Bieberbach groups and flat manifolds with finite abelian holonomy from Artin braid groups

Let [Formula: see text]. In this paper, we show that for any abelian subgroup [Formula: see text] of [Formula: see text] the crystallographic group [Formula: see text] has Bieberbach subgroups [Formula: see text] with holonomy group [Formula: see text]. Using this approach, we obtain an explicit description of the holonomy representation of the Bieberbach group [Formula: see text]. As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of [Formula: see text] and determine the existence of Anosov diffeomorphisms and Kähler geometry of the flat manifold [Formula: see text] with fundamental group the Bieberbach group [Formula: see text].

Twisted crystallographic T-duality via the Baum–Connes isomorphism

We establish the twisted crystallographic T-duality, which is an isomorphism between Freed–Moore twisted equivariant K-groups of the position and momentum tori associated to an extension of a crystallographic group. The proof is given by identifying the map with the Dirac homomorphism in twisted Chabert–Echterhoff KK-theory. We also illustrate how to exploit it in K-theory computations.

On symmetry breaking of dual polyhedra of non-crystallographic group H 3

The study of the polyhedra described in this paper is relevant to the icosahedral symmetry in the assembly of various spherical molecules, biomolecules and viruses. A symmetry-breaking mechanism is applied to the family of polytopes {\cal V}_{H_{3}}(\lambda) constructed for each type of dominant point λ. Here a polytope {\cal V}_{H_{3}}(\lambda) is considered as a dual of a {\cal D}_{H_{3}}(\lambda) polytope obtained from the action of the Coxeter group H 3 on a single point \lambda\in{\bb R}^{3}. The H 3 symmetry is reduced to the symmetry of its two-dimensional subgroups H 2, A 1 × A 1 and A 2 that are used to examine the geometric structure of {\cal V}_{H_{3}}(\lambda) polytopes. The latter is presented as a stack of parallel circular/polygonal orbits known as the `pancake' structure of a polytope. Inserting more orbits into an orbit decomposition results in the extension of the {\cal V}_{H_{3}}(\lambda) structure into various nanotubes. Moreover, since a {\cal V}_{H_{3}}(\lambda) polytope may contain the orbits obtained by the action of H 3 on the seed points (a, 0, 0), (0, b, 0) and (0, 0, c) within its structure, the stellations of flat-faced {\cal V}_{H_{3}}(\lambda) polytopes are constructed whenever the radii of such orbits are appropriately scaled. Finally, since the fullerene C20 has the dodecahedral structure of {\cal V}_{H_{3}}(a,0,0), the construction of the smallest fullerenes C24, C26, C28, C30 together with the nanotubes C20+6N , C20+10N is presented.

The algebra of crystallographic groups in dimension two

Starting with the two semi-direct products of two copies of the group of integers, a description of all possible semi-direct products of these two groups with finite groups, under a faithful action, is obtained. The results are groups that are isomorphic to 16 crystallographic groups in dimension two. An extra effort gives us a group isomorphic to the 17th crystallographic group. Four of the crystallographic groups are shown to be the only solutions to a natural generalization of the semi-direct product of two copies of the group of integers. Finally, we verify that the two additional groups, that are solution to the generalization of the semi-direct product of two copies of the integers, preserve stability in the sense that their semi-direct products with finite groups under a faithful action give rise only to crystallographic groups, if we disregard some trivial outcomes.

Hydrogen order in hydrides of Laves phases

Many Laves phases AM2 takes up hydrogen to form interstitial hydrides in which hydrogen atoms partially occupy A2M2, AM3, and/or M4 tetrahedral interstices. They often exhibit temperature-driven order-disorder phase transitions, which are triggered by repulsion of hydrogen atoms occupying neighboring tetrahedral interstices. Because of the phase widths with respect to hydrogen a complete ordering, i.e., full occupation of all hydrogen positions is usually not achieved. Order-disorder transitions in Laves phase hydrides are thus phase transitions between crystal structures with different degrees of hydrogen order. Comparing the crystal structures of ordered and disordered phases reveals close symmetry relationships in all known cases. This allows new insights into the crystal chemical description of such phases and into the nature of the phase transitions. Structural relationships for over 40 hydrides of cubic and hexagonal Laves phases ZrV2, HfV2, ZrCr2, ZrCo2, LaMg2, CeMg2, PrMg2, NdMg2, SmMg2, YMn2, ErMn2, TmMn2, LuMn2, Lu0.4Y0.6Mn2 YFe2, and ErFe2 are concisely described in terms of crystallographic group-subgroup schemes (Bärnighausen trees) covering 32 different crystal structure types, 26 of which represent hydrogen-ordered crystal structures.

The Exterior Square of a Bieberbach Group with Quaternion Point Group of Order Eight

A Bieberbach group is defined to be a torsion free crystallographic group which is an extension of a free abelian lattice group by a finite point group. This paper aims to determine a mathematical representation of a Bieberbach group with quaternion point group of order eight. Such mathematical representation is the exterior square. Mathematical method from representation theory is used to find the exterior square of this group. The exterior square of this group is found to be nonabelian. Keywords: mathematical structure; exterior square; Bieberbach group; quaternion point group

Formation of the cementite crystal in austenite by transformation of triangulated polyhedra

A mechanism is proposed for the nucleus formation at the mutual transformation of austenite and cementite crystals. The mechanism is founded on the interpretation of the considered structures as crystallographic tiling onto non-intersecting rods of triangulated polyhedra. A 15-vertex fragment of this linear substructure of austenite (cementite) can be transformed by diagonal flipping in a rhombus consisting of two adjacent triangular faces into a 15-vertex fragment of cementite (austenite). In the case of the mutual austenite–cementite transformation, the mutual orientation of the initial and final fragments coincides with the Thomson–Howell orientation relationships which are experimentally observed [Thompson & Howell (1988). Scr. Metall. 22, 229–233] in steels. The observed orientation relationship between f.c.c. austenite and cementite is determined by a crystallographic group–subgroup relationship between transformation participants and noncrystallographic symmetry which determines the transformation of triangulated clusters of transformation participants. Sequential fulfillment of diagonal flipping in the 15-vertex fragments of linear substructure (these fragments are equivalent by translation) ensures the austenite–cementite transformation in the whole infinite crystal. The energy barrier for diagonal flipping in the rhombus with iron atoms in its vertices has been calculated using the Morse interatomic potential and is found to be equal to 162 kJ mol−1 at the face-centered cubic–body-centered cubic transformation temperature in iron.

The symmetry origin of the austenite-cementite orientation relationships in steels

The connection between austenite/cementite orientation relationships and crystal structure of both phases has been established. The nucleus formation mechanism at the mutual transformation of austenite and cementite structures has been proposed. Mechanism is based on the interpretation of the considered structures as crystallographic tiling onto triangulated polyhedra, and the said tiling can be transformed by diagonal flipping in a rhombus consisting of two adjacent triangular faces. The sequence of diagonal flipping in the fragment of the initial crystal determines the orientation of the fragment of the final crystal relative to the initial crystal. In case of the mutual austenite/cementite transformation the mutual orientation of the initial and final fragments is coinciding to the experimentally observed in steels Thomson-Howell orientation relationships: ${\left\{ {\bar 103} \right\}_{\rm{C}}}||{\left\{ {111} \right\}_{\rm{A}}};{\rm{}} < {\kern 1pt} 010{\kern 1pt} { > _{\rm{C}}}{\rm{||}} < {\kern 1pt} 10\bar 1{\kern 1pt} { > _{\rm{A}}};\; < {\kern 1pt} 30\bar 1{\kern 1pt} { > _{\rm{C}}}\;||\,\, < {\kern 1pt} \bar 12\bar 1{\kern 1pt} { > _{\rm{A}}}{\rm{}}$ The observed orientation relationship between FCC austenite and cementite is determined by crystallographic group-subgroup relationship between transformation participants, and non-crystallographic symmetry which is determining the transformation of triangulated clusters of transformation participants.

The R∞ property for crystallographic groups of Sol14

For any automorphism of a crystallographic group of [Formula: see text], we prove that its Reidemeister number is infinite.