Quasi-crystals

Any scientific discovery is new achievement committed during scientific cognition of the nature and society. However, so-called revolutionary discoveries refuted some steady scientific theories and outlooks has a special significance for the science progress. Examples of such discoveries are the famous Kepler's Laws given the beginning of the new astronomy, the non-Euclidean geometry offered by the Russian geometer Lobatchevski in mathematics, Einstein's relativity theory in physics.

The discovery of the quasi-crystals made in 1984 by the Israeli physicist Dan Shechtman belongs to the category of such discoveries in the modern physics, in particular, in the crystallography.

To understand and to evaluate a significance of this discovery, it is necessary to remember the fundamental laws of the classic crystallography.

As is known, the crystals are called all solids, in which the composing them fragments (atoms, ions, molecules) are arranged strictly regularly like clusters of space lattices.

Within the long centuries the geometry of crystals seems by a mysterious and insoluble riddle. In 1619 the great German mathematician and astronomer Johannes Kepler paid his attention on the sextuple symmetry of snowflakes. He attempted to explain of its nature by the fact that the crystals are constructed from the smallest identical marbles, which are connected closely one to another (around of the central marble it is possible to place densely only six identical marbles). Subsequently many great minds (Guk, Lomonosov, Gajui, Brave etc.) made many efforts to uncover the secret of crystals. The discovery of the main law of the crystallographic symmetry was the result of these researches. According to this law it is possible for the crystals only the symmetry axis's of the first, second, third, fourth and sixth orders. The main crystallography law rejects a possibility of the symmetry axis of the fifth order in the crystallographic lattices.

So there was found the main crystallography law giving the sharp difference between the crystal symmetry and the plants and animals symmetry, allowing the symmetry of the fifth order.

The above-considered facts were the canons of the traditional crystallography before Shechtman's discovery.

The alloy of the aluminum and the manganese discovered by Shechtman is formed at the super-fast cooling of the melt with the speed 106 K per second. Thus there is formed the alloy ordered in the pattern, which is characteristic for the symmetry of the regular icosahedron having alongside with the dodecahedron the symmetry axes of the 5th order.

For the theoretical explanation of Shechtman's discovery the researchers paid their attention so-called "Penrose's tiles". The English mathematician Penrose was engaged in the "parquet's problem" consisting of the dense filling of the plane with the help of polygons. In 1972 he found the method to cover flatness only with two simple polygons arranged non-periodically. In their simplest form "Penrose's tiles" represent a nonrandom set of the diamond-shaped figures of two types, one of them is with the interior angle of 36°, the other one with the angle of 72°.

Penrose's tiles

To understand the mathematical nature of "Penrose's rhombuses" let's return again to the "pentagram".

As is known, in the pentagram there is a number of characteristic isosceles triangles. The triangle of the kind of ADC is the first of them. The acute angle at the vertex of A is equal to 36° and the ratio of the side AC = AD to the base DC is equal to the golden proportion, that is, the given triangle is the "golden" one. If now to take such two triangles and to connect them together, that their bases coincided, we will get "Penrose's rhombus" in figure (a).

Let's consider now one more type of the isosceles triangle available in the pentagram, for example, EBK. In such triangle the acute angles at the vertex of E and B are equal to 36° and the obtuse angle at the vertex of K is equal to 108°. Note that the ratio of the base EB of the triangle EBK to its side EK is equal to the golden proportion, that is this triangle also is the "golden" one. If now to connect such two triangles together so that their bases coincided we will get other "Penrose rhombus" in Figure (b).

"Penrose's tile" in Figure (c) can be formed by using the "golden" rhombuses in Figures (a) and (b). Figure (c) demonstrates the beginning of Penrose's tile construction. Let's take the 5 "golden" rhombuses of the kind of (b) and form from them the pentagonal star. After that we add to the pentagonal star the 5 "golden" rhombuses of the kind of (a). In outcome we get a decahedron. Continuing this process, that is, adding to the decahedron the new "golden" rhombuses we can cover the plane with usage only of two "golden" rhombuses of the kinds of (a) and (b). At that there is some non-periodic structure called "Penrose's tile". Is was proved, that the ratio of the number of the "thick" rhombuses (a) to the number of the "thin" rhombuses (b) in such structure strives for in the limit to the golden proportion!

The parquet (mosaic) can be good clone of the crystal. The elementary space cells fill the three-dimensional space of the crystal like the two-dimensional plane is filled by Penrose's tiles. Penrose's idea about dense filling of the plane with the help of the "golden" rhombuses of the kinds of (a) and (b) was transformed on the three-dimensional space. However, in the space structure the regular icosahedrons play a role of the cells filling densely the space. These spatial structures represent themselves the quasi-crystals.

As Gratia states in the article "Quasi-crystals" (1988) "the concept of the quasi-crystal presents a fundamental interest because it extends and completes the definition of the crystal. The theory based on this concept replaces the traditional idea about the "structural unit repeated in the space by the strictly periodic mode" by the key concept of the distant order. This concept resulted in widening the crystallography and we only begin to study newly uncovered wealth's. Its significance in the mineral world can be put in one row with attachment of the irrational numbers concept to the rational ones in mathematics".

This discovery considerably stimulated researches in this interesting area. In the recent years it was discovered many kinds of the quasi-crystal alloys. It appeared, that except for the quasi-crystals having the "icosaedral" symmetry (of the fifth order) there are also the alloys with the "decagonal" symmetry (of the 10th order) and the "decagonal" symmetry (of the 12th order).

What is of practical significance of the quasi-crystals discovery? As Gratia notes "the mechanical strength of the quasi-crystal alloys increases sharply; the absence of periodicity results in slowing down of the dislocations distribution in comparison to the traditional metals... This property has the great applied significance: the application of the "icosahedral" phase will allows to get the light and very strong alloys by means of the intrusion of the small-sized fragments of quasi-crystals in the aluminum matrix".

What is the significance of the quasi-crystals discovery since the point of our Museum view? First of all, this discovery is the instant of the great celebration of the "icosahedro-dodecahedral doctrine", which penetrates through all history of natural sciences and is the source of steep and useful scientific ideas. Secondly, the quasi-crystals shattered the conventional presentation about the insuperable watershed between the mineral world where the "pentagonal" symmetry was prohibited, and the alive world, where the "pentagonal" symmetry is one of most widespread.

And it is one more historical remark in conclusion. The quasi-crystals discovery made in 1984 is a worthy gift to the hundredth anniversary of the issuance of the book "Lectures on the Icosahedron .." (1884) by the famous German geometer Felix Klein who made exactly 100 years ago before the quasi-crystals discovery the brilliant prediction as to the role played by the icosahedron in future science.