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Основанный на системах Делоне пространств постоянной кривизны дискретный подход к описанию структуры Вселенной позволяет охватить все ее иерархические уровни, начиная от элементарных частиц и кончая упаковкой сверхскоплений галактик. В самом общем виде системы Делоне представляют расположения центров молекул в идеальном газе, в наиболее вырожденном виде – идеальные кристаллические структуры. Все остальные состояния материи находятся в промежутке между двумя этими предельными состояниями. Большая часть этого промежутка занята дефектными кристаллическими структурами, т.е. на долю остальных состояний материи приходится весьма незначительный интервал. Полученную извне энергию система тратит на упорядочение. А идеальный порядок может быть только в кристаллических структурах. Поэтому открытые системы на любых иерархических уровнях стремятся к кристаллическому состоянию.


Рисунки к данной работе, а также новые ссылки [5-9,15], отражающие современное развитие темы, добавлены Р.В.Галиулиным по просьбе редакции.  – Прим. Ред.






While simulating the given state of the matter (gas, liquid, solid state, plasma) its structure is supposed to be either continuous (in the mechanics of the continuous medium) or discrete (in the statistical physics and crystallography). The main goal of this paper is to create a new method of discrete mathematical simulation for the main states of the matter the space-time distribution of particles which is defined by the points of so-called Delone sets.

The space-time discrete models appeared in modern physics in 1930 [1]. In 1937 a famous geometer B.N. Delone (Delaunay) [2] (Fig. 1) proposed the general model of discrete systems for the study of positive quadratic forms. The term "Delone sets" and its interpreta-tion as the arrangement of atomic centers for each real atomic structure was introduced in 1980 [3]. An ideal gas and ideal crystal are two opposite states which have exact description. The first one is simulated by the most general case of Delone systems, the second one by most singular Delone sets. All other states of the matter can be considered as intermediate states between them. It may be fractal systems of points, which finite sets of hierarchical levels also are Delone sets.

Fig. 1. Fragment of Delone set.


Delone sets in non-Euclidean spaces make it possible to consider new unusual substances - quasicrystals and fullerenes, lately discovered, as ideal crystals in hyperbolic and spherical spaces respectively. It is impossible in Euclidean space. They allow to describe more wider class of substances including these which had no mathematical description before and to simulate the extremal states of matter such as quark-gluon plasma [4] - the main component of neutron stars, the black holes, packing Galaxies and Supergathering of Galaxies in Universe, and substance of the Universe evolution initial stage (in accordance with a hypothesis). Last investigation shows that Fermi-Dirac and Bose-Einstein condensates beeng considering as nanocrystals and electron orbitals [5,6,7, 8,9] are regular Delone sets too.

Definition 1. Delaney set - a set of points in space with constant curvature which arc satisfying two following conditions:

discreteness - the distance between any two of its points is not less than some fixed length r;

overlapping - the distance between any point of the space and the nearest point of the set is not more than some fixed length R.

The first condition does not allow the points of the system lie too densely, the second - too rarely. Despite the generality of the requirements imposed by these two axioms, the general theory of Delone sets has wide implications. In its general case Delone sets may be considered as a geometrical model of ideal gas. Let us consider three very important properties of general Delone sets.

Lemma 1. It holds that r<2R.

Lemma 2. A sphere of radius 2R described around any point of a Delone set will contain at least a tetrahedron with vertices which are points of the set.

Lemma 3. Any two points of Delone set can be connected by a broken line with vertices at the point of this set and the length of segments not exceeding 2R.
Definition 2. We call a Delone set locally regular if all of its points have identical environment within a sphere of radius 2R.

Stogrin's theorem: The order of any rotation axis of the locally regular Delone system does not exceed six.
Let us take an arbitrary point of the Delone set and join it to all the remaining points of the set by segments. We thus obtain an infinite "spider" for the given point. The set is regular if and only if these "spiders" are congruent for all points of the set. In other words, for any two points of a regular set there exists a motion that takes the first into the second and matches the whole set with itself. The full set of such transformations form a group - a discrete group with finite independent regions – Fedorov’s groups (the space group in international terminology). This and only this groups form a long-range order.

But the bad side of this definition is, that it has no connection with physical reasons which lead to crystallization. That is why where was worked out another, physical evidently, definition of crystal [10].

This definition helped us to understand the situation with quasi-crystals [11] (Fig. 2) and fullerenes (Fig. 3) and explain these я лично познакомился с профессором Д.Д. Иваненко в феврале 1975 года.

Fig. 2. Quasicrystals.


Fig. 3. Fullerenes.


Of course, not every Delone set is regular. It needs congruence global "spiders" of Delone set. Show yet that it is sufficient for Delone set to be regular that certain small finite "spiders" of all its points should be congruent.

Take any point A of a Delone set and circumscribe around it concentric spheres with radii 2R, 4R, ... . Join the point A with the points of the Delone set which are located inside these concentric spheres, and designate by HI, H2 , ... the point groups of these obtained stars. Obviously, with the increase of the radius of such spheres the order of the point groups of the corresponding stars may decrease or keep constant. Due to the finiteness of the order of such groups there exist stars with identical groups H. Such stars are called prestable and stable, respectively.

Local theorem. If the stable stars of all points of a Delone set are equal then such a set is regular.


The interpretation of atomic formations giving X-ray patterns with icosahedral symmetry as a new type of atomic order (called the quasicrystalline order) equally excited both physicists and crystallographers. Physicists were excited to learn about the possible existence of atomic formations with long-range order and non-crystallographic symmetry.

A non-crystallographic long-range order would undoubtedly open up new horizons in physics where most laws are based (explicitly or implicitly) on periodicity. However, a number of scientists (in particular, L.Pauling [12]) were skeptical about long-range non-crystallographic order because the diffraction patterns mentioned above could also be obtained with any required accuracy produced from ideal crystals called quasicrystals approximants.

The relation between quasicrystals and there approximants is similar to the one between irrational and rational numbers, and it cannot be established experimentally whether it is a crystal or a quasicrystal. A widely used method of constructing Penrose-like three-dimensional quasicrystalline structures by projecting the layers of a certain thickness of a 6-dimensional lattice onto the three-dimensional irrational section of the lattice (i.e., sections whose coefficients are irrational numbers in the unit cell) [13,14] reflects the geometric essence of this problem.

Thus, the existence of the diffraction patterns of the icosahedral symmetry does not prove by itself the existence of a long-range order different from the crystallographic order [15]. Then, the question arises if an alternative to the long-range order exists or if it is possible, in principle, to obtain a system with long-range order different from crystallographic?

This question was brought to L.Dancers attention and he proved the theorem [16] from which follows that if the system is growing from its finite piece only by single way it is the crystal. Therefore if quasicrystals have long order then they are approximants.

The study of quasicrystals showed [17], that all experimental facts on quasicrystals are not in contradiction with the hypothesis that quasicrystal is an ideal crystal in a hyperbolic space. Such way we went to geometries with constant curvature: the Euclidean, the spherical one, the hyperbolic one. These geometries can’t be distin-guish locally. Because when the crystal appears there are three ways to reach his death ("Crystals are death" - wrote Fedorov [18] because they have no opportunity to change). But why we used only Euclidean geometry? It is very silly not to use all opportunities of geometry in its whole.

“Concepts, - wrote Lobachevsky [19] - for example geometrical, were created by our mind artificially ... no contradiction is seen in that some forces in the nature obey one geometries, while other forces, other, particular geometries".

Against this is usually mentioned that our space is Euclidean and because of enough week atomic interaction solid state physics isn't needed in the other geometries. But fullerenes show that this is not true because we can consider them as regular Delone sets on the sphere, i.e. as a crystal. Consider yet a Christmas tree ball which is broken into very small pieces. Each piece looks flat. But when we stick it together without gaps we receive a sphere again. Such experiment we can do with the Beltrami surface which is a part of hyperbolic plane. Why it is impossible to imbed molecules on this surface?

As connection with spherical space, in crystallography it is used from the old times. The crystal classes there are integral groups on the sphere. In the last issue of the International Tables simple forms begin to classify using Wyckoff positions. They are usually sorry for the fact that such beauty figures as Platonic and Archimedean solids with icosahedral symmetry do not belong to crystallography. Yet this is not so. What is not possible in Euclidean space is possible in non-Euclidean crystallography. A substantial difference of hyperbolic and Euclidean crystallography is that any isogons (polyhedron with group-equivalent vertices) can be represented by the points of the same regular system. In Euclidean space for example it is impossible for isogons with icosahedral symmetries. Therefore, the atoms of one regular system in a hyperbolic crystal can have coordination polyhedra of with every point group symmetry. This provides high stability for any atomic formation. It is also possible that the vertices of the Penrose tilling also form a multiregular system on the hyperbolic plane. A more general hypothesis is the following: each projection of n-dimensional (m<n) irrational intersection of the lattice is regular set in m-dimensional hyperbolic space.

From this standpoint, a model of quasicrystal based on a division of a three-dimensional Euclidean space into congruent the same kind (left or right) polyhedra, the so-called Schmidt-Conway-Dancer biprisms, is of great of interest. Among such divisions, there are some with a global irrational screw axis. This proves their aperiodicity . Nevertheless [20], in the hyperbolic space such a division it seems to us can be regular because the space groups of this space can have irrational axes.

Fig. 4. The convex polyhedra that divide the hyperbolic space.


However, it was noticed [21] that in the hyperbolic space there are convex polyhedra that divide the space (Fig. 4), but can not be used for constructing a regular division. In other words, if atoms had the shapes of such polyhedra, then no crystal structure could be formed. Such structures can be called ideal amorphous structures. Note that the biprism mentioned above was discovered long before quasicrystals . Such biprism (both right and left) being packed differrently, can also make regular divisions of the Euclidean space. This biprism is one of 180 kind of stereohedra N.174 of the Pi space group (the so-called Stogrin polyhedra [22]). It is interesting to find this sterehoedron in the other space groups.

Among the regular division of a hyperbolic space, there are some similar to the Kelvin divisions [23] that consists of tetrahedra and icosahedra (which can model quasicrystal structures). These divisions are dual to those of a hyperbolic space into rhombic triacontahedra already discussed in the literature as possible constituent parts of three-dimensional analogs of Penrose patterns [13]. At each node of the divided space, either four or twelve rhombic triacontahedra share their vertices (tetrahedrally and dodecahedrally, respectively). The dual division is made up of tetrahedra and icosahedra sharing the vertices at the nodes of the regular set of a truncated icosahedron. Note that a truncated icosahedron has the same properties as the cuboctahedron; namely, it can be bisected by a section containing a regular decagon. The halves rotated with respect to one another by the action of a tenfold axis and glued form a convex polyhedron analogous to a hexagonal cuboctahedron in Euclidean space (Fig. 5, b). This operation can be repeated an infinite number of times in the motion along the separated tenfold axis. Thus, icosahedral analogs of Kelvin packings in a hyperbolic space were obtained.

Such rotations can be performed not only along a separated axis (as for analogous Euclidean packings, Fig. 5), but also about any non-parallel tenfold axes. In other words, the twin formation for crystal in a hyperbolic space is much more probable, and, if researchers are not acquainted with models of this type, they have almost no chances to recognize them among real atomic formations. Note also that this is not the only example of packings in a hyperbolic space similar to Kelvin packings [23].

Fig. 5. Different presentations of 2 regular close packings in 3-dimensional Euclidean space.


We emphasize that usual crystals are not so close to ideal as it would seem in theory. The fact that any crystal consist of individual blocks also confirms the fact that the Euclidean space is not quite convenient for the crystal and that a sufficiently regular structure is formed only under extreme physical conditions, which seem to cause a space curvature. Clusters, from such point of view are the splinters of failure crystals curvatures of which not to size with the curvature of the space.

In its pure way hyperbolic symmetry take place in reduction theory: the points in cone of positive quadratic, forms which are in connection with integral unimodular forms form a hyperbolic, plane [24].

Such geometrical approach has opportunity to describe not only usual crystals but some unusual states of the matter for example as we have supposed the structures of neutron stars because In hyperbolic space in the case of densest packing of the balls each ball may have every number of the neighbors i.e. any density [25]. If the number of balls of the environment tended to infinity, then the density of a neutron star would have drastically increased, and, as a result, the neutron star could have turned into an object in spirit of black hole. This object can be considered to be an ideal crystal, having an infinity high density.

Crystal-like structures may be meet and in the galaxies and supergathering of galaxies packings in the Universe. We were starting from the position that ideal gas and ideal crystals are two limiting stable states of discrete matter. All other states of the matter are situated between these two. As follows from Stogrin's theorem, the gap between locally regular sets and an ideal gas is small enough (2R). If we take into account that the packing of galaxies has a symmetry [26] which satisfies this condition then the Galaxy structure is situated in a very small gap [27]. The ratio r/R in the case of super-gathering of galaxies packing is equal 27/26 [28]. The value of this ratio (it is >1) is conform stability corresponding packings. Fig. 6 show, that some galaxies have twofold axis.

Lemma 4. If each point of Delone set is centrosymmetric relativity to the another points then this set is the lattice [29].

Lemma 5. If the order of stabilizer of the points of the Delone set higher or equal to 12 then such Delone set is regular.

         Fig. 6. Twofold axis galaxy.

This facts and the twofold symmetry of galaxies let us to hope that the packing of the supergathering of galaxies look like as local regular Delone sets.
These results may be in connection with Logunow's relativistic theory of gravitation [30]. The main idea follows from the fact that Noether's conservations laws have place only in spaces with constant curvature. If we add that the matter is r-discrete (but not epsilon-discrete as following from [31], [32]), then the most parts of today physics is based on discrete groups in the spaces with constant curvature i.e. on the Space Groups.

Notice yet that the conditions for crystallization are so week that it isn't understand which way the World save itself from general crystallization? It seems to us that it not happened because pf fractal nature the second range of interactions.

Regular linear sets are skeletons for all stable structures. The interactions of the second range are described by nonlinear equations for which to receive an exact solution is usually very difficult. But these solutions have very impressive geometrical representations through fractals [33] (Fig. 9). It seems that the Universe in whole may be presented as fractal. But till now we do not know the groups connecting the hierarchical levels of this fractal. The methods of handling fractals can be compared with working as an artist. Fractals are at last connected with both, science and arts.

Fig. 9. Fractals.



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