 1. Introduction
 2. Exposition
 3. Discussion
 4. Conclusion
 A1. Twodimensional model
 A2. Threedimensional model
 A3. Inclination function as a binary case distinction
 A4. Inclination function as a binary constant
 A5. Inclination function as a binary string
 A6. Properties of the generalized alternating function αμ(j)
 A7. Properties of the generalized baseline function βμ(j)
 A8. Properties of the generalized baseline parameter μ
 A9. Heuristics for the discovery of yetunknown triples
 A10. Further refinement of the baseline function
 A11. Interpretation: the case for odd moduli
 References
 1. Introduction
 2. Exposition
 3. Discussion
 4. Conclusion
 A1. Twodimensional model
 A2. Threedimensional model
 A3. Inclination function as a binary case distinction
 A4. Inclination function as a binary constant
 A5. Inclination function as a binary string
 A6. Properties of the generalized alternating function αμ(j)
 A7. Properties of the generalized baseline function βμ(j)
 A8. Properties of the generalized baseline parameter μ
 A9. Heuristics for the discovery of yetunknown triples
 A10. Further refinement of the baseline function
 A11. Interpretation: the case for odd moduli
 References
research papers
Chiral spiral cyclic twins
^{a}Institute of Physics of the Academy of Sciences of the Czech Republic, Na Slovance 2, 182 21 Praha 8, Czech Republic
^{*}Correspondence email: hornfeck@fzu.cz
A formula is presented for the generation of chiral mfold multiply twinned twodimensional point sets of even twin modulus m > 6 from an integer inclination sequence; in particular, it is discussed for the first three nondegenerate cases m = 8, 10, 12, which share a connection to the aperiodic crystallography of axial quasicrystals exhibiting octagonal, decagonal and dodecagonal longrange orientational order and symmetry.
Keywords: chiral; spiral; cyclic twins.
1. Introduction
The binary intermetallic compound NiZr exhibits a remarkable tendency of forming cyclic tenfold twins from either a vitrified amorphous matrix upon heating (Jiang et al., 1985) or a deeply undercooled melt upon solidification (Hornfeck et al., 2014, 2018).
Similar distinctive cyclic _{20}Cu_{2}Mn_{3} (Feng et al., 2014; Wang et al., 2016). In nature, cyclic twins are commonly found in rutiletype structures, such as in the mineral cassiterite (SnO_{2}), as has been studied recently both in the context of the reticular theory of (Nespolo & Souvignier, 2015), as well as in electron backscatter diffraction (EBSD) experiments performed on synthetic samples doped with CoO and Nb_{2}O_{5} (PadrónNavarta et al., 2020). A geometric theory of cyclic growth twins (Ericksen, 2006) has been applied not only to multiple twins of rutile (TiO_{2}), but also to those found in aragonite (CaCO_{3}), marcasite (FeS_{2}) and quartz (SiO_{2}). The concept of cyclic or cyclic intergrowth plays an important role in the description and classification of inorganic crystal structures (Hyde et al., 1979; Andersson & Hyde, 1982; Andersson & Stenberg, 1982; Stenberg & Andersson, 1982; Andersson, 1983; Hyde & Andersson, 1989), with phenomena in general, due to their variety and profundity, arguably forming their own subfield of crystallography: geminography (Grimmer & Nespolo, 2006).
phenomena occur in other intermetallic alloys, too, a particularly pronounced example being AlIn order to better understand the twin formation process an atomistic model was developed (Hornfeck et al., 2014, 2018; Hornfeck, 2018), based on some idealized geometric properties of NiZr's orthorhombic including not only its special axial ratio, in the form of , being the prerequisite of tenfold but also taking into account special values for the atomic coordinates, governed by a module. This model has been experimentally confirmed at atomic resolution (Hornfeck et al., 2018) by means of highangle annular darkfield (HAADFSTEM), thereby making NiZr the first example of module (Quiquandon et al., 2016; Sirindil et al., 2017, 2018). The atomistic model found (Fig. 1) exhibits some remarkable properties: (i) the structure in the bulk of the twin domains is identical to the structure extending across the twin boundaries; (ii) the structure is chiral, featuring an irrational shift between adjacent twin domains; (iii) the structure is solely determined by the choice of the twin modulus, m = 10, determining a corresponding module (apart from a global scaling factor fixing the minimal interatomic distance); (iv) the structure features a unique description based on the union of ten spirals.
The latter point deserves particular attention. One important aspect to note about spiral patterns is their ubiquitous occurrence in nature [see Hammer (2016) for an excellent overview], in particular as growth forms in the plant and animal kingdoms, yet also in the form of screw dislocation based growth spirals (helices), in the inanimate realm of crystals. Spiral microstructures have been observed in some eutectic alloys as well (Fullman & Wood, 1954). Astonishing geometric regularity can thus be found in biology as well as chemistry, such as in the spatial arrangement of seeds in the sunflower floret (Vogel, 1979), as a specific example of phyllotaxis (Jean, 1995; Adler et al., 1997; Pennybacker et al., 2015), or the atoms in polymeric chain molecules (Müller, 2017), in which their spiral arrangement extends along a common screw axis into that of closely related threedimensional helices, with the doublehelical structure of DNA arguably the most important representative (Watson & Crick, 1953). In the field of crystallography botanical phyllotaxis inspired the study of spiral lattices (Rivier, 1988; Rothen & Koch, 1989a,b; Kunz & Rothen, 1992; Sadoc et al., 2012), a concept pioneered and greatly expanded upon by Bursill and coworkers, both in direct and and with relations to aperiodic crystals (Bursill et al., 1987; Fan, Peng et al., 1988; Fan, Bursill et al., 1988; Bursill, 1990).
For most of these examples, the geometric properties of a particular spiral type (Archimedean, logarithmic etc.) frequently match in a natural fashion with simple growth principles (equidistant layering, selfsimilarity/scale invariance etc.), thereby rationalizing the selforganization of matter on its way from chaos to order. By their observed morphology and microstructure, and following a simple geometrical argument, the formation of cyclic twins is naturally thought to originate in the nucleation from a single common seed. Yet, how exactly the which can be seen as a disadvantageous defect occurring in an otherwise undistorted bulk crystal, is induced first, and furthermore preserved during crystal growth, is much less clear (Shahani & Voorhees, 2016). In the case of NiZr, however, the spiral description suggests an intrinsic mechanism, in the form of a lateral growth program, explaining effortlessly the formation of a cyclically twinned microstructure and its spatial continuation into a macroscopically observable twinned morphology. The resulting twin structure is a highly correlated arrangement of atoms, and due to its perfect twin boundaries matching the bulk crystal, energetically favorable.
These favorable properties recommend an investigation into the potential of transcending the NiZr model into one of chiral spiral cyclic twins (CSCTs) of arbitrary m.
2. Exposition
The key to the desired generalization is given by exploring and exploiting the most suitable parametrization of the CSCT in order to prepare the stage for subsequent generalizations. In the following we state the obtained results and refer the interested reader to the extensive appendix containing a neverdaredtoaskfor discussion about the derivation and further optimization of this parametrization.
2.1. Parametrization
Twodimensional CSCT point patterns can be parametrized by three numbers: (i) an integer modulus m, (ii) an integer multiplier and (iii) a real scaling factor . To cut things extremely short for the moment, the parametrization is based upon the use of the complex mth roots of unity
and their combination according to
with Cartesian coordinates resulting as points
suitably indexed as the lth spiral node of the kth spiral branch. Here,
denotes an integer inclination function, itself described by an integer baseline function determined by the rule
which, for the cases under discussion hereafter, can be expressed algebraically as
Here, and denote the integer modulo and floor function, respectively. More details are given in the appendix.
Empirically it is found that only certain combinations of parameters yield a meaningful, uniformly distributed point pattern composed of selfavoiding discrete spirals. Noteworthy are the following triples , which will be discussed in detail hereafter: (6,1,1), [8,1,1+(2 )^{1/2}], and [12,2,1+(3 )^{1/2}]. Here, denotes the golden ratio. The triples with m = 8,10,12 are especially interesting because their rotational symmetry matches those of octagonal, decagonal and dodecagonal axial quasicrystals.
2.2. The case m = 6
As a prelude we briefly discuss the case m = 6. As it turns out, this is a degenerate case, not matching the subsequently described twin structures for m = 8,10,12, yet still of some interest in terms of its crystallographic interpretation. The sixfold `chiral' spiral `twin' is the hexagonal honeycomb grid (Fig. 2).
Thus, for indistinguishable points in an infinite pattern, it is neither chiral, nor a twin, since there are no observable boundaries, which would induce both properties to the point pattern. Instead, the hexagonal honeycomb grid is the vertex set of one of the three tessellations of the plane by regular convex polygons, with the other two tilings being based on the vertex set as defined by the twodimensional triangular and square lattice.
The case m = 6 is also interesting, insofar as a cyclic twin nucleation has been reported to occur in tinbased solder alloys (Lehman et al., 2010), which strongly resembles the cyclic twin nucleation phenomena observed for NiZr (Hornfeck et al., 2018), yet differs in that the symmetry of the twin is sixfold.
2.3. The cases m = 8, 10, 12
The exploration of the parameter space , with the already known case (10,2,1.618), and following the rationale described in the appendix, leads to the discovery of a regular CSCT described by the parameter triple (8,1,2.414) with . This seemed to suggest a correspondence with the well known (Niizeki, 1989; Burdik et al., 1998) and experimentally observed quadratic irrational inflation factors (see the appendix for the concept and notation)
for intermetallic (IM) axial quasicrystals with eight, ten and 12fold rotational symmetry, including the silver, golden and platinum ratio, respectively. Thus, a triple (12,3,3.732) was checked for, yet did not yield a CSCT. Neither did a triple (12,3,0.822) obtained by a linear extrapolation of the established octagonal and decagonal parameters, although its regularity was higher than in the aforementioned case. Instead, it was found empirically that the triple (12,2,2.732) with and yields a perfect solution! Figs. 3, 4 and 5 show the corresponding point patterns.
Notably, the observed factors and can be associated with a single formula,
which is known to describe the inflation factors of certain quasiperiodic rhombic tilings of octagonal and dodecagonal symmetry as obtained by the projection method (Steurer & Deloudi, 2009; p. 38 for the octagonal Ammann–Beenker tiling and p. 40 for the dodecagonal rhomb tiling). Note that the inflation factor , while not to be regarded the most characteristic inflation factor for dodecagonal tilings, occurs in several distinct instances (Schaad & Stampfli, 2021; see also https://geometricolor.wordpress.com/2012/07/29/anothertilingofdodecagonalsymmetry/ and https://geometricolor.wordpress.com/2021/03/04/yetanothertilingwith12foldrotationalsymmetry/).
Now, equation (8) has a simple geometric interpretation as the circumcircle radius of the corresponding regular mgon (Fig. 6), spanned by m triples of mutually adjacent thin and thick rhombs of unit edge length e = 1 [compare this with equation (58) in the appendix, in which the unit edge length E = 1 refers to the regular mgon instead].
2.4. The general scheme
Thus, from the small number of observations made so far, and taking the approach detailed in the appendix into account, one might boldly generalize the existence of two series of CSCTs, one for and another one for . In particular, if is an odd number, one obtains (see the appendix)
while one obtains
for being an even number. By using the alternating functions
in which , one can combine both parameter evolutions into a single set
of formulas, which inherently include the case distinctions.
Eventually, it is observed that all scaling factors also follow an alternative formula, common for both series,
in which denotes the ith out of the distinct diagonals of a regular jgon of unit edge length (Fontaine & Hurley, 2006; cf. Fig. 7). Here, the observed value of j follows the splitting pattern for being an odd or even number, with j = m/2^{M}, whereas the value of i shows an individual pattern alternating between a linear increase for the odd cases and a constant one for the even ones. The parameters for the CSCTs up to m = 24 have been collected for easy reference in Table 1.

As a summary, any given triple fully characterizes a CSCT, with each parameter on its own determining certain of its geometrical features: (i) the modulus , denoting the order of the rotation axis, determines the number of twin domains and the successive angular inclinations of the spiral branches. (The number of distinct orientation states, the twin multiplicity, is only half as big, since opposite domains share the same orientation state, see Fig. 1.) (ii) The multiplier determines the location of the inclination points occurring on a spiral branch, by controlling the repeat number of alternating steps occurring inbetween the inclination points. (iii) The multiplier defines the circumcircle radius of the innermost mgon, in relation to the unit moduli along the spiral, together forming a pair of multipliers.
3. Discussion
After showing how the CSCTs are generated, a conceptualization of what has been found is given in the following discussion, highlighting the CSCTs' relation to Archimedean spirals and circle involutes, as well as modules.
3.1. Archimedean spiral and circle involute description
Although the spirals used for the description of the CSCTs are discrete in nature, they can be approximated by continuous curves, which is of particular interest for a more comprehensive characterization of CSCTs in terms of their general geometric properties and their largescale asymptotic behavior. While a direct interpolation by some general curve is always possible, it is never unique, thereby limiting deeper insights (Davis, 1993; p. 34). Instead, it is rather more important to note the type of continuous spiral, some common examples of which are given by Bursill et al. (1987), that will represent a best fit to the discrete set of spiral nodes P_{0,l}. The most common types of spirals, often known since antiquity, are usually fully described by very few parameters only, thus being ideal candidates for the aforementioned purpose. A number of textbooks have compiled information about spirals and other curves, their construction, properties and applications (Eagles, 1885; Lockwood, 1961; Zwikker, 1963; Lawrence, 1972; Yates, 1974; Shikin, 1995; Rutter, 2000).
Now, the layerbylayer growth of the discrete spiral gives the crucial clue for the unique type of continuous spiral featuring a constant separation distance between successive turns, thereby ruling out a kind of spiral similarity symmetry in the sense of Shubnikov (1961). In fact, two closely related candidates exist, namely the Archimedean spiral and the involute of a circle. In the case of the Archimedean spiral a constant separation distance between consecutive convolutions is observed along rays radiating from its origin, but with the caveat that the intersections of these rays with the spiral are not perpendicular. For the involute of the circle, however, a constant separation distance is measured along common normals of consecutive convolutions of the curve.
Despite this subtle yet important difference, both curves are intimately related to one another, and thus can be treated within a general system of parametric equations, suitably formulated in matrixvector form
in which R is a common scalar reference radius, and the matrix is defined as
with the choice of matrix parameters a, b discriminating the cases
Here, the additional fixed parameter designates a single spiral's starting point in terms of a cyclic phase shift. In particular, the value represents the starting points (0,0) and (R,0) for the Archimedean spiral and the involute of the circle, respectively, for the parameter t = 0. For spirals starting at phase angles the parameter t starts from a shifted starting point, t = s, as well.
In a dynamic perspective an Archimedean spiral is defined as the locus of a point that moves away from another, fixed point at uniform linear velocity v (radially) and uniform angular velocity ω (azimuthally). As a consequence of the described uniform motion, Archimedean spirals feature a constant separation distance of .
In order to obtain the best fit curve, it is actually better to use a polar plot of a point's radius vector length r versus its polar angle θ, yielding the simple linear function
in which the slope is given as the quotient of the radial and azimuthal velocities, and in which the ordinate intercept s is related to the starting point . Since a point in our chosen parametrization is represented by a complex number, both the radius and the angle have a simple representation as a point's complex absolute value and absolute argument, respectively.
Polar plots for the CSCTs with m = 8,10,12 are shown in Fig. 7 together with their approximations with
determined by a simple linear leastsquares procedure (Mathematica).
Note that while the slopes increase in the unnatural order m = 10,8,12, this matches the ascending order of their scaling factors , and , respectively.
A plot of the CSCT for the case m = 8 is shown in Fig. 8 together with its circle of radius R = R_{8}, circle involutes for and Archimedean spiral for s = 0.
Note that the above parametrization has been chosen so as to account for a fixed phase shift between the circle involute and the Archimedean spiral, such that any given pair of a circle involute and its corresponding Archimedean spiral can be associated with the same variable phase shift s, thereby highlighting their asymptotic equivalence for .
3.2. module description
For a single spiral branch, each node's position in the complex plane is, by definition, a linear combination of unit basis vectors represented by a complex summation over the roots of unity. By this means a single spiral branch traces a path in what is known by mathematicians as a module, a higherdimensional generalization of a lattice (Steurer & Deloudi, 2009, p. 61; Sirindil et al., 2017). In particular, a module of rank n, existing in a of dimension , can be described by the set of points
which form an integer lattice , spanned by n linear independent basis vectors , in the ndimensional . For d = n this describes the special case of a lattice, while for this corresponds to the projection of a higherdimensional lattice in n dimensions into the module in d dimensions.
Now, the generation of the full CSCT structure demands the additional complex multiplication with all possible roots of unity , which leads to the question of whether this relation to a module still holds true for the CSCT as a whole, despite the occurrence of the central radius vectors of length in the parametrization. While it is conjectured to hold true for general m, for the most interesting cases m = 8,10,12 this question can be answered in the affirmative, with a visual proof given in Fig. 9. Here, the proof consists of finding a continuous path, solely constructed of unit length segments, connecting the origin P_{00} of a single spiral branch with the origin (0,0) of the CSCT, and thus, by the action of the mfold rotational symmetry, connecting all spiral branches with each other.
This relation to a module implies that every point on a single spiral branch, with in our case d = 2 real coordinates, can be indexed (in real space) instead by n integers z_{i}, where n = 4 for the octagonal, decagonal and dodecagonal cases. Apart from the formalism this means that the CSCT structures are exceptionally highly correlated structures, in which every position is precisely in registry with the underlying module. The module description thus highlights a hidden algebraic order among the coordinates with the presented parametrization by a single formula acting as a quantitative descriptor (Hornfeck, 2012).
Moreover, since all points of the octagonal, decagonal and dodecagonal CSCTs are located on a module, we expect a possible description of these patterns by means of a higherdimensional cutandproject scheme, as well as their diffraction patterns to exhibit perfect octagonal, decagonal and dodecagonal symmetry, arising from the longrange orientational order present in the point patterns, similar to the crystallography of quasicrystals. Thus, the CSCTs with m = 8,10,12, while not being quasicrystals themselves (and experimentally discernible from them by the diffraction of a single domain), share a common characteristic of this special class of aperiodic crystals. Indeed, they are role models for the twins proposed by Pauling in his argument against the existence of quasicrystals.
4. Conclusion
A theoretical study has been made on chiral spiral cyclic twins with an emphasis on twins of eightfold, tenfold and 12fold symmetry generated by the application of a unique integervalued inclination function in a parametrization based on complex roots of unity. The underlying association of each generated point pattern's coordinates with those represented by a module highlights the highly correlated nature of this spatial arrangement, illustrates the importance of hidden algebraic relations presenting a form of symmetry going beyond the grouptheoretical one, and establishes a connection to the crystallography of octagonal, decagonal and dodecagonal axial quasicrystals, respectively.
APPENDIX A
Modeling of chiral spiral cyclic twins
In the following a detailed description of the derivation of the fundamental formula modeling CSCT structures is given, with each section discussing specific details of the modeling process.
A1. Twodimensional model
For the parametrization the focus is laid on a purely twodimensional point pattern, acting as an abstract model for the CSCT structure of NiZr as illustrated in Fig. 1, and obtained from it by a projection of the structure along the tenfold axis.
This is possible, because the tenfold twin structure of NiZr is essentially a (2+1)dimensional structure, with the cyclic
occurring in the plane perpendicular to the tenfold axis, along which the threedimensional twin structure is periodic.This is similar to the case of axial quasicrystals, in which the
also is restricted to a twodimensional subspace.Indeed, a periodicity in the twin plane is only observed within and strictly limited to any of the ten single twin domains, while the infinite pattern as a whole is aperiodic. Its symmetry is restricted to a
composed of pure rotations, singling out the origin as a fixed point and implying the point pattern's The pattern's is reflected in a spiral arrangement of atoms, akin to the phyllotactic arrangement of seeds in a sunflower floret, except for the presence of spirals of one handedness only.The parametrization can take advantage of the abstract model's twodimensional nature by the use of complex coordinates, in particular using the tenth roots of unity ω, defined by , as basis vectors. Anticipating the general case of an mfold twin pattern, a general mth root of unity is given by the complex number, written in exponential form,
in which k runs from zero to in order to produce all distinct roots.
Exponential sums of the general form
in which f(j) is a realvalued function defined on the positive integers, show a rich variety of geometric forms, including spiral ones, in plots of all their partial sums up to the given n (Lehmer & Lehmer, 1979, 1980; Loxton, 1983; Deshouilliers, 1985; Chamizo & Raboso, 2015). Sums of the aforementioned type are commonly referred to by the name of Weyl sums, as they feature prominently in the study of uniform distribution modulo one, as pioneered by Hermann Weyl (Dekking & Mendèz France, 1981).
In our case, each point located on a single spiral branch of a CSCT is parametrized by the complex number
obtained by adding the individual contributions of the mth roots of unity in a consecutive manner. Here, is a nonnegativeintegervalued inclination function (see below). The full mfold point pattern is generated from a single spiral branch's points by multiplying the with the appropriate complex mth root of unity:
Here, k and l denote spiral coordinate indices, designating the lth spiral node on the kth spiral branch.
Cartesian coordinates (x,y) are obtained as the real and imaginary parts of the complex number , namely as the twodimensional point . Note that P_{0,0} is not denoting the origin, but the point (, 0).
The real number denotes a fixed scaling factor, in particular = [1+(5 )^{1/2}]/2 is the golden ratio, and is chosen such that the distance between consecutive spiral nodes is unity, including, in the case of NiZr, the edge length of the central mgon. Apart from an arbitrary general scaling factor, not stated explicitly, this choice fixes the model.
Finally, the mirrorsymmetric twin of opposite et al., 1988).
is obtained by replacing every instance of a complex root of unity by its complex conjugate counterpart. Here, a plus (minus) sign in the exponent corresponds to a counterclockwise (clockwise) oriented, righthanded (lefthanded) spiral, following the usual convention (Fan, PengA2. Threedimensional model
In a realistic threedimensional model further scaling factors have to be taken into account, following NiZr's orthorhombic unitcell metrics (Hornfeck, 2018). In addition, atom types (Ni or Zr) and atom heights (1/4 or 3/4) in the c direction have to be specified. This can be done as a function of the twodimensional spiral coordinates k and l as
Notably, the assignment of alternating atomic heights, apart from creating a twolayer structure, reduces the tenfold symmetry to a fivefold one. Thus, alternating twin domains appear flipped, in accordance, however, with a relative shift of atomic positions by z = 1/2. As is the case for axial quasicrystals, the structure can be continued periodically in the c direction.
A3. Inclination function as a binary case distinction
The inclination function of equation (23) encodes all the information about the consecutive coordinate evolution tracing out a given spiral branch k up to a given spiral node l for a specific CSCT. For the case of NiZr it was empirically found (Hornfeck, 2018) to be defined as
with
Here, , and are the integeryielding ceiling, floor and modulo functions, respectively, while
denotes the Iverson bracket, checking its argument C is true or false, thus forming a generalization of an indicator function [see Graham et al. (1994) for the definition and properties of all integer functions]. In the case of equation (27), the condition to check for is given by the question of set membership of the variable j in the set . The use of the Iverson bracket allows for an alternative definition of the complex number
in the way that the summation of the complex unit vectors is not carried out on a consecutive basis, as in equation (23), but instead by counting the total number of occurrences of a given ith root of unity , out of m distinct ones, up to the number l of nodes. Note that the modulo operation can be skipped for the application of in the exponent of a complex mth root of unity [cf. equation (23)], since these form a cyclic group, yet not in the condition of the Iverson bracket used in equation (30).
The two summands in equation (27) give rise to the following sequences:
As can be seen from this comparison, the sequence defines a baseline, increasing by one at certain steps of increasing interval length, while the sequence is alternating between the values of zero and one, in an intricate, yet certainly not irregular fashion.
Both sequences added (modulo 10) yield the spiral sequence
the numbers of which, ranging from zero to nine, specify one out of ten basis vector directions.
The twin model of NiZr was constructed by adhering to certain geometrical constraints imposed by NiZr's et al., 2014). Its parametrization by the spiral sequence of equation (33) followed only afterwards (Hornfeck et al., 2018; Hornfeck, 2018) and by empirically finding the suitable integer sequences for the baseline and alternating part in the Online Encyclopedia of Integer Sequences (OEIS; Sloane, 2018). This has several drawbacks.
(HornfeckFirst, seen from a technical point of view, the construction of the alternating sequence could not be made in a direct way, i.e. the sequence of equation (32) could not be found in the OEIS, and accordingly no formula for its generation is yet known. Only by transforming its information, by stating the position of ones, and thereby forming another sequence, encoded in the set of equation (28), can its description be allowed in an indirect way.
Second, any generating formula found faces the problem of being nonunique – usually several immediate alternatives exist, and, in principle, an infinitude of them. This is, of course, not a mathematical problem. The description of the spiral sequence by one formula or another mathematically equivalent one makes no practical difference whatsoever. However, the conceptual difference is huge, thinking in terms of explaining not how the formula works, but why it does. Without knowing this, there seem also no opportunities for obvious generalizations to other than tenfold symmetric cases.
A4. Inclination function as a binary constant
For the purpose of a more direct generation of the alternating sequence of equation (32), one idea is to identify it with a binary number by means of concatenating zeros and ones according to the set membership condition and in the order in which they appear in the sequence:
The knowledge of then allows for an alternative representation of the inclination function as
An analytic expression of the constant is given by the infinite reciprocal sum [cf. equation (28)]
which, after a combination of manual and computerassisted (Mathematica) algebraic transformations, evaluates to
in the case of general base b. Here, is a shorthand notation for Jacobi's second and third theta function,
stated for the special case in which the complex argument z is equal to zero. The socalled nome q is restricted according to .
Jacobi theta functions are interrelated by numerous algebraic identities, and can be expressed in a multitude of infinite series and product representations, with a rich numbertheoretic background. In particular, they appear in the closedform expressions for similar constants, e.g. for the infinite sums of reciprocals of the Fibonacci and Lucas numbers (Borwein & Borwein, 1987). With the complex argument z being zero, they also occur in the exact evaluation of lattice sums, of which the Madelung constant is the most prominent example (Borwein et al., 2013; Zucker, 2017).
However, the aforementioned approach is still not well suited for generalization, since for each case it requires the derivation of another specific constant.
A5. Inclination function as a binary string
In order to generate the alternating sequence of equation (32) in a more direct way, we analyze the pattern of zeros and ones of which the sequence is made. For this purpose we switch to a representation of the sequence by a string by concatenating zeros and ones, just as we did for the representation as a binary number, yet skipping the leading zero and decimal point. Now, the baseline sequence [equation (31)] suggests a splitting pattern according to , due to the incremental change of digits and with respect to the interval lengths in between. A comparison with equation (31) shows that we have the following multiplicities times digits: 1×0, 1×1, 3×2, 3×3, 5×4, 5×5, and so on. The multiplicities run over the odd integers, yet with a period of two, which could be highlighted by introducing a second marker  at the appropriate positions: . Obviously the pattern between the markers is very simple – just the alternation of zeros and ones – with everincreasing odd number interval lengths while one proceeds along the string. In the limit of an infinite string this would result in a periodic alternation of zeros and ones, , which could be thought of the average string underlying the inclination pattern. In a plot this would just correspond to a rectified `spiral', so any of the observed differences are really caused by the occasionally interspersed additional steps preferring one direction over the other, thereby creating a true spiral and a chiral twin structure. Now, the average sequence is generated by
and it is clear that the need to adjust the sequence, in such a way that the alternating sequence is recovered, is connected to the incremental increase of the baseline sequence , in particular to the positions when an incremental step occurs. In these cases the average sequence has to be shifted/extended by one bit, thereby creating the intermittant repeats. Taking these arguments into account one finds
as the correct function. Here, the baseline generating function
was adjusted for an offset of one, in order to let the sequence start with an index j of zero, as is the case for the average sequence . With this we have
where j starts from zero. This formula has the advantage of using only simple integer functions, while effectively coupling the alternating sequence to the baseline one. As such it can be very simply generalized to
with the general baseline function
and two fixed parameters, a multiplier μ and a modulus m.
A6. Properties of the generalized alternating function α_{μ}(j)
In order to see why
gives a uniquely alternating sequence of zeros and ones, as a function of , let us use the identity and thus
Since both j and are integers by definition [cf. equation (45)] we can include each of them separately into one of the two floor functions
and further simplify, by using the identity , to
It is now clear that can only assume the values of zero (if x is an integer) or one (otherwise), as implied by the modulo function of modulus two. For the argument to be distinct from an integer, the sum of the integers j and has to be odd, which is to say that either one of the summands has to be odd, while the other one has to be even. Since j runs consecutively over the integers, thereby alternating between odd and even values, the result will be one, whenever fails to follow this alternation, which happens for every other value within the resulting baseline function's plateaus.
A7. Properties of the generalized baseline function β_{μ}(j)
The core of the presented parametrization is given by the baseline function , which governs the path traced out by the spiral, in terms of its inclination points and their relative distance on the spiral arc. As such, this restricts its mathematical properties rather rigorously.
A function under consideration has to adhere to the following requirements. It has to map an integer input, the intermediate node index j, to an integer output, the intermediate baseline level, . In the presented baseline function this is achieved by the floor function. Furthermore, the baseline levels have to start at zero and increase monotonically, so as to represent the increasing radial distance of a spiral node from the spiral's origin. The floor function guarantees a stepwise increase of the baseline levels, thereby yielding straight spiral segments within each twin domain [here, we refer to the spiral segments being straight, apart from the alternating pattern induced by the function , which is disregarded for the sake of argument]. The increase of the baseline levels has to decrease with increasing intermediate node index j, since the farther apart a straight spiral segment is situated, the larger has to be its length, i.e. the relative distance between consecutive inclination points on the spiral.
Naturally, the combination of these requirements already rules out many of the fundamental mathematical functions, while the floored square root function just happens to fulfill all of these requirements. This baseline function is a step function of increment one and plateaus constantly increasing in their length, due to the monotonously, yet ever more slowly increasing continuous function . The multiplier μ is a constant for any fixed m, but its functional dependence for varying m has yet to be clarified.
A8. Properties of the generalized baseline parameter μ
While the mathematical appearance of the baseline function can be rationalized in a rather straightforward manner, a physical interpretation of the baseline parameter μ appears to be less evident. In order to get a feeling about its influence on the twin point pattern one can, however, study its extremal behavior in the limits and .
For the case one observes an everexpanding length of the straight spiral segments until, at , the spiral branch is fully rectified. In this case the baseline function becomes zero everywhere, , which means that the inclination and the alternating function become equal, , resulting in a strictly alternating sequence, , of baseline level values. Upon increasing μ again a single spiral branch with a fixed number of nodes first gets more curled up, with its winding number around the origin increasing and its straight segment perpendicular separation distance decreasing accordingly. At some point, however, the regularity of the point pattern of a single spiral branch gets lost, since for the inclination sequence defined by becomes more and more indistinguishable from a random sequence of baseline level values ranging between zero and , with any given value occurring in the sequence with its natural uniform density 1/m for . A single spiral thus eventually transforms into a random walk about the origin. This behavior is known for the graphs of the partial sums of exponential sums in which the phase values of the summands are proportional to a square root function (Loxton, 1983; Chamizo & Raboso, 2015).
A9. Heuristics for the discovery of yetunknown triples
Which triples generate a CSCT? One way to answer this question would be an exhaustive search by applying a bruteforce sampling of the parameter space. Since point patterns of already rather moderate modulus, say , are becoming less and less interesting, due to the diminishing size of individual twin domains, this task seems feasible. However, one does not know exactly which, maybe noninteger, values the multiplier might take, and since the multiplier is observed to be an irrational number, any search by an incremental variation of the parameters will anyhow only yield an approximate solution, which, in the best case, could result in an educated guess regarding the ideal value of this constant. If one assumes a variation of the parameters in the intervals by an incremental resolution of this amounts to roughly 10^{9} trial cases.
In practice, however, this approach failed not due to any computational intractability of the search itself, but due to a lack of algebraic methods for automatic checking of the geometric regularity of the candidate solutions. A check by visual inspection, although taking advantage of the supreme
abilities of the human visual system, the eyes and the brain, proved to be utterly tedious, since the overwhelming number of trial cases turns out to be far away from being regular. At least there is a lesson learned here, too: any parameter triple, if it exists, has to be finetuned, since even the slightest deviations from the ideal parameter values quickly destroy the geometric regularity of the point pattern.Instead of an exhaustive approach, a better way seems to be to find some heuristics for focusing the search on promising candidates. While the modulus m might simply be chosen to correspond to some of the known numbers m = 8,10,12 occurring for axial quasicrystals, a good guess for the other parameters is more difficult. However, since is the golden ratio, one might assume suitable candidates for other factors to be similar well known numbers, possibly occurring as members of some general number families.
For instance, the golden ratio occurs as the member of the family of infinite, nested radicals:
On the other hand, the member is a constant known under the name plastic ratio, and both ratios are Pisot–Vijayaraghavan (PV) numbers, the plastic ratio also being the smallest such number. A PV number is a real algebraic integer α greater than unity, all of whose conjugate elements have absolute value less than unity (PV property): . The PV property proves to be of particular importance in the theory of quasiperiodic substitution tilings in defining their selfsimilar inflation factors and other of their properties, and thus seems to be a good heuristic for which to check.
Like the golden ratio, the plastic ratio is an algebraic number, which means that it is a complex (real) root of a nonzero polynomial in one variable, with rational (integer) coefficients, where the polynomial is usually assumed to be of minimal degree. The minimal polynomial for the golden ratio is given by (of degree 2), while the one for the plastic ratio is given by (of degree 3). Being the root of a minimal polynomial of degree two makes the golden ratio an algebraic number of degree two, and in particular a quadratic irrational number
in which a,b,c,d are integers, with b,c,d being nonzero and c being squarefree. The property of being a quadratic (or cubic) irrational number was used to explain the occurrence of quasicrystals with mfold rotational order only for m = 5,8,10,12 (Levitov, 1988). The golden ratio corresponds to . The silver ratio, appearing as the selfsimilar inflation factor for the octagonal Ammann–Beenker quasiperiodic tiling, corresponds to . Both the golden and the silver ratio are part of a larger series of metallic ratios of the general form (n,1,n^{2}+4,2), which are defined by their infinite continuous fraction representation of
or by being a root of the quadratic equation , with n = 1 and n = 2 corresponding to the golden and silver ratio, respectively. The next, n = 3, member of the series, representing the bronze ratio has also been associated with a twodimensional of hexagonal symmetry, formed in simulations of hardcore/squareshoulder colloidal particles (Dotera et al., 2017; Nakakura et al., 2019).
In general, the inflation factors of quasiperiodic tilings can be calculated from the tiling's substitution rules, which relate the number of tiles before and after an inflation step. The possibility of repeating an inflation step over and over again ad infinitum, from an aperiodic finite patch until its ultimate extension to the infinite plane itself, proves the of the tiling. The substitution rules can be formulated as a system of linear equations, with the number of individual tile multiplicities upon inflation forming a coefficient matrix. The eigenvalues of this coefficient matrix, being the roots of its characteristic (possibly minimal) polynomial, encode the information about the number of tiles after the nth inflation step, as well as the asymptotic proportions of tiles in the limit . In this limit only the largest root is dominant and thus defines the inflation factor.
Despite the general theory for selfsimilar inflation factors of quasiperiodic substitution tilings, it is actually not a trivial task to find the exact values for a given modulus m, for several reasons: (i) many more tilings are informally known than formally published, although there exists an extensive monograph (Grünbaum & Shephard, 1987) as well as promising attempts at an allencompassing atlas of tilings (Frettlöh et al., 2020); (ii) no coherent standard seems to exist in reporting the properties of tilings; (iii) existing results are scattered few and far between in a large body of literature; (iv) existing results are hard to search for, since synonymous expressions prevail, such as substitution factor, deflation factor or length expansion; (v) values for the inflation factor are not always stated explicitly.
For certain nonperiodic rhombic substitution tilings with mfold symmetry, however, Brown (2018) has described an easy and systematic way of obtaining the characteristic polynomials from Pascal's triangle (Fig. 10).
For this purpose the triangle is leftjustified and sliced diagonally, with the integers within a slice forming the coefficients, alternating by their sign, of the soughtfor polynomial. For instance, one obtains the following five first polynomials
of general formula
having only positive, real roots. The degree of the polynomial is related to the order m of the rotational symmetry of the twin pattern by . The scaling factor is given as the square root of the largest polynomial root, yielding
These scaling factors show a linear relationship with μ, which was estimated for the first 100 values to be equal to
with a coefficient of determination R^{2} almost equal to unity. This approximate linear relationship should facilitate an interpolation to the scaling factors of intermediate cases for which . Moreover, by using Plouffe's inverse symbolic calculator (Plouffe, 1998), we find all these numbers to be equivalent to
thus establishing a relation between and m. Here, the superscript PT denotes the Pascal's triangle family of tilings. Since m was related to before, the triple of parameters needed to specify a cyclic twin has effectively reduced to the choice of a single parameter. This could be , as it is the natural index variable used in the construction before, or m since it corresponds to the order of the rotational symmetry of the cyclic twin under consideration. In any case the coupling of the model's parameters reduces any arbitrariness in the twin's construction.
Another aspect is remarkable. The formula found by inverse symbolic calculation and subsequent computer algebraic simplification is identical to the formula for the circumcircle radius
of a regular mgon of unit edge length E = 1. This relates the above series of scale multipliers to a simple property of the central polygon of the CSCT.
A drawback of the approach just presented is that it misses out on the interesting values of m = 8,12. Thus, for these cases another solution had to be found empirically.
A10. Further of the baseline function
The investigation of CSCTs with is interesting as well, because quasicrystals with a corresponding higher mfold symmetry, for instance m = 18 or m = 24, exist in soft condensedmatter systems (Dotera et al., 2014). Some general trends can be inferred from plots of their point patterns (not shown). On the one hand, the overall appearance of the CSCTs, expressed by their convex hulls, becomes more and more indistinguishable from a circle, as is to be expected for a compounded point pattern consisting of regular polygons with an everincreasing number of edges. Individual spiral branches progressively approximate circle involutes, due to both the straight spiral segments decreasing in their relative length and the alternating inclinations differing by a decreasing angle, respectively. As a consequence, the individual wedgeshaped twin domains, their facetting and boundaries, become less and less visually recognizable. On the other hand, the increasing baseline parameter value counteracts the emerging regularity, in that jumps of the baseline function become more frequent and their sequence more erratic. In summary, the point patterns appear less regular than those for the cases m = 6,8,10,12. Eventually, the inclination pattern results in collisions, thus rendering the approach ultimately futile for both large m and l, respectively. Indeed, it seems rather remarkable that the approach works perfectly for the cases m = 6,8,10,12.
The lack of regularity can be attributed to a breakdown occurring in the spiral sequences going from
to
While for , with and starting at j = 0, the pattern can be described according to the rule stated in equation (5), this is not any longer true for with . Replacing the function with the verbatim rule, however, restores the regularity. Note that the remaining parametrization stays the same. The rule of equation (5) can be cast into an analytic expression, too, by noting the pattern of deviations, in particular taking into account integer points on the continuous square root function. Assuming μ is an integer, these points share the general form . It turns out that, for the rule of equation (5) to hold true, these points can be taken as support for a continuous, piecewise linear substitution of the continuous, smooth square root function by a family of integerparametrized lines (Fig. 11).
It happens that for every interval the segment of the line exhibiting the smallest slope connects the supporting points. Taking the floor function on these piecewise linear segments as baseline function
restores the regularity of the twin structure, as, for instance, in
Now, by the rule of equation (5), the meaning of a preferentially integer baseline parameter μ as a repeat period becomes obvious.
A11. Interpretation: the case for odd moduli
The CSCT models developed so far share a common feature in that their twin modulus m is always an even number. While odd moduli are not excluded by the mathematical rules of CSCTs, they are troublesome with respect to the stereochemistry of the CSCTs. Decorated CSCTs as illustrated in Fig. 1 are incompatible with odd moduli, since they cause a geometrical frustration between atoms located at the twin domain boundary following a spiral's full turn. Such a geometrical frustration cannot be resolved even by considering the presence of a screw dislocation, which would, in principle, allow for a recovery of the cyclic periodicity of the alternating twin domains after two full turns instead of one, yet also yield another geometrical frustration, now created between twin domains of alternating polarity along the dislocation axis. Accordingly, a spiral growth model that has been constructed based on the rhombic Penrose tiling (Baranidharan, 1990) does not take into account any atomic decoration of its tiles.
Apart from this the observed dichotomy of two series existing for , as described in the main part of this work, makes it seem advantageous to consider the fundamental twin modulus not being equal to m, but rather to instead, for the following reasons: (i) if is taken to be of fundamental importance, the lower limit of becomes meaningful, since an equilateral triangle is the regular polygon with the smallest possible number of vertices; (ii) the classification into two series indexed by odd and even is emphasized; (iii) CSCTs for which m is an odd number are excluded, just by definition; (iv) if CSCTs are decorated with distinct atoms of alternating z coordinates, thereby forming alternating twin domains breaking the mfold rotation symmetry, the number reflects the reduced order of the after the symmetry reduction. This makes appear to be the more natural parameter to consider.
Acknowledgements
The author thanks Peter Zeiner for his independent proof of equation (37), Peter Stampfli for giving the geometric explanation of equation (8), Walter Steurer for providing literature on the inflation factors of dodecagonal quasiperiodic tilings, and Michal Dušek for his endorsement.
Funding information
This work was supported by the Czech Science Foundation through research grant No. 1810438S.
References
Adler, I., Barabe, D. & Jean, R. V. (1997). Ann. Bot. 80, 231–244. CrossRef Google Scholar
Andersson, S. (1983). Angew. Chem. Int. Ed. Engl. 22, 69–81. CrossRef Web of Science Google Scholar
Andersson, S. & Hyde, B. G. (1982). Z. Kristallogr. 158, 119–131. CrossRef CAS Google Scholar
Andersson, S. & Stenberg, L. (1982). Z. Kristallogr. 158, 133–139. CrossRef CAS Google Scholar
Baranidharan, S. (1990). Pramana – J. Phys. 35, L593–L598. CrossRef Google Scholar
Borwein, J. M. & Borwein, P. B. (1987). Pi and the AGM: a Study in Analytic Number Theory and Computational Complexity, Section 3.7, pp. 91–101. New York: Wiley. Google Scholar
Borwein, J. M., Glasser, M. L., McPhedran, R. C., Wan, J. G. & Zucker, I. J. (2013). Lattice Sums Then and Now. Cambridge: Cambridge University Press. Google Scholar
Brown, K. (2018). NonPeriodic Tilings With Nfold Symmetry, https://www.mathpages.com/home/kmath539/kmath539.htm. Google Scholar
Burdik, Č., Frougny, Ch., Gazeau, J. P. & Krejcar, R. (1998). J. Phys. A Math. Gen. 31, 6449–6472. CrossRef Google Scholar
Bursill, L. A. (1990). Int. J. Mod. Phys. B, 04, 2197–2216. CrossRef Google Scholar
Bursill, L. A., Lin, F. J. & Fan, X. (1987). Mod. Phys. Lett. B, 01, 195–206. CrossRef Google Scholar
Chamizo, F. & Raboso, D. (2015). Indagationes Math. 26, 723–735. CrossRef Google Scholar
Davis, P. J. (1993). Spirals – From Theodorus to Chaos. Wellesley: A. K. Peters. Google Scholar
Dekking, F. M. & Mendèz France (1981). J. Reine Angew. Math. 329, 143–153. Google Scholar
Deshouilliers, J.M. (1985). In Elementary and Analytic Theory of Numbers, Banach Center Publications Vol. 17. Warsaw: PWN – Polish Scientific Publishers. Google Scholar
Dotera, T., Bekku, S. & Ziherl, P. (2017). Nat. Mater. 16, 987–992. CrossRef CAS PubMed Google Scholar
Dotera, T., Oshiro, T. & Ziherl, P. (2014). Nature, 506, 208–211. Web of Science CrossRef CAS PubMed Google Scholar
Eagles, T. H. (1885). Constructive Geometry of Plane Curves. London: Macmillan. Google Scholar
Ericksen, J. L. (2006). Math. Mech. Solids, 11, 3–22. CrossRef Google Scholar
Fan, X., Bursill, L. A. & Peng, J. L. (1988). Int. J. Mod. Phys. B, 02, 131–146. CrossRef Google Scholar
Fan, X., Peng, J. L. & Bursill, L. A. (1988). Int. J. Mod. Phys. B, 02, 121–129. CrossRef Google Scholar
Feng, Z. Q., Yang, Y. Q., Huang, B., Li, M. H., Chen, Y. X. & Ru, J. G. (2014). J. Alloys Compd. 583, 445–451. CrossRef CAS Google Scholar
Fontaine, A. & Hurley, S. (2006). Forum Geom. 6, 97–101. Google Scholar
Frettlöh, D., Harriss, E. & Gähler, F. (2020). Tilings Encyclopedia, https://tilings.math.unibielefeld.de/. Google Scholar
Fullman, R. L. & Wood, D. L. (1954). Acta Metall. 2, 188–193. CrossRef CAS Google Scholar
Graham, R. L., Knuth, D. L. & Patashnik, O. (1994). Concrete Mathematics – a Foundation for Computer Science, 2nd ed. New York: Addison–Wesley. Google Scholar
Grimmer, H. & Nespolo, M. (2006). Z. Kristallogr. 221, 28–50. Web of Science CrossRef CAS Google Scholar
Grünbaum, B. & Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman and Company. Google Scholar
Hammer, Ø. (2016). The Perfect Shape – Spiral Stories. New York, Berlin, Heidelberg: Springer. Google Scholar
Hornfeck, W. (2012). Acta Cryst. A68, 167–180. Web of Science CrossRef CAS IUCr Journals Google Scholar
Hornfeck, W. (2018). Acta Cryst. A74, 659–672. CrossRef IUCr Journals Google Scholar
Hornfeck, W., Kobold, R., Kolbe, M., Conrad, M. & Herlach, D. (2018). Nat. Commun. 9, 4054. CrossRef PubMed Google Scholar
Hornfeck, W., Kobold, R., Kolbe, M. & Herlach, D. (2014). arXiv:1410.2952. Google Scholar
Hyde, B. G. & Andersson, S. (1989). Inorganic Crystal Structures. New York: John Wiley and Sons. Google Scholar
Hyde, B. G., Andersson, S., Bakker, M., Plug, C. M. & O'Keeffe, M. (1979). Prog. Solid State Chem. 12, 273–327. CrossRef CAS Web of Science Google Scholar
Jean, R. V. (1995). Math. Biosci. 127, 181–206. CrossRef CAS PubMed Google Scholar
Jiang, W. J., Hei, K., Guo, Y. X. & Kuo, K. H. (1985). Philos. Mag. A, 52, L53–L58. CrossRef CAS Web of Science Google Scholar
Kunz, M. & Rothen, F. (1992). J. Phys. I Fr. 2, 2131–2172. CrossRef Google Scholar
Lawrence, J. D. (1972). A Catalog of Special Plane Curves. New York: Dover Publications. Google Scholar
Lehman, L. P., Xing, Y., Bieler, T. R. & Cotts, E. J. (2010). Acta Mater. 58, 3546–3556. CrossRef CAS Google Scholar
Lehmer, D. H. & Lehmer, E. (1979). Am. Math. Mon. 86, 725–733. CrossRef Google Scholar
Lehmer, D. H. & Lehmer, E. (1980). J. Reine Angew. Math. 318, 1–19. Google Scholar
Levitov, L. S. (1988). Europhys. Lett. 6, 517–522. CrossRef CAS Google Scholar
Lockwood, E. H. (1961). A Book of Curves. Cambridge: Cambridge University Press. Google Scholar
Loxton, J. H. (1983). Mathematika, 30, 153–163. CrossRef Google Scholar
Müller, U. (2017). Acta Cryst. B73, 443–452. Web of Science CrossRef IUCr Journals Google Scholar
Nakakura, J., Ziherl, P., Matsuzawa, J. & Dotera, T. (2019). Nat. Commun. 10, 4235. CrossRef PubMed Google Scholar
Nespolo, M. & Souvignier, B. (2015). J. Mineral. Petrol. Sci. 110, 157–165. Web of Science CrossRef CAS Google Scholar
Niizeki, K. (1989). J. Phys. A Math. Gen. 22, 193–204. CrossRef Google Scholar
PadrónNavarta, J. A., Barou, F. & Daneu, N. (2020). Acta Cryst. B76, 875–883. CrossRef IUCr Journals Google Scholar
Pennybacker, M. F., Shipman, P. D. & Newell, A. C. (2015). Physica D, 306, 48–81. CrossRef Google Scholar
Plouffe, S. (1998). Inverse Symbolic Calculator, https://wayback.cecm.sfu.ca/projects/ISC/ and https://web.archive.org/web/20050812010306/https://pi.lacim.uqam.ca/Eng/. Google Scholar
Quiquandon, M., Gratias, D., Sirindil, A. & Portier, R. (2016). Acta Cryst. A72, 55–61. Web of Science CrossRef IUCr Journals Google Scholar
Rivier, N. (1988). Mod. Phys. Lett. B, 02, 953–960. CrossRef Google Scholar
Rothen, F. & Koch, A.J. (1989a). J. Phys. Fr. 50, 1603–1621. CrossRef Google Scholar
Rothen, F. & Koch, A.J. (1989b). J. Phys. Fr. 50, 633–657. CrossRef Google Scholar
Rutter, J. W. (2000). Geometry of Curves. Boca Raton: Chapman & Hall/CRC. Google Scholar
Sadoc, J.F., Rivier, N. & Charvolin, J. (2012). Acta Cryst. A68, 470–483. CrossRef IUCr Journals Google Scholar
Schaad, T. P. & Stampfli, P. (2021). arXiv:2102.06046. Google Scholar
Shahani, A. J. & Voorhees, P. W. (2016). J. Mater. Res. 31, 2936–2947. CrossRef CAS Google Scholar
Shikin, E. V. (1995). Handbook and Atlas of Curves. Boca Raton: CRC Press. Google Scholar
Shubnikov, A. V. (1961). Sov. Phys. Cryst. 5, 469–476. Google Scholar
Sirindil, A., Kobold, R., Mompiou, F., LartigueKorinek, S., Perriere, L., Patriarche, G., Quiquandon, M. & Gratias, D. (2018). Acta Cryst. A74, 647–658. CrossRef IUCr Journals Google Scholar
Sirindil, A., Quiquandon, M. & Gratias, D. (2017). Acta Cryst. A73, 427–437. Web of Science CrossRef IUCr Journals Google Scholar
Sloane, N. J. A. (2018). Editor. The OnLine Encyclopedia of Integer Sequences, https://oeis.org/. Google Scholar
Stenberg, L. & Andersson, S. (1982). Z. Kristallogr. 158, 205–212. CrossRef CAS Google Scholar
Steurer, W. & Deloudi, S. (2009). Crystallography of Quasicrystals – Concepts, Methods and Structures, Springer Series in Materials Science, p. 126. New York, Berlin, Heidelberg: Springer. Google Scholar
Vogel, H. (1979). Math. Biosci. 44, 179–189. CrossRef Google Scholar
Wang, J., Zhang, B., He, Z. B., Wu, B. & Ma, X. L. (2016). Philos. Mag. 96, 2457–2467. Web of Science CrossRef CAS Google Scholar
Watson, J. D. & Crick, F. H. C. (1953). Nature, 171, 737–738. CrossRef PubMed CAS Web of Science Google Scholar
Yates, R. C. (1974). Curves and Their Properties. Reston, Virginia, USA: The National Council of Teachers of Mathematics. Google Scholar
Zucker, I. J. (2017). Symmetry, 9, 314. CrossRef Google Scholar
Zwikker, C. (1963). The Advanced Geometry of Plane Curves and Their Applications. New York: Dover Publications. Google Scholar
This article is published by the International Union of Crystallography. Prior permission is not required to reproduce short quotations, tables and figures from this article, provided the original authors and source are cited. For more information, click here.