СИСТЕМЫ ДЕЛОНЕ В ПРОСТРАНСТВАХ ПОСТОЯННОЙ КРИВИЗНЫ КАК ДИСКРЕТНЫЕ
МАТЕМАТИЧЕСКИЕ МОДЕЛИ ДЛЯ ОПИСАНИЯ СТРУКТУР ВСЕХ СОСТОЯНИЙ МАТЕРИИ (ОТ
ЭЛЕМЕНТАРНЫХ ЧАСТИЦ ДО КРУПНО-МАСШТАБНОЙ СТРУКТУРЫ ВСЕЛЕННОЙ)
ИВАНЕНКО, РАВИЛ ВАГИЗОВИЧ ГАЛИУЛИН
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.
DELONE SETS IN THE SPACES OF CONSTANT CURVATURE AS DISCRETE MATHEMATICAL
MODELS FOR DESCRIPTION THE STRUCTURES OF ALL STATES OF MATTER (FROM THE
ELEMENTARY PARTICLES TO THE LARGE SCALE OF THE UNIVERSE)
IVANENKO, RAVIL VAGIZOVICH GALIULIN
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 . In 1937 a famous geometer B.N. Delone
(Delaunay)  (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 . 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
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  - 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
Definition 1. Delaney set - a set of
points in space with constant curvature which arc satisfying two
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 .
This definition helped us to understand
the situation with quasi-crystals  (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.
From the local theorem follows that THE
LONG ORDER OF THE CRYSTAL FOLLOWS FROM ITS SHORT ORDER.
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
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 ) 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 . 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  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
The study of quasicrystals showed ,
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  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  - 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
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 , 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  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 ). 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  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 .
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 .
Fig. 5. Different
presentations of 2 regular close packings in 3-dimensional Euclidean
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 .
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 .
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  which satisfies this condition then the Galaxy structure
is situated in a very small gap . The ratio r/R in the case of
super-gathering of galaxies packing is equal 27/26 . 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
Lemma 5. If the order of stabilizer of the
points of the Delone set higher or equal to 12 then such Delone set is
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 . 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 , ), 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  (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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