within a few years, discovered four more pentagons. Finally, in 1985, Rolf Stein increased the number of figures to fourteen.
Pentagons remain the only figure about which uncertainty and mystery remain. In 1963, it was proven that there are only three types of hexagons covering the plane. There are no such triangles among convex heptagonal, octagonal, and so on. But with the Pentagons, not everything is completely clear yet.
Until today, only 14 types of such pentagons were known. They are shown in the illustration. The formulas for each of them are given at the link.
For 30 years no one could find anything new, and finally the long-awaited discovery! It was made by a group of scientists from the University of Washington: Casey Mann, Jennifer McLoud and David Von Derau. This is what the little handsome guy looks like.
“We discovered the shape by computer searching through a large but limited number of variations,” says Casey Mann. “Of course, we are very excited and a little surprised that we were able to discover a new type of pentagon.”
The discovery seems purely abstract, but it may actually have practical applications. For example, in the production of finishing tiles.
The search for new pentagons covering the plane will certainly continue.
Why do some human organs come in pairs (for example, lungs, kidneys), while others come in one copy?
Caustics are ubiquitous optical surfaces and curves created by the reflection and refraction of light. Caustics can be described as lines or surfaces along which light rays are concentrated.
Shabbat G.B.
We now know about the same amount about the structure of the Universe as ancient people knew about the surface of the Earth. More precisely, we know that the small part of the Universe accessible to our observations is structured in the same way as a small part of three-dimensional Euclidean space. In other words, we live on a three-dimensional manifold (3-manifold).
Victor Lavrus
A person distinguishes objects around him by their shape. Interest in the shape of an object can be dictated by vital necessity, or it can be caused by the beauty of the shape. The form, the construction of which is based on a combination of symmetry and the golden ratio, contributes to the best visual perception and the appearance of a feeling of beauty and harmony. The whole always consists of parts, parts of different sizes are in a certain relationship to each other and to the whole. The principle of the golden ratio is the highest manifestation of the structural and functional perfection of the whole and its parts in art, science, technology and nature.
The documentary "Dimensions" is two hours of mathematics that gradually takes you into the fourth dimension.
Sergey Stafeev
The most knowledge-intensive task of the ancient peoples was orientation in space and time. For this purpose, from time immemorial, mankind has erected numerous megalithic structures - cromlechs, dromos, dolmens and menhirs. Incredibly ingenious devices were invented that made it possible to count time with an accuracy of minutes or to visualize directions with an error of no more than half a degree. We will show how on all continents people created traps for the sun's rays, built temples, as if "strung" on astronomical directions, dug inclined tunnels for daytime stargazing, or erected gnomon obelisks. Incredibly, our distant ancestors, for example, managed to follow not only the solar or lunar shadows, but even the shadow of Venus.
To explore and describe volume, people use the method of projecting a volumetric body onto a plane. It looks something like this:
Knowing what projections look like, you can recognize, explore, and construct a true three-dimensional object.
This is a research method common in classical crystallography. Researchers first study one projection or plane, “paving it” with calculated elements as tightly as parquet, and at the same time studying symmetry and other features in the paved plane.
Then the entire three-dimensional volume is filled with these planes, just as books fill a cubic packing box. This method is called the tiling method.
Interest in tiling arose in connection with the construction of mosaics, ornaments and other patterns based on regular polyhedra: triangles, squares and hexahedrons.
It has never been possible to tile a plane from a regular pentagon or pentagon. It leaves gaps—unfilled cracks. And therefore, in classical crystallography, pentagonal symmetry is considered prohibited to this day.
And finally, such a method was found.
In 1976, the English mathematician Roger Penrose, actively working in various fields of mathematics, general relativity and quantum theory, gave a mathematical description of the “Penrose mosaic” named after him.
She made it possible, with the help of just two tiles of a very simple shape, to pave an endless plane with a never-repeating pattern.
To understand the mathematical essence of “Penrose diamonds”, let us turn to the pentagram.
In their simplest form, “Penrose tiles” are a set of two types of diamond shapes, some with an internal angle of 36°, others with an internal angle of 72°. Each consists of two triangles that fill the corresponding pentagram model.
The ratios of the elements of the pentagram fully reflect the Fibonacci golden proportion. Its basis is the irrational number = 1.6180339...
Penrose's idea of densely filling a plane with the help of “golden” rhombuses was transformed into three-dimensional space.
In this case, the role of “Penrose rhombuses” in new spatial structures can be played by icosahedrons and dodecahedrons.
It was a beautiful find, just one of the many inventions of the bright and tenacious mind of Roger Penrose, who is fascinated by spatial paradoxes. His impeccable understanding of the Fibonacci golden ratio is present here, which brought his research closer to art.
And it was this that served as the basis for further research and the discovery of quasicrystals in chemical laboratories and a new, more creative understanding of three-dimensional space, both for science and art.
One of the striking examples of creative exploration that caught my attention was the young Slovenian artist Matyushka Teija Krašek.
She received her BA in Painting from the College of Visual Arts (Ljubljana, Slovenia). Her theoretical and practical work focuses on symmetry as a bridging concept between art and science.
Her artwork has been presented at many international exhibitions and published in international magazines .
M.T. Krašek at his exhibition ‘Kaleidoscopic Fragrances’, Ljubljana, 2005
The artistic creativity of Mother Teia Krashek is associated with various types of symmetry, Penrose tiles and rhombuses, quasicrystals, the golden ratio as the main element of symmetry, Fibonacci numbers, etc.
With the help of reflection, imagination and intuition, it tries to find new relationships, new levels of structure, new and different types of order in these elements and structures.
In her work, she makes extensive use of computer graphics as a very useful tool for creating artwork, which is a link between science, mathematics and art.
If we choose one of the Fibonacci numbers (for example, 21 cm) for the side length of the Penrose diamond in this palpably unstable composition, we can observe how the lengths of some of the segments in the composition form a Fibonacci sequence.
A large number of the artist’s artistic compositions are dedicated to Shekhtman quasicrystals and Penrose lattices.
In these amazing compositions, manifestations of circular symmetry can be observed in the relationships between Penrose rhombuses:
Every two adjacent Penrose diamonds form a pentagonal star. You can see the Decagon formed by the edges of 10 adjacent Penrose rhombuses, creating a new regular polyhedron.
And in the last picture there is an endless interaction of Penrose rhombuses - pentagrams, pentagons, decreasing towards the central point of the composition. Golden ratio ratios are represented in many different ways on different scales.
The artistic compositions of Mother Teia Krashek attracted great attention from representatives of science and art.
Penrose mosaic is a great example of how a beautiful construction, located at the intersection of various disciplines, necessarily finds its application.
We will talk about tiling the plane. Tessellation is the covering of an entire plane with non-overlapping shapes. Probably, interest in paving first arose in connection with the construction of mosaics, ornaments and other patterns. There are many known ornaments composed of repeating motifs. One of the simplest tilings is shown in Figure 1.
The plane is covered with parallelograms, and all parallelograms are identical. Any parallelogram of this tiling can be obtained from the pink parallelogram by shifting the latter by a vector (vectors and are determined by the edges of the selected parallelogram, n and m are integers). It should be noted that the entire tiling as a whole is transformed into itself when shifted by a vector (or). This property can be taken as a definition: namely, a periodic tiling with periods is a tiling that transforms into itself when shifted by a vector and by a vector. Periodic tilings can be quite intricate, some of them are very beautiful.
Quasiperiodic tilings of the plane
There are interesting and non-periodic tessellations of the plane. In 1974 The English mathematician Roger Penrose discovered quasiperiodic tilings of the plane. The properties of these tilings naturally generalize the properties of periodic ones. An example of such tiling is shown in Figure 2.
The entire plane is covered with rhombuses. There are no gaps between the diamonds. Any rhombus tessellation can be obtained using only two tessellations using shifts and rotations. This is a narrow rhombus (36 0, 144 0) and a wide rhombus (72 0, 108 0), shown in Figure 3. The length of the sides of each of the rhombuses is 1. This tiling is not periodic - it obviously does not transform into itself under any shifts . However, it has some important property, which brings it closer to periodic tilings and forces it to be called quasiperiodic. The point is that any finite part of a quasiperiodic tiling occurs countless times throughout the entire tiling. This tiling has a symmetry axis of order 5, while such axes do not exist for periodic tilings.
Another quasiperiodic tiling of the plane, constructed by Penrose, is shown in Figure 4. The entire plane is covered with four polygons of a special type. This is a star, a rhombus, a regular pentagon.
A) Conversion of inflation and deflation
Each of the three examples of quasiperiodic tiling shown above is a covering of a plane using translations and rotations of a finite number of figures. This covering does not transform into itself under any shifts; any finite part of the covering occurs throughout the entire covering countless times, moreover, equally often throughout the entire plane. The tilings described above have some special property, which Penrose called inflation. Studying this property allows us to understand the structure of these coatings. Moreover, inflation can be used to construct Penrose patterns. Inflation can be most clearly illustrated using the example of Robinson triangles. Robinson triangles are two isosceles triangles P, Q with angles (36 0, 72 0, 72 0) and (108 0, 36 0, 36 0) respectively and side lengths, as in Figure 6. Here φ is the golden ratio:
These triangles can be cut into smaller ones so that each of the new (smaller) triangles is similar to one of the original ones. The cutting is shown in Figure 7: the straight line ac is the bisector of the angle dab, and the segments ae, ab and ac are equal. It is easy to see that triangle acb and ace are congruent and similar to triangle P, and triangle cde is similar to triangle Q. Triangle Q is cut like this. The length of the segment gh is equal to the length of the segment ih (and is equal to 1). Triangle igh is similar to triangle P, and triangle igf is similar to triangle Q. The linear dimensions of the new triangles are t times smaller than those of the original ones. This cutting is called deflation.
The reverse transformation - gluing - is called inflation.
The figure shows us that from two P - triangles and one Q - triangle we can glue a P - triangle, and from a P and Q triangle we can glue a Q triangle. The new (glued) triangles have linear dimensions t times larger than the original triangles.
So, we have introduced the concept of transformations of inflation and deflation. Clearly, the inflation transformation can be repeated; this will result in a pair of triangles whose dimensions are t 2 times larger than the original ones. By successively applying inflation transformations, you can obtain a pair of triangles of arbitrarily large size. In this way, you can pave the entire plane.
It can be shown that the tiling described above by Robinson triangles is not periodic
Proof
Let us outline the proof of this statement. Let's argue by contradiction. Suppose that the tiling of the plane with Robinson triangles is periodic with periods u and w. Let's cover the plane with a network of parallelograms with sides u, w. Let's denote by p the number of P - triangles whose lower left vertex (relative to our network) is located in a shaded parallelogram; Let's define the number q in a similar way. (Selected p+q triangles form the so-called fundamental region of a given periodic tiling.) Consider a circle with radius R with center O. Let us denote by PR (actually QR) the number of P-triangles (respectively, Q-triangles) lying inside this circle.
Let's prove that
1) Indeed, the number of triangles intersecting a circle of radius R is proportional to R, while the number of triangles inside a circle of radius R is proportional to R 2. Therefore, in the limit, the ratio of the number of P - triangles to the number of Q - triangles in a circle is equal to this ratio in the fundamental region.
Let's now take our tessellation and perform deflation transformations. Then in the original fundamental region there will be pґ = 2p + q smaller P - triangles and qґ = p + q smaller Q - triangles. Let us denote by pґR and qґR the number of smaller triangles in a circle of radius R. Now it is easy to obtain a contradiction. Indeed,
= = = = (L'Hopital's rule)
From where, solving the equation
p/q=(2p+q)/(p+q),
while p and q are integers! The contradiction shows that tiling with Robinson triangles is not periodic.
It turns out that this covering by Robinson triangles is not the only one. There are infinitely many different quasiperiodic coverings of the plane by Robinson triangles. Roughly speaking, the reason for this phenomenon lies in the fact that during deflation the bisector in Figure 7 can be drawn from vertex b, and not from vertex a. Using this arbitrariness, it is possible to achieve, for example, that the covering with triangles turns into a covering of triangles with rhombuses
B) Transformation of duality
The method for constructing quasiperiodic tilings given above looks like a guess. However, there is a regular way to construct quasiperiodic coverings. This is a duality transformation method, the idea of which belongs to the Dutch mathematician de Braun.
Let us explain this method using the example of constructing the replacement of a plane with rhombuses (see Fig. 3). First, let's build a grid G. To do this, take a regular pentagon and number its sides (j = 1,2,3,4,5; Fig. 10). Let's look at the side numbered j. Let's construct an infinite set of lines parallel to this side, so that the distance between the two nearest lines is equal to 1.
Let's carry out a similar construction for each of the sides of the pentagon; We will draw straight lines so that they intersect only in pairs. The result is a set of lines that is not periodic (Fig. 9). The lines in this set will be denoted by the letters l. Let's renumber the lines with two indices: l j (n). Here j indicates the direction of the line (which side of the pentagon it is parallel to). The integer n numbers different parallel lines, runs through all integer values (both positive and negative). This set of lines divides the plane into an infinite set of polygons. These polygons are called mesh faces. We will call the sides of the polygons the edges of the mesh, and the vertices of the polygons the vertices of the mesh. (Similarly for a quasiperiodic covering Q: rhombuses are faces of Q, sides of rhombuses are edges of Q, vertices of rhombuses are vertices of Q)
Thus, the grid G is constructed. Let us now perform the transformation of duality. Each face of the mesh G is comparable to a vertex of a quasiperiodic covering Q (the vertex of a rhombus). We denote the vertices by letters (these are vectors). First, we associate each face M of the mesh with five integers n j = (M), j - 1,2, ....5 according to the following rule. Internal points of M lie between some line l j (n) and a line parallel to it l j (n+1).
This integer n we will match the faces of M. Since the mesh has straight lines in five directions, then in this way we will match five integers n j (M) of each M of the mesh G. The vertex of the quasi-periodic covering Q, corresponding to a given face M of the mesh G, is constructed as follows:
(M) = n 1 (M) + + … +
Here is a vector of unit length directed from the center of a regular pentagon to the middle of side number j. Thus, we associated a covering vertex with each face of the mesh. This way we can construct all the vertices of Q.
Now let's connect some vertices with straight line segments. These will be the edges of the covering Q (the sides of the rhombuses). To do this, consider a pair of faces M1 and M2 that have a common edge. We will connect the vertices of the coating corresponding to these faces and with segments.
Then it turns out that the difference
Maybe equal to only one out of ten vectors.
Thus, each mesh edge is associated with a cover face Q. Each mesh vertex is associated with a cover face Q (rhombus). Indeed, each mesh vertex is adjacent to four faces M R (R = 1,2,3,4). Let us consider the four covering vertices (M R) corresponding to them. From the difference property (2) it follows that the edges of the covering passing through these vertices form the boundary of the rhombus. A quasiperiodic covering of the plane with rhombuses is constructed.
We have illustrated the duality transformation method. This is a general way to construct a method for quasiperiodic coverings. In this construction, the regular pentagon can be replaced by any regular polygon. The result will be a new quasi-periodic coating. The duality transformation method is also applicable for constructing quasiperiodic structures in space.
B) Quasiperiodic filling of three-dimensional space
There is a three-dimensional generalization of Penrose patterns. Three-dimensional space can be filled with parallelepipeds of a special type. Parallelepipeds do not have common internal points and there are no gaps between them. Each parallelepiped of this filling can be obtained from only two parallelepipeds using shifts and rotations. These are the so-called Amman-Mackay parallelepipeds. In order to define a parallelepiped, it is enough to specify three edges emerging from one vertex. For the first Amman-Mackay parallelepiped these vectors have the form:
= (0; 1; φ), = (-φ; 0; -1)
And for the second parallelepiped:
= (0; -1;f), = (f; 0;1), = (0;1; f)
The filling with these parallelepipeds does not transform into itself under any shifts, however, any finite part of it occurs throughout the entire filling countless times. The filling of space with these parallelepipeds is associated with the symmetries of the icosahedron. The icosahedron is a Platonic solid. Each of its faces is a regular triangle. The icosahedron has 12 vertices, 20 faces and 30 edges
Application
It turned out that the rapidly cooled aluminum-manganese melt (discovered in 1984) has precisely these symmetries. Thus, Penrose patterns helped to understand the structure of the newly discovered substance. And not only this substance, other real quasicrystals have also been found, their experimental and theoretical study is at the forefront of modern science.
An example of tiling on a hyperbolic plane
French mathematician Michael Rao from the University of Lyon has completed the solution to the problem of tiling a plane with convex polygons. A preprint of the work can be found on the scientist’s page.
A polygon is called convex if all its angles are less than 180 degrees or, which is the same, along with any pair of points, such a polygon also contains a segment connecting them. The tiling problem (also called the parquet problem) is formulated as follows: let the plane be divided into polygons so that any two polygons either do not have common points or have only common boundary points. If all the polygons of such a partition are the same (that is, one can be translated into another by a composition of translation, rotation or axial symmetry), then the polygon is said to tile the plane. The problem goes like this: describe all the convex polygons that tile the plane.
Using some combinatorial reasoning, one can prove that such a polygon can only have 3, 4, 5 or 6 sides. It is easy to check that the plane can be tiled with any tri- or quadrilateral. You can read more about this in our material.
To describe all hexagons, let's denote their angles as A, B, C, D, E, F, and their sides as a, b, c, d, e, f. In this case, we assume that side a is adjacent to angle A on the right and all sides and angles are named clockwise. In the 60s, it was proven that all hexagons that can be used to tile a plane belong to at least one of three classes (the classes intersect here; say, a regular hexagon belongs to all three):
- A + B + C = 360
- A + B + D = 360, a = d, c = e
- A = C = E = 120, a = b, c = d, e = f.
All 15 known pentagonal tessellations
The most difficult case is that of pentagonal parquet. In 1918, mathematician Karl Reinhardt described five classes of such parquets, the simplest of which was the class of pentagons with the condition that there is a side whose sum of adjacent angles is equal to 180 degrees. In 1968, Robert Kershner found three more such classes, and in 1975, Richard James found another. A magazine wrote about James' discovery Scientific American, The article was seen by the American housewife and amateur mathematician Marge Rice, who manually found 5 more families over 10 years.
The latest progress in the tiling problem occurred in August 2015. Then mathematicians from the University of Washington in Bothell used a computer program to grade 15 pentagonal parquets. In his new work, Michael Rao reduced the problem of classifying pentagonal parquet floors to a search of 371 options. He went through the options on the computer and showed that nothing but 15 already known tiling classes existed. Thus, he finally closed the tiling problem.
Andrey Konyaev