Chiral spiral cyclic twins

Abstract

A formula is presented for the generation of chiral m-fold multiply twinned two-dimensional point sets of even twin modulus m > 6 from an integer inclination sequence; in particular, it is discussed for the first three non-degenerate cases m = 8, 10, 12, which share a connection to the aperiodic crystallography of axial quasicrystals exhibiting octagonal, decagonal and dodecagonal long-range orientational order and symmetry.

Chiral spiral cyclic twins. II. A two-parameter family of cyclic twins composed of discrete circle involute spirals

“…Naturally, knowledge of such an interpolation formula for the case of chiral spiral cyclic twins would be very pleasing. It has been shown before (Hornfeck, 2022) that there does indeed exist a relation to continuous curves, namely involutes of the circle. These spirals are characterized by the unique geometric property of having a constant orthogonal distance of magnitude 2R I between successive turns of the curve, R I being the radius of the associated base circle.…”

“…following movement instructions from a finite set of potential choices differing in their relative orientation (say, leftward or rightward, as a binary choice). The nature of the turtle's movement is that of a non-random spiral self-avoiding walk on some lattice (Privman, 1983;Lin, 1985;Joyce & Brak, 1985;Seitz & Klein, 1992) or, generalized, a Z module [see Hornfeck (2022) for the definition]. In our case, the inclination sequence acts as a predetermined program set out for a turtle's spiralling movement into infinity, yielding just the right combination of alternating steps, coming in pairs and thereby defining a straight movement, intermingled with the occasional one step more in the same direction, which guarantees the creation of the spiral's curvature (Fig.…”

“…Those cases for which solutions exist (Fig. 11 shows cases up to m = 40 inclusive) are special in the way that all nodes of the chiral spiral cyclic twin pattern can be indexed by a tuple of up to m separate integers, thereby representing a connection to socalled Z modules and the higher-dimensional crystallography of aperiodic crystals (Hornfeck, 2022).…”

“…are perpendicular, thereby fixing the ideal value of m, as a function of the parameters m and . This case represents the ideal tenfold twin structure of NiZr (Hornfeck, 2018(Hornfeck, , 2022Hornfeck et al, 2018), in which the colours are associated with the atom types and the filling styles are associated with the atom displacements AE z.…”

“…Here, abstraction in combination with idealization led to the construction of an atomistic model, and its confirmation with respect to the structure of the twin boundaries by high-angle annular dark-field scanning transmission electron microscopy (Hornfeck et al, 2018). Forming the basis of an extensive theoretical study into tenfold twins of the CrB structure type (Hornfeck, 2018), this has been generalized into the concept of chiral spiral cyclic twins in Part I of this work (Hornfeck, 2022).…”