A. L. Talis

Structural realization of the polytope approach for the geometrical description of the transition of a quasicrystal into a crystalline phase

A geometrical model is proposed for the recently observed transformation under the action of highly localized stresses during the surface scratch test of the icosahedral phase into a body-centred cubic (BCC) phase, the disordered version of the B2 phase with the CsCl structure, which occupies a large portion of the Al–Cu–Fe phase diagram around the Al50(Cu, Fe)50 concentration. The model is founded on a polytope concept and the concept of the eight-dimensional root lattice E8. In accordance with these concepts many crystalline or quasicrystalline structures can be derived from the polytope (the four-dimensional polyhedron) by fulfilling operations of lowering its local curvature with subsequent mapping of decurved polytope fragments onto Euclidian three-dimensional space. The properties of the E8 lattices give foundation to the possibility of mapping a quasicrystalline structure on a crystalline structure. The structural transformation is effected through intermediate atomic configurations coinciding with both structures, which are determined by a four-dimensional icosahedron (the {3, 3, 5} polytope). For the transformation of the icosahedron of the icosahedral phase into a rhombic dodecahedron of the cubic B2 phase, the cubic A15 structure plays the role of an intermediate configuration since it can be represented as a three-dimensional packing of linearly interlaced chains of Frank–Kasper polyhedra with coordination numbers Z = 12 (icosahedron) and Z = 14. The transition between the rhombic dodecahedron of a B2 structure and the Frank–Kasper polyhedron with Z = 14 requires insertion of disclination quadruplets into some facets of the rhombic dodecahedron. The proposed geometrical model can be applied also to the polymorphic BCC–FCC transformation since the Miller indices of the Frank–Kasper polyhedron with Z = 14 coincide with the observed indices of habit planes of iron martensites.

FULL EQUIVALENCE BETWEEN SOCOLAR’S TILINGS AND THE (A, B, C, K)-TILINGS LEADING TO A RATHER NATURAL DECORATION

In 1986 Socolar and Steinhardt introduced a family of quasiperiodic tilings of the euclidean 3-space E3 by four rhombic zonohedra, which admits a local matching rule. In 1989 the first of the present authors introduced another family with similar properties and four skew tetrahedra A, B, C, K as prototiles. Now we show, that either family can be obtained from the other by a unique construction; hence they are mutually derivable in the sense of Baake et al. Some additional observations are induced by this equivalence.

Finite noncrystallographic groups, 11-vertex equi-edged triangulated clusters and polymorphic transformations in metals

The one-to-one correspondence has been revealed between a set of cosets of the Mathieu group M11, a set of blocks of the Steiner system S(4, 5, 11) and 11-vertex equi-edged triangulated clusters. The revealed correspondence provides the structure interpretation of the S(4, 5, 11) system: mapping of the biplane 2-(11, 5, 2) onto the Steiner system S(4, 5, 11) determines uniquely the 11-vertex tetrahedral cluster, and the automorphisms of the S(4, 5, 11) system determine uniquely transformations of the said 11-vertex tetrahedral cluster. The said transformations correspond to local reconstructions during polymorphic transformations in metals. The proposed symmetry description of polymorphic transformation in metals is consistent with experimental data.

A special class of simple 24-vertex polyhedra and tetrahedrally coordinated structures of gas hydrates

It is established that the eight-dimensional lattice E(8) and the Mathieu group M(12) determine a unique sequence of algebraic geometry constructions which define a special class of simple 24-vertex, 14-face polyhedra with four-, five- and six-edge faces. As an example, the graphs of the ten stereohedra that generate most known tetrahedrally coordinated water cages of gas hydrates have been derived a priori. A structural model is proposed for the phase transition between gas hydrate I and ice.

Gosset helicoids: I. 8D crystallographic lattice E 8 and crystallographic, quasi-crystallographic, and fractional helicoidal axes determined by this lattice

It is shown that mapping of substructures of a semiregular Gosset polytope, whose 240 vertices form the first coordination sphere of a 8D lattice E8, determines the orders of p/d axes of helicoids that are set only by invariants of (sub)algebras. Axes of such (Gosset) helicoids are derived. These axes perform rotation by an angle (360°/p(d; can be crystallographic, quasi-crystallographic (p = 2, 3, 4, 5, 6, …; d = 1), or fractional (1 < d < p/2); and may belong to regular polytopes (conventional, starlike, etc.). Formation of ordered structures is considered as a formation of Gosset helicoids (rods) with their subsequent assembly. Helicoids with axes 15/4 and 15/7 are considered as examples. They correspond to crystallographic approximants—helicoids with axes 41 and 21 composed of deformed icosahedra (dodecahedra)—in a β-Mn crystal (clathrate IX).

Generalized Crystallography of Diamond-Like Structures: II. Diamond Packing in the Space of a Three-Dimensional Sphere, Subconfiguration of Finite Projective Planes, and Generating Clusters of Diamond-Like Structures

An enantiomorphous 240-vertex “diamond structure” is considered in the space of a three-dimensional sphere S3, whose highly symmetric clusters determined by the subconfigurations of finite projective planes PG(2, q), q = 2, 3, 4 are the specific clusters of diamond-like structures. The classification of the generating clusters forming diamond-like structures is introduced. It is shown that the symmetry of the configuration, in which the configuration setting the generating clusters is embedded, determines the symmetry of diamond-like structures. The sequence of diamond-like structures (from a diamond to a BC8 structure) is also considered. On an example of the construction of PG(2, 3), it is shown with the aid of the summation and multiplication tables of the Galois field GF(3) that the generalized crystallography of diamond-like structures provides more possibilities than classical crystallography because of the transition from groups to algebraic constructions in which at least two operations are defined.

Symmetry foundations of a polymer model for close-packed metallic liquids and glasses

The atomic packing density of metallic melts and glasses is too high for their structures to be considered as chaotic. To remove this contradiction, we propose to describe the structures of metallic liquids and the glasses that form from them using (i) a base set of three spirals made of regular tetrahedra with specific noncrystallographic symmetry and (ii) combinatorial permutations of the vertices of a set of the coordination polyhedra that describe the polymorphic transformations in metals. The symmetry base of the proposed model of the structures of liquids and glasses is represented by projective linear groups PSL(2, p), where the order of the Galois field is p = 3, 7, and 11. These groups uniquely determine a tetrahedron, the 7-vertex joining of four tetrahedra along their faces (tetrablock), the 11-vertex joining of two tetrablocks into a spiral, and the throwing over of the diagonals in a rhombus from two triangular faces of neighboring tetrahedra. The throwing over of the diagonals in a rhombus is considered as a unit act of any structural transformation and ensures the melt–crystal, melt–glass, and glass-crystal transitions and the structural relaxation of metallic glasses. In terms of the proposed scheme, the high density of melts and glasses is caused by tetrahedral packing (up to 78%), and the absence of a diffraction pattern of melts and glasses is explained by the absence of translation along the spiral axis. The suggested polymer model also explains the collective effects (string vibrations) that were detected upon measuring the shear modulus relaxation of a metallic glass.

Structural regularities of helicoidally-like biopolymers in the framework of algebraic topology: II. α-Helix and DNA structures

The developed apparatus of the “structural application” of algebraic geometry and topology makes it possible to determine topologically stable helicoidally-like packings of polyhedra (clusters). A packing found is limited by a minimal surface with zero instability index; this surface is set by the Weierstrass representation and corresponds to the bifurcation point. The symmetries of the packings under consideration are determined by four-dimensional polyhedra (polytopes) from a closed sequence, which begins with diamondlike polytope {240}. One example of these packings is a packing of tetrahedra, which arises as a result of the multiplication of a peculiar starting aggregation of tetrahedra by a fractional 40/11 axis with an angle of helical rotation of 99°. The arrangement of atoms in particular positions of this starting aggregation allows one to obtain a model of the α-helix. This apparatus makes it possible to determine a priori the symmetry parameters of DNA double helices.

Transformations of Gosset Rods as the Structural Basis of Gas-Hydrate-Ice Phase Transitions

define the special class of helicoids (rods) to which belong also the rods of gas hydrates formed by polyhedron cavities. The “invariant” basis of such rods causes an additional “topological” stability of structures assembled from them, which is necessary for taking into account when calculating the thermodynamic characteristics. The last statement is especially significant for the gas hydrates, which, as a rule, are in an unstable equilibrium state [2]. The local approach is necessary also for determining the structural mechanism of the gas-hydrate‐ice phase transition (the bound-water structure) caused by the exit of hydrocarbon guest molecules from polyhedron cavities of the gas hydrate.

Gosset helicoids: II. Second coordination sphere of eight-dimensional lattice E8 and ordered noncrystalline tetravalent structures

This paper completes a series of works in which the three-dimensional Euclidean realization of the system of constructions of algebraic geometry determines the three-level (cluster-helicoid-union of helicoids) scheme of the assembly of ordered tetravalent structures. The algebraic and topological properties of the second coordination sphere of the eight-dimensional lattice E8 have been used to derive an ordered noncrystalline diamond-like structure. An a priori constructed model represents a helicoidal assembly of finite tubes (from allowably distorted 〈110〉 diamond chains) with transverse pentacycles, hexacycles, and heptacycles and nonintegral screw axes. It has been assumed that similar structures can be formed in diamond-like thin films, gas hydrates, biological structures, nanostructures, and other ordered systems.