Spatio-temporal Symmetry - Point Groups with Time Translations
Haricharan Padmanabhan, Maggie L. Kingsland, Jason M. Munro, Daniel B. Litvin, Venkatraman Gopalan
SSpatio-temporal Symmetry - Point Groups withTime Translations
Haricharan Padmanabhan , Maggie L. Kingsland , Jason M.Munro , Daniel B. Litvin , and Venkatraman Gopalan Department of Materials Science and Engineering, The Pennsylvania StateUniversity, University Park, PA 16801, USA; [email protected] (J.M.M.);[email protected] (V.G.) Department of Physics, University of South Florida, Tampa, FL 33620, USA;[email protected] Department of Physics, The Eberly College of Science, The Pennsylvania StateUniversity, Penn State Berks, PO Box 7009, Reading, PA 19610, USA; [email protected]
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Abstract
Spatial symmetries occur in combination with temporal symmetries ina wide range of physical systems in nature, including time-periodic quan-tum systems typically described by the Floquet formalism. In this context,groups formed by three-dimensional point group symmetry operations incombination with time translation operations are discussed in this work.The derivation of these ’spatio-temporal’ groups from conventional pointgroups and their irreducible representations is outlined, followed by acomplete listing. The groups are presented in a template similar to spacegroup operations, and are visualized using a modified version of conven-tional stereographic projections. Simple examples of physical processesthat simultaneously exhibit symmetry in space and time are identifiedand used to illustrate the application of spatio-temporal groups.
Spatial symmetries are ubiquitous in nature, ranging from atoms and moleculesto crystals and biological systems. The mathematical groups corresponding tothese symmetries, i. e. point groups and space groups, have been listed exhaus-tively and in great detail [1], and are indispensable in the study of matter. Inthis work, we consider the groups formed by spatial symmetries in combinationwith temporal symmetries.There are different ways in which temporal symmetries occur in physicalsystems. Most notably, strongly driven time-periodic quantum systems are1 a r X i v : . [ c ond - m a t . m t r l - s c i ] S e p ypically described by the Floquet formalism, which involves a time-periodicHamiltonian with its corresponding time-periodic solutions. Examples includeproblems that consider interaction of matter with strong electromagnetic fields,such as in high-harmonic generation of light [2]. Separately, an idea proposedby Wilczek et al. [3], [4] considers time-independent Hamiltonians that sponta-neously break time-translational symmetry, leading to the idea of ’time crystals’.This is a topic that has experienced a flurry of activity [5]–[7] and debate [8],[9] in recent years. In all these examples, with the addition of such temporalsymmetries to the spatial symmetries intrinsic to these systems, it is appropriateto describe them using symmetries that combine operations in space and time,i.e. spatio-temporal symmetries, rather than conventional spatial symmetryoperations. Much like in other areas of science, symmetry can be a power-ful tool in the study of these systems, such as in labeling Floquet states [10],deriving selection rules for high-harmonic generation spectra [11], [12], iden-tifying symmetry-protected topological Floquet phases [13], deriving the formof property tensors of space-time crystals, and so on. A systematic listing ofspatio-temporal groups would facilitate their use in such applications.This paper presents the derivation and listing of groups that combine spa-tial operations with time-translations. While spatio-temporal groups have beenpreviously listed [14], they have not found widespread use, perhaps becausethey have not been sufficiently comprehensible to the general reader, unlike thewidely used conventional point group and space group listings [1]. In this work,the listing of groups is reformulated with the intention to remedy this problem.This includes outlining a straightforward derivation using character tables ofconventional point groups, representing them using a template similar to spacegroup operations, and devising a simple way to represent these using standardcrystallographic diagrams. Furthermore, some simple examples are shown todemonstrate how these groups can be applied to physical systems.While the spatio-temporal groups corresponding to the 32 crystallographicpoint groups are listed explicitly, formulas are listed to generate the spatio-temporal groups corresponding to the non-crystallographic point groups. Define a point ( r | t ) and an operation ( R | τ ) in four-dimensional space-time,where r is the vector of three-dimensional spatial coordinates, t is the timecoordinate, R is a proper or improper rotation, and τ is a time translation, suchthat ( R | τ )( r | t ) = ( R r | t + τ ). The objective is to list all possible groups of suchoperations.Consider the group of all spatial symmetry operations in three-dimensions, E s (3), and the group of all time translations E t (1). The stated objective isequivalent to listing all the subgroups of the direct product E s (3) × E t (1). The isomorphism theorem [15] can be used to do this. Consider two groups A and B , and the direct product A × B . Choose two arbitrary normal subgroups(with different subgroups indexed by j ), a j and b j , of A and B respectively.2erforming a coset decomposition, A = a j + A a j + A a j + ... + A n a j B = b j + B b j + B b j + ... + B n b j . (1)The isomorphism theorem states that if the factor groups A / a j and B / b j areisomorphic to each other, X j = ( a j | b j ) { (1 | , ( A | B ) , ( A | B ) , ... } is a subgroupof A × B .The above derivation is illustrated with an example. Consider A = z = { , z , z , − z } , where n λ represents an anti-clockwise n-fold rotation about the λ -axis, and B = T = { ... − , , ... } , the set of all integral time translations,i.e. translations by integral multiples of unit time. Choosing the normal sub-groups a = z = { , z } and b = = { ... − , , ... } , it is easy to verifythat the factor groups z / z and T / are isomorphic to each other. Usingthe isomorphism theorem, X j = ( z | ) { (1 | , (4 z | } is a subgroup of thedirect product z × T . Rearranging the terms, this group can be written as(1 | ) { (1 | , (4 z | , (2 z | , (4 − z | } . Equivalent spatio-temporal groups are de-fined in this work as two groups that can be transformed from one to the othereither by rescaling the unit of time or by a proper spatial rotation. In the caseof X j , multiplying the unit of time by a factor of 2 gives the equivalent group(1 | T ) { (1 | , (4 z | ) , (2 z | , (4 − z | ) } . Repeating this process with the other nor-mal subgroups a j and b j , all the spatio-temporal subgroups X j of z × T maybe listed. In general, this process may still result in spatio-temporal groups thatcan be transformed from one to the other by a proper spatial rotation, whichare hence equivalent. In this work, one group is listed from each set of suchequivalent groups.An alternative but mathematically equivalent approach was shown by Boyleet al. [16]. Each one-dimensional irrep of a group G is associated with a uniquespatio-temporal subgroup of G × T , where T = { ... − , , ... } , the group ofintegral time translations. Given the i th one-dimensional irrep χ i of a group G ,each element g ij of the irrep is mapped to τ i ( g ij ) using g ij = exp(2 πiτ i ( g ij )),and the subgroup corresponding to this irrep can be listed as X i = { ( g i | τ i ( g i )) , ( g i | τ i ( g i )) , ( g i | τ i ( g i )) , ... } . (2)For example, consider the group z = { , z , z , − z } , and its second one-dimensional irrep, χ = { , − , , − } . The irrep can be expressed as χ = { , − , , − } = { exp (2 πi ( n )) , exp (2 πi ( n + 12 )) , exp (2 πi ( n )) , exp (2 πi ( n + 12 )) } = ⇒ τ = { n, n + 12 , n, n + 12 } = n { , , , } . (3)The subgroup corresponding to this irrep is then (1 | T ) { (1 | , (4 z | ) , (2 z | , (4 − z | ) } ,which is the same as that obtained using the isomorphism theorem, with the3ormal subgroups z and . Running through all the one-dimensional irrepsof a point group in this manner is equivalent to going through all the sets ofnormal subgroups.As stated by Boyle et al. [16], the above method can be used to generatea complete list of spatio-temporal point groups, which we do by going throughthe one-dimensional irreps of each of the 32 crystalline point groups, as wellas non-crystalline point groups. Those obtained from crystallographic pointgroups are explicitly listed, while formulas are listed for the spatio-temporalgroups obtained from non-crystallographic point groups. Note that the crys-tallographic spatio-temporal groups are necessarily obtained from finite spatialgroups, whereas the non-crystallographic spatio-temporal groups include bothfinite as well as infinite spatial groups. The groups are listed in sets according to the underlying point groups usedto generate them. Each group in the set is also assigned a serial number foridentification. Positions in these are separated by a period, and from left toright represent the underlying point group of the translation group, the numberof the group in the set of groups listed under a specific point group, and theoverall index of the group with respect to all possible time translation groups,respectively. For example, the group 11.3.29 in Table 1 refers to the twentyninth listed spatio-temporal group, which is the third group in the series ofgroups constructed from the eleventh point group (which is m ).The elements of the group are expressed as ( R | τ ), where R is a proper orimproper rotation, and τ is a time translation. The standard crystallographicnotation for spatial symmetry as found in the International Tables for Crystal-lography, Volume A [1] is used to express the proper and improper rotations.Further, non-zero time-translations are shown in blue.Because of the infinite nature of the time translation groups, they are listedusing the coset representatives of their decomposition with respect to the normalsubgroup of all integral time translations. For example, the group(1 | T ) { (1 | , (4 z |
12 ) , (2 z | , (4 − z |
12 ) } is given by listing its coset representatives with respect to T , which are { (1 | , (4 z |
12 ) , (2 z | , (4 − z |
12 ) } . Finally, a simple method is devised to help visualize these groups, by modi-fying conventional point group stereographic projections. Time translations areindicated in the diagram in a manner similar to how spatial translations per-pendicular to the plane are indicated in space-group diagrams. Non-zero timetranslations are visually indicated by numbers in blue. The spatial element as-sociated with a time translation is located within the plane by proximity, and4igure 1: The stereographic projection of the group 11.3.29 from Table 1, whichis (1 | T ) { (1 | , (4 z | ) , (2 z | ) , (4 − z | ) , (¯1 | , (¯4 − z | ) , ( m z | ) , ( ¯4 z | ). The timetranslations are indicated visually on the stereographic projection with bluefractions, and the + ( − ) superscript is used to specify that the time translationis associated with a spatial element above (below) the plane.outside the plane using the superscript. An example is shown in Fig. 1, and thesupplementary information contains stereographic projections for the remaininggroups as well, listed according to their serial numbers.In addition to the explicit listing of spatio-temporal groups obtained from the32 crystallographic point groups, formulas to generate spatio-temporal groupscorresponding to the non-crystallographic point groups have also been listedin Table 2. The method of listing and notation used is similar to that of thecrystallographic spatio-temporal groups. Spatio-temporal symmetries are seen in many complex physical systems, asoutlined in the introduction, but the simplest example of one is the ubiquitousclassical harmonic oscillator. Indeed, its temporal symmetry is simple enoughthat it is universally described using just spatial groups, as in molecular and lat-tice vibrations. It does however, exhibit non-trivial spatio-temporal symmetry.Consider an oscillator which is described by the equation x = x sin ( ωt + φ ),where ω = 2 π/τ , with τ being the time-period of oscillation. It is clearthat applying the spatial operation m x ( x → − x ) in combination with thetime translation operation τ ( t → t + τ ) leaves the equation invariant. Inother words, ( m x | τ ) is a symmetry of this system. Since it has no other non-trivial spatio-temporal symmetries, the spatio-temporal group that describesthis oscillation is (1 | τ ) { (1 | , ( m x | τ ) } . The equivalent spatio-temporal group(1 | T ) { (1 | , ( m x | ) } is obtained by dividing the unit of time by τ . This corre-sponds to the group 4 . . τ coupled to spatial symmetry operations.More complex harmonic systems can exhibit higher order symmetries. Aparticular physical example of this is the motion in k-space, of electrons in asolid, within the semiclassical model of electron dynamics [17]. Electrons under auniform magnetic field follow an orbit in k-space given by the intersection of theFermi surface with planes normal to the magnetic field. Depending on the sym-metry of the crystal and the direction of the magnetic field, these orbits can havedifferent symmetries. Consider the schematic orbit shown in Fig. 2. The opera-tion (4 z | τ ) is a symmetry of this motion, as described in the figure. Using thisas a generator, the spatio-temporal group (1 | T ) { (1 | , (4 z | τ ) , (2 z | τ ) , (4 − z | τ ) } is obtained, which is equivalent to the group 9 . .
23 in Table 1.Much like conventional spatial symmetry, spatio-temporal symmetry canalso be applied to derive properties of physical systems. For example, the se-lection rules for high-harmonic generation spectra can be derived using spatio-temporal symmetry. This has been shown in previous works [11], [12], [18], andthe simplest case of this process is outlined below. It can be shown [18] thatunder the influence of linearly polarized light E = E o cos ( ωt + φ ) x , (using thesemiclassical picture of light-matter interaction) the probability to generate the n th harmonic from a system in a Floquet state ψ (cid:15) = exp ( − i(cid:15)t ¯ h ) φ (cid:15) is given by σ ( n ) (cid:15) ∝ n |(cid:104)(cid:104) φ (cid:15) | ˆ µe − inωt | φ (cid:15) (cid:105)(cid:105)| , (4)where ˆ µ is the dipole moment operator, ω is the frequency of the incidentlight, and (cid:104)(cid:104) .. (cid:105)(cid:105) stands for integration over spatial variables and time. Notethat the electric field, and hence the Hamiltonian is invariant under the spatio-temporal symmetry operation ( m x | τ ). Hence, if there is no degeneracy inthe Floquet states, | φ (cid:15) (cid:105)(cid:105) are simultaneous eigenstates of the Floquet Hamil-tonian as well as elements of the group generated by the operation ( m x | τ ), i. e.(1 | τ ) { ( | ) , ( m x | τ ) } , which is equivalent to the group 4 . . m x | τ ) has eigenvalues of ±
1. Applying a6patio-temporal coordinate transformation ˆ M = ( m x | τ ) to the matrix elementin (4), n |(cid:104)(cid:104) φ (cid:15) | ˆ µe − inωt | φ (cid:15) (cid:105)(cid:105)| = n |(cid:104)(cid:104) ˆ M φ (cid:15) | ˆ M ˆ µe − inωt ˆ M − | ˆ M φ (cid:15) (cid:105)(cid:105)| (cid:54) = 0 (5)for a non-vanishing probability of obtaining the n th harmonic. Using theeigenvalues of ˆ M given by ˆ M | φ (cid:15) (cid:105)(cid:105) = ±| φ (cid:15) (cid:105)(cid:105) , it is inferred thatˆ µ ( x ) e − inωt = ˆ M ˆ µ ( x ) e − inωt ˆ M − = ˆ µ ( − x ) e ( − inω ( t + τ/ . (6)It is clear from (6) that the matrix element is non-vanishing only for odd n , resulting in the selection rule that under linearly polarized light, only theharmonics given by odd n are allowed in this Floquet state. Such selectionrules can be derived for more complex systems, such as crystals with non-trivialspatial symmetry, and elliptically polarized incident light. A parallel can be drawn between spatio-temporal groups and the distortionantisymmetry groups formulated by VanLeeuwen and Gopalan [19]. Certainphysical systems that can be described by spatio-temporal groups with timetranslations of τ can also be described by distortion groups obtained from thecorresponding point group. The simple harmonic oscillator is a simple exam-ple of this. By parameterizing the oscillation using λ , where − < λ < +1,and λ = 0 defines the equilibrium position, the distortion group of this sys-tem is m ∗ x = { , m ∗ x } , while the spatio-temporal group is (1 | T ) { (1 | , ( m x | τ ) } .Furthermore, borrowing from the concept of distortion symmetry, where λ is a’time-like’ coordinate rather than the time-coordinate itself, these point groupswith time-translations can be extended to include ’time-like’ translations, or’distortion’ translations. This opens up the possibility of describing a wholerange of problems using these groups, such as diffusion in materials, which maynot be periodic in time, but still exhibit symmetry in a ’time-like’ coordinate.Finally, in order to extend the scope of spatio-temporal symmetry, addi-tional groups can be derived using space groups, magnetic space groups, and byconsidering time reversal symmetry. These groups could describe spatial trans-lation and time reversal symmetries in addition to the point group operationsand time translations described by the listing in this work, pushing the possibleboundaries of application. 7 able 1 - List of crystallographic spatio-temporalgroups This table includes a complete listing of the crystallographic spatio-temporalpoint groups. The first column assigns a serial number for each spatio-temporalgroup. The second column specifies which point group the corresponding spatio-temporal group was derived from. The third column lists a coset representativeof each non-equivalent spatio-temporal point group, as described in the section’Listing’. Each element ( R | τ ) in this column consists of a spatial component R and a time translation τ , with non-zero time-translations shown in blue.For the point groups with three-fold and six-fold axial symmetry, the follow-ing convention is used - the axis ’1’ is chosen to be in the in-plane horizontaldirection, and the axis ’x’ makes an angle of − π with respect to it. The setsof axes 1, 2, and 3, and x, y, and xy are each generated by threefold rotationsabout the out-of-plane direction.Serial Number Point Group Spatio-temporal Group1.1.1 1 (1 | |
0) (¯1 | |
0) (¯1 | )3.1.4 2 (1 |
0) (2 y | |
0) (2 y | )4.1.6 m (1 |
0) ( m y | |
0) ( m y | )5.1.8 2/m (1 |
0) (2 y |
0) (¯1 |
0) ( m y | |
0) (2 y | ) (¯1 |
0) ( m y | )5.3.10 (1 |
0) (2 y |
0) (¯1 | ) ( m y | )5.4.11 (1 |
0) (2 y | ) (¯1 | ) ( m y | |
0) (2 z |
0) (2 y |
0) (2 x | |
0) (2 z |
0) (2 y | ) (2 x | )7.1.14 mm2 (1 |
0) (2 z |
0) ( m y |
0) ( m x | |
0) (2 z |
0) ( m y | ) ( m x | )7.3.16 (1 |
0) (2 z | ) ( m y |
0) ( m x | )8.1.17 mmm (1 |
0) (2 z |
0) (2 y |
0) (2 x |
0) (¯1 |
0) ( m z | m y |
0) ( m x | |
0) (2 z |
0) (2 y | ) (2 x | ) (¯1 |
0) ( m z | m y | ) ( m x | )8.3.19 (1 |
0) (2 z |
0) (2 y |
0) (2 x |
0) (¯1 | ) ( m z | )( m y | ) ( m x | )8.4.20 (1 |
0) (2 z |
0) (2 y | ) (2 x | ) (¯1 | ) ( m z | )( m y |
0) ( m x | |
0) (4 z |
0) (2 z |
0) (4 − z | |
0) (4 z | ) (2 z |
0) (4 − z | )9.3.23 (1 |
0) (4 z | ) (2 z | ) (4 − z | )10.1.24 ¯4 (1 |
0) ( ¯4 z |
0) (2 z |
0) (¯4 − z | |
0) ( ¯4 z | ) (2 z |
0) (¯4 − z | )80.3.26 (1 |
0) ( ¯4 z | ) (2 z | ) (¯4 − z | )11.1.27 4/m (1 |
0) (4 z |
0) (2 z |
0) (4 − z |
0) (¯1 |
0) (¯4 − z | m z |
0) ( ¯4 z | |
0) (4 z | ) (2 z |
0) (4 − z | ) (¯1 |
0) (¯4 − z | )( m z |
0) ( ¯4 z | )11.3.29 (1 |
0) (4 z | ) (2 z | ) (4 − z | ) (¯1 |
0) (¯4 − z | )( m z | ) ( ¯4 z | )11.4.30 (1 |
0) (4 z |
0) (2 z |
0) (4 − z |
0) (¯1 | ) (¯4 − z | )( m z | ) ( ¯4 z | )11.5.31 (1 |
0) (4 z | ) (2 z |
0) (4 − z | ) (¯1 | ) (¯4 − z | m z | ) ( ¯4 z | |
0) (4 z | ) (2 z | ) (4 − z | ) (¯1 | ) (¯4 − z | )( m z |
0) ( ¯4 z | )12.1.33 422 (1 |
0) (4 z |
0) (4 − z |
0) (2 z |
0) (2 y |
0) (2 x | xy |
0) (2 − xy | |
0) (4 z |
0) (4 − z |
0) (2 z |
0) (2 y | ) (2 x | )(2 xy | ) (2 − xy | )12.3.35 (1 |
0) (4 z | ) (4 − z | ) (2 z |
0) (2 y | ) (2 x | )(2 xy |
0) (2 − xy | |
0) (4 z |
0) (4 − z |
0) (2 z |
0) ( m x |
0) ( m y | m xy |
0) ( m − xy | |
0) (4 z |
0) (4 − z |
0) (2 z |
0) ( m x | ) ( m y | )( m xy | ) ( m − xy | )13.3.38 (1 |
0) (4 z | ) (4 − z | ) (2 z |
0) ( m x | ) ( m y | )( m xy |
0) ( m − xy | |
0) ( ¯4 z |
0) (¯4 − z |
0) (2 z |
0) (2 y |
0) (2 x | m xy |
0) ( m − xy | |
0) ( ¯4 z |
0) (¯4 − z |
0) (2 z |
0) (2 y | ) (2 x | )( m xy | ) ( m − xy | )14.3.41 (1 |
0) ( ¯4 z | ) (¯4 − z | ) (2 z |
0) (2 y |
0) (2 x | m xy | ) ( m − xy | )14.4.42 (1 |
0) ( ¯4 z | ) (¯4 − z | ) (2 z |
0) (2 y | ) (2 x | )( m xy |
0) ( m − xy | |
0) (4 z |
0) (4 − z |
0) (2 z |
0) (2 y |
0) (2 x | xy |
0) (2 − xy |
0) (¯1 |
0) ( ¯4 z |
0) (¯4 − z |
0) ( m z | m y |
0) ( m x |
0) ( m xy |
0) ( m − xy | |
0) (4 z |
0) (4 − z |
0) (2 z |
0) (2 y | ) (2 x | )(2 xy | ) (2 − xy | ) (¯1 |
0) ( ¯4 z |
0) (¯4 − z |
0) ( m z | m y | ) ( m x | ) ( m xy | ) ( m − xy | )15.3.45 (1 |
0) (4 z | ) (4 − z | ) (2 z |
0) (2 y | ) (2 x | )(2 xy |
0) (2 − xy |
0) (¯1 |
0) ( ¯4 z | ) (¯4 − z | ) ( m z | m y | ) ( m x | ) ( m xy |
0) ( m − xy | |
0) (4 z |
0) (4 − z |
0) (2 z |
0) (2 y |
0) (2 x | xy |
0) (2 − xy |
0) (¯1 | ) ( ¯4 z | ) (¯4 − z | ) ( m z | )( m y | ) ( m x | ) ( m xy | ) ( m − xy | )95.5.47 (1 |
0) (4 z |
0) (4 − z |
0) (2 z |
0) (2 y | ) (2 x | )(2 xy | ) (2 − xy | ) (¯1 | ) ( ¯4 z | ) (¯4 − z | ) ( m z | )( m y |
0) ( m x |
0) ( m xy |
0) ( m − xy | |
0) (4 z | ) (4 − z | ) (2 z |
0) (2 y |
0) (2 x | xy | ) (2 − xy | ) (¯1 | ) ( ¯4 z |
0) (¯4 − z |
0) ( m z | )( m y | ) ( m x | ) ( m xy |
0) ( m − xy | |
0) (3 z |
0) (3 − z | |
0) (3 z | ) (3 − z | )17.1.51 ¯3 (1 |
0) (3 z |
0) (3 − z |
0) (¯1 |
0) (¯3 − z |
0) (¯3 z | |
0) (3 z | ) (3 − z | ) (¯1 |
0) (¯3 − z | ) (¯3 z | )17.3.53 (1 |
0) (3 z |
0) (3 − z |
0) (¯1 | ) (¯3 − z | ) (¯3 z | )17.4.54 (1 |
0) (3 z | ) (3 − z | ) (¯1 | ) (¯3 − z | ) (¯3 z | )18.1.55 32 (1 |
0) (3 z |
0) (3 − z |
0) (2 x |
0) (2 y |
0) (2 xy | |
0) (3 z |
0) (3 − z |
0) (2 x | ) (2 y | ) (2 xy | )19.1.57 3m (1 |
0) (3 z |
0) (3 − z |
0) ( m x |
0) ( m y |
0) ( m xy | |
0) (3 z |
0) (3 − z |
0) ( m x | ) ( m y | ) ( m xy | )20.1.59 ¯3m (1 |
0) (3 z |
0) (3 − z |
0) (2 |
0) (2 |
0) (2 | |
0) (¯3 z |
0) (¯3 − z |
0) ( m x |
0) ( m y |
0) ( m xy | |
0) (3 z |
0) (3 − z |
0) (2 | ) (2 | ) (2 | )(¯1 |
0) (¯3 z |
0) (¯3 − z |
0) ( m x | ) ( m y | ) ( m xy | )20.3.61 (1 |
0) (3 z |
0) (3 − z |
0) (2 |
0) (2 |
0) (2 | | ) (¯3 z | ) (¯3 − z | ) ( m x | ) ( m y | ) ( m xy | )20.4.62 (1 |
0) (3 z |
0) (3 − z |
0) (2 | ) (2 | ) (2 | )(¯1 | ) (¯3 z | ) (¯3 − z | ) ( m x |
0) ( m y |
0) ( m xy | |
0) (6 z |
0) (3 z |
0) (2 z |
0) (3 − z |
0) (6 − z | |
0) (6 z | ) (3 z |
0) (2 z | ) (3 − z |
0) (6 − z | )21.3.65 (1 |
0) (6 z | ) (3 z | ) (2 z | ) (3 − z | ) (6 − z | )21.4.66 (1 |
0) (6 z | ) (3 z | ) (2 z |
0) (3 − z | ) (6 − z | )22.1.67 ¯6 (1 |
0) (3 z |
0) (3 − z |
0) ( m z |
0) (¯6 z |
0) (¯6 − z | |
0) (3 z | ) (3 − z | ) ( m z |
0) (¯6 z | ) (¯6 − z | )22.3.69 (1 |
0) (3 z |
0) (3 − z |
0) ( m z | ) (¯6 z | ) (¯6 − z | )22.4.70 (1 |
0) (3 z | ) (3 − z | ) ( m z | ) (¯6 z | ) (¯6 − z | )23.1.71 6/m (1 |
0) (6 z |
0) (3 z |
0) (2 z |
0) (3 − z |
0) (6 − z | |
0) (¯3 − z |
0) (¯6 − z |
0) ( m z |
0) (¯6 z |
0) (¯3 z | |
0) (6 z | ) (3 z |
0) (2 z | ) (3 − z |
0) (6 − z | )(¯1 |
0) (¯3 − z |
0) (¯6 − z | ) ( m z | ) (¯6 z | ) (¯3 z | |
0) (6 z | ) (3 z | ) (2 z | ) (3 − z | ) (6 − z | )(¯1 |
0) (¯3 − z | ) (¯6 − z | ) ( m z | ) (¯6 z | ) (¯3 z | )23.4.74 (1 |
0) (6 z | ) (3 z | ) (2 z |
0) (3 − z | ) (6 − z | )(¯1 |
0) (¯3 − z | ) (¯6 − z | ) ( m z |
0) (¯6 z | ) (¯3 z | )23.5.75 (1 |
0) (6 z |
0) (3 z |
0) (2 z |
0) (3 − z |
0) (6 − z | | ) (¯3 − z | ) (¯6 − z | ) ( m z | ) (¯6 z | ) (¯3 z | )23.6.76 (1 |
0) (6 z | ) (3 z |
0) (2 z | ) (3 − z |
0) (6 − z | )(¯1 | ) (¯3 − z | ) (¯6 − z |
0) ( m z |
0) (¯6 z |
0) (¯3 z | )23.7.77 (1 |
0) (6 z | ) (3 z | ) (2 z | ) (3 − z | ) (6 − z | )10¯1 | ) (¯3 − z | ) (¯6 − z | ) ( m z |
0) (¯6 z | ) (¯3 z | )23.8.78 (1 |
0) (6 z | ) (3 z | ) (2 z |
0) (3 − z | ) (6 − z | )(¯1 | ) (¯3 − z | ) (¯6 − z | ) ( m z | ) (¯6 z | ) (¯3 z | )24.1.79 622 (1 |
0) (6 z |
0) (6 − z |
0) (3 z |
0) (3 − z |
0) (2 z | x |
0) (2 |
0) (2 xy |
0) (2 |
0) (2 y |
0) (2 | |
0) (6 z |
0) (6 − z |
0) (3 z |
0) (3 − z |
0) (2 z | x | ) (2 | ) (2 xy | ) (2 | ) (2 y | ) (2 | )24.3.81 (1 |
0) (6 z | ) (6 − z | ) (3 z |
0) (3 − z |
0) (2 z | )(2 x |
0) (2 | ) (2 xy |
0) (2 | ) (2 y |
0) (2 | )25.1.82 6mm (1 |
0) (6 z |
0) (6 − z |
0) (3 z |
0) (3 − z |
0) (2 z | m x |
0) ( m |
0) ( m xy |
0) ( m |
0) ( m y |
0) ( m | |
0) (6 z |
0) (6 − z |
0) (3 z |
0) (3 − z |
0) (2 z | m x | ) ( m | ) ( m xy | ) ( m | ) ( m y | ) ( m | )25.3.84 (1 |
0) (6 z | ) (6 − z | ) (3 z |
0) (3 − z |
0) (2 z | )( m x |
0) ( m | ) ( m xy |
0) ( m | ) ( m y |
0) ( m | )26.1.85 ¯62m (1 |
0) (3 z |
0) (3 z − |
0) (2 x |
0) (2 xy |
0) (2 y | m z |
0) (¯6 z |
0) (¯6 − z |
0) ( m |
0) ( m |
0) ( m | |
0) (3 z |
0) (3 z − |
0) (2 x | ) (2 xy | ) (2 y | )( m z |
0) (¯6 z |
0) (¯6 − z |
0) ( m | ) ( m | ) ( m | )26.3.87 (1 |
0) (3 z |
0) (3 z − |
0) (2 x |
0) (2 xy |
0) (2 y | m z | ) (¯6 z | ) (¯6 − z | ) ( m | ) ( m | ) ( m | )26.4.88 (1 |
0) (3 z |
0) (3 z − |
0) (2 x | ) (2 xy | ) (2 y | )( m z | ) (¯6 z | ) (¯6 − z | ) ( m |
0) ( m |
0) ( m | |
0) (6 z |
0) (6 − z |
0) (3 z |
0) (3 − z |
0) (2 z | x |
0) (2 |
0) (2 xy |
0) (2 |
0) (2 y |
0) (2 | |
0) (¯3 z |
0) (¯3 − z |
0) (¯6 z |
0) (¯6 − z |
0) ( m z | m x |
0) ( m |
0) ( m xy |
0) ( m |
0) ( m y |
0) ( m | |
0) (6 z |
0) (6 − z |
0) (3 z |
0) (3 − z |
0) (2 z | x | ) (2 | ) (2 xy | ) , (2 | ) (2 y | ) (2 | )(¯1 |
0) (¯3 z |
0) (¯3 − z |
0) (¯6 z |
0) (¯6 − z |
0) ( m z | m x | ) ( m | ) ( m xy | ) ( m | ) ( m y | ) ( m | )27.3.91 (1 |
0) (6 z | ) (6 − z | ) (3 z |
0) (3 − z |
0) (2 z | )(2 x |
0) (2 | ) (2 xy | , (2 | ) (2 y |
0) (2 | )(¯1 |
0) (¯3 z |
0) (¯3 − z |
0) (¯6 z | ) (¯6 − z | ) ( m z | )( m x |
0) ( m | ) ( m xy |
0) ( m | ) ( m y |
0) ( m | )27.4.92 (1 |
0) (6 z |
0) (6 − z |
0) (3 z |
0) (3 − z |
0) (2 z | x |
0) (2 |
0) (2 xy |
0) (2 |
0) (2 y |
0) (2 | | ) (¯3 z | ) (¯3 − z | ) (¯6 z | ) (¯6 − z | ) ( m z | )( m x | ) ( m | ) ( m xy | ) ( m | ) ( m y | ) ( m | )27.5.93 (1 |
0) (6 z |
0) (6 − z |
0) (3 z |
0) (3 − z |
0) (2 z | x | ) (2 | ) (2 xy | ) , (2 | ) (2 y | ) (2 | )(¯1 | ) (¯3 z | ) (¯3 − z | ) (¯6 z | ) (¯6 − z | ) ( m z | )( m x |
0) ( m |
0) ( m xy |
0) ( m |
0) ( m y |
0) ( m | |
0) (6 z | ) (6 − z | ) (3 z |
0) (3 − z |
0) (2 z | )(2 x | ) (2 |
0) (2 xy | ) (2 |
0) (2 y | ) (2 | | ) (¯3 z | ) (¯3 − z | ) (¯6 z | , (¯6 − z |
0) ( m z | m x |
0) ( m | ) ( m xy |
0) ( m | ) ( m y |
0) ( m | )28.1.95 23 (1 |
0) (3 xyz |
0) (3 xy − z |
0) (3 − xyz |
0) (3 x − yz | − xyz |
0) (3 − xy − z |
0) (3 − − xyz |
0) (3 − x − yz |
0) (2 z |
0) (2 y |
0) (2 x | |
0) (3 xyz | ) (3 xy − z | ) (3 − xyz | ) (3 x − yz | )(3 − xyz | )(3 − xy − z | ) (3 − − xyz | ) (3 − x − yz | ) (2 z |
0) (2 y |
0) (2 x | |
0) (3 xyz |
0) (3 xy − z |
0) (3 − xyz |
0) (3 x − yz | − xyz |
0) (3 − xy − z |
0) (3 − − xyz |
0) (3 − x − yz |
0) (2 z |
0) (2 y |
0) (2 x | |
0) (¯3 xyz |
0) (¯3 xy − z |
0) (¯3 − xyz |
0) (¯3 x − yz | − xyz |
0) (¯3 − xy − z |
0) (¯3 − − xyz |
0) (¯3 − x − yz |
0) ( m x |
0) ( m y | m z | |
0) (3 xyz | ) (3 xy − z | ) (3 − xyz | ) (3 x − yz | )(3 − xyz | ) (3 − xy − z | ) (3 − − xyz | ) (3 − x − yz | ) (2 z |
0) (2 y |
0) (2 x | |
0) (¯3 xyz | ) (¯3 xy − z | ) (¯3 − xyz | ) (¯3 x − yz | ) (¯3 − xyz | )(¯3 − xy − z | ) (¯3 − − xyz | ) (¯3 − x − yz | ) ( m x |
0) ( m y | m z | |
0) (3 xyz |
0) (3 xy − z |
0) (3 − xyz | , (3 x − yz | − xyz |
0) (3 − xy − z |
0) (3 − − xyz |
0) (3 − x − yz |
0) (2 z |
0) (2 y |
0) (2 x | | ) (¯3 xyz | ) (¯3 xy − z | ) (¯3 − xyz | ) (¯3 x − yz | )(¯3 − xyz | ) (¯3 − xy − z | ) (¯3 − − xyz | ) (¯3 − x − yz | ) ( m x | ) ( m y | )( m z | )29.4.100 (1 |
0) (3 xyz | ) (3 xy − z | ) (3 − xyz | ) (3 x − yz | )(3 − xyz | ) (3 − xy − z | ) (3 − − xyz | ) (3 − x − yz | ) (2 z |
0) (2 y |
0) (2 x | | ) (¯3 xyz | ) (¯3 xy − z | ) (¯3 − xyz | ) (¯3 x − yz | )(¯3 − xyz | ) (¯3 − xy − z | ) (¯3 − − xyz | ) (¯3 − x − yz | ) ( m x | ) ( m y | )( m z | )30.1.101 432 (1 |
0) (3 xyz |
0) (3 xy − z |
0) (3 − xyz |
0) (3 x − yz | − xyz |
0) (3 − xy − z |
0) (3 − − xyz |
0) (3 − x − yz |
0) (2 z |
0) (2 y |
0) (2 x | z |
0) (4 − z |
0) (4 y |
0) (4 − y |
0) (4 x |
0) (4 − x | − xy |
0) (2 xy |
0) (2 − yz |
0) (2 yz |
0) (2 xz |
0) (2 − xz | |
0) (3 xyz |
0) (3 xy − z |
0) (3 − xyz |
0) (3 x − yz | − xyz |
0) (3 − xy − z |
0) (3 − − xyz |
0) (3 − x − yz |
0) (2 z |
0) (2 y |
0) (2 x | z | ) (4 − z | ) (4 y | ) (4 − y | ) (4 x | ) (4 − x | )(2 − xy | ) (2 xy | ) (2 − yz | ) (2 yz | ) (2 xz | ) (2 − xz | )31.1.103 ¯43m (1 |
0) (3 xyz |
0) (3 xy − z |
0) (3 − xyz |
0) (3 x − yz | − xyz |
0) (3 − xy − z |
0) (3 − − xyz |
0) (3 − x − yz |
0) (2 z |
0) (2 y |
0) (2 x | z |
0) (¯4 − z |
0) (¯4 y |
0) (¯4 − y |
0) (¯4 x |
0) (¯4 − x | m − xy |
0) ( m xy |
0) ( m − yz |
0) ( m yz |
0) ( m xz |
0) ( m − xz | |
0) (3 xyz |
0) (3 xy − z |
0) (3 − xyz |
0) (3 x − yz | − xyz |
0) (3 − xy − z |
0) (3 − − xyz |
0) (3 − x − yz |
0) (2 z |
0) (2 y |
0) (2 x | z | ) (¯4 − z | ) (¯4 y | ) (¯4 − y | ) (¯4 x | ) (¯4 − x | )( m − xy | ) ( m xy | ) ( m − yz | ) ( m yz | ) ( m xz | ) ( m − xz | )32.1.105 m¯3m (1 |
0) (3 xyz |
0) (3 xy − z |
0) (3 − xyz |
0) (3 x − yz | − xyz |
0) (3 − xy − z |
0) (3 − − xyz |
0) (3 − x − yz |
0) (2 z |
0) (2 y |
0) (2 x | z |
0) (4 − z |
0) (4 y | , (4 − y |
0) (4 x |
0) (4 − x | − xy |
0) (2 xy |
0) (2 − yz |
0) (2 yz |
0) (2 xz |
0) (2 − xz | |
0) ( ¯4 z |
0) (¯4 − z |
0) (¯4 y |
0) (¯4 − y |
0) (¯4 x | − x |
0) (¯3 xyz |
0) (¯3 xy − z |
0) (¯3 − xyz |
0) (¯3 x − yz |
0) (¯3 − xyz | − xy − z |
0) (¯3 − − xyz |
0) (¯3 − x − yz |
0) ( m z |
0) ( m y |
0) ( m x | m − xy |
0) ( m xy |
0) ( m − yz |
0) ( m yz |
0) ( m xz |
0) ( m − xz | |
0) (3 xyz |
0) (3 xy − z |
0) (3 − xyz |
0) (3 x − yz | − xyz |
0) (3 − xy − z |
0) (3 − − xyz |
0) (3 − x − yz |
0) (2 z |
0) (2 y |
0) (2 x | z | ) (4 − z | ) (4 y | ) (4 − y | ) (4 x | ) (4 − x | )(2 − xy | ) (2 xy | ) (2 − yz | ) (2 yz | ) (2 xz | ) (2 − xz | )(¯1 |
0) ( ¯4 z | ) (¯4 − z | ) , (¯4 y | ) (¯4 − y | ) (¯4 x | )(¯4 − x | ) (¯3 xyz |
0) (¯3 xy − z |
0) (¯3 − xyz |
0) (¯3 x − yz |
0) (¯3 − xyz | − xy − z |
0) (¯3 − − xyz |
0) (¯3 − x − yz |
0) ( m z |
0) ( m y |
0) ( m x | m − xy | ) ( m xy | ) ( m − yz | ) ( m yz | ) ( m xz | ) ( m − xz | )32.3.107 (1 |
0) (3 xyz |
0) (3 xy − z |
0) (3 − xyz |
0) (3 x − yz | − xyz |
0) (3 − xy − z |
0) (3 − − xyz |
0) (3 − x − yz |
0) (2 z |
0) (2 y | x |
0) (4 z |
0) (4 − z |
0) (4 y |
0) (4 − y |
0) (4 x | − x |
0) (2 − xy |
0) (2 xy |
0) (2 − yz |
0) (2 yz |
0) (2 xz | − xz |
0) (¯1 | ) ( ¯4 z | ) (¯4 − z | ) (¯4 y | ) (¯4 − y | )(¯4 x | ) (¯4 − x | ) (¯3 xyz | ) (¯3 xy − z | )(¯3 − xyz | ) (¯3 x − yz | ) (¯3 − xyz | ) (¯3 − xy − z | ) (¯3 − − xyz | )(¯3 − x − yz | ) ( m z | ) ( m y | )( m x | ) ( m − xy | ) ( m xy | ) ( m − yz | ) ( m yz | ) ( m xz | )( m − xz | )32.4.108 (1 |
0) (3 xyz |
0) (3 xy − z |
0) (3 − xyz |
0) (3 x − yz | − xyz |
0) (3 − xy − z |
0) (3 − − xyz |
0) (3 − x − yz |
0) (2 z |
0) (2 y | x |
0) (4 z | ) (4 − z | ) (4 y | ) (4 − y | ) (4 x | )(4 − x | ) (2 − xy | ) (2 xy | ) (2 − yz | ) (2 yz | ) (2 xz | )(2 − xz | ) (¯1 | ) ( ¯4 z |
0) (¯4 − z |
0) (¯4 y |
0) (¯4 − y | x |
0) (¯4 − x |
0) (¯3 xyz | ) (¯3 xy − z | ) (¯3 − xyz | ) (¯3 x − yz | )(¯3 − xyz | ) (¯3 − xy − z | ) (¯3 − − xyz | ) (¯3 − x − yz | ) ( m z | ) ( m y | )( m x | ) ( m − xy |
0) ( m xy |
0) ( m − yz |
0) ( m yz |
0) ( m xz | m − xz | able 2 - Formulas to generate spatio-temporalpoint groups In this table, formulas used to generate both crystallographic as well as non-crystallographic spatio-temporal point groups are shown.The first and second columns specify in Sch¨onflies and International [1] nota-tion respectively, the point group from which the corresponding spatio-temporalgroup is derived. The third column shows a coset representative of each spatio-temporal group. In these groups, n denotes an n-fold rotation, or a rotationby angle φ = πn in radians. In the case of limiting (or infinite) point groups, ∞ is used to denote an ∞ -fold rotation, while φ denotes the correspondinginfinitesimal angle of rotation in radians.A shorthand notation is used to list the coset representative of each group.For example, the set of elements generated by an n-fold rotation with zerotime translation, that is { ( n | , ( n | , ( n | , ..., ( n n − | } , is represented by( n j | | j =0 , ,...,n − . In the case of limiting point groups, the index j is dropped.For example the set of elements generated by an ∞ -fold rotation with zero timetranslation is represented simply by ( ∞|
0) ... .Sch¨onflies Notation International Notation Spatio-temporal Group C n n ( n j | | j =0 , ,...,n − ( n j | jn ) | j =0 , ,...,n − ...( n j | j ( n − n ) | j =0 , ,...,n − C nv nmm (even n) ( n j | | j =0 , ,...,n − ( m x | ... ( m xy | ... ( n j | | j =0 , ,...,n − ( m x | ) ... ( m xy | ) ... ( n j | | j =0 , ,...,n − ( n j | ) | j =1 , ,...,n − ( m x | ... ( m xy | ) ... ( n j | | j =0 , ,...,n − ( n j | ) | j =1 , ,...,n − ( m x | ) ... ( m xy | ... nm (odd n) ( n j | | j =0 , ,..., ( n − ( m x | ... ( n j | | j =0 , ,..., ( n − ( m x | ) ...C nh n/m (even n) ( n j | | j =0 , ,...,n − ( n j m z | | j =0 , ,...,n ( n j | jn ) | j =0 , ,...,n − ( n j m z | jn ) | j =0 , ,...,n ...( n j | j ( n − n ) | j =0 , ,...,n − ( n j m z | j ( n − n ) | j =0 , ,...,n n (odd n) ( n j | | j =0 , ,...,n − ( n j | | j =0 , ,...,n − ( m x | ... ( n j | jn ) | j =0 , ,...,n − ( n j | jn ) | j =0 , ,...,n − ( m x | ... ..14( n j | jn ) | j =0 , ,...,n − ( n j | j ( n − n ) | j =0 , ,...,n − ( m x | ...S n n (even n) (2 n j | | j =0 , ,...,n − (2 n j | | j =0 , ,...,n − (2 n j | jn ) | j =0 , ,...,n − (2 n j | jn ) | j =0 , ,...,n − ...(2 n j | j ( n − n ) | j =0 , ,...,n − { (2 n j | j ( n − n ) | j =0 , ,...,n − n (odd n) (2 n j | | j =0 , ,...,n − (2 n j | jn ) | j =0 , ,...,n − ...(2 n j | j ( n − n ) | j =0 , ,...,n − D n n22 (even n) ( n j | | j =0 , ,...,n − (2 x | ... (2 xy | ... ( n j | | j =0 , ,...,n − (2 x | ) ... (2 xy | ) ... ( n j | | j =0 , ,...,n − ( n j | ) | j =1 , ,...,n − (2 x | ... (2 xy | ) ... ( n j | | j =0 , ,...,n − ( n j | ) | j =1 , ,...,n − (2 x | ) ... (2 xy | ... n2 (odd n) ( n j | | j =0 , ,...,n − (2 x | ... ( n j | | j =0 , ,...,n − (2 x | ) ...D nh n/mmm (even n) ( n j | | j =0 , ,...,n − (2 x | ... (2 xy | ... ( m z | n j m z | | j =1 , ,...,n − (2 x m z | ... (2 xy m z | ... ( n j | | j =0 , ,...,n − (2 x | ) ... (2 xy | ) ... ( m z | n j m z | | j =0 , ,...,n − (2 x m z | ) ... (2 xy m z | ) ... ( n j | | j =0 , ,...,n − ( n j | ) | j =1 , ,...,n − (2 x | ... (2 xy | ) ... ( m z |
0) ( n j m z | ) | j =0 , ,...,n − (2 x m z | ... (2 xy m z | ) ... ( n j | | j =0 , ,...,n − ( n j | ) | j =1 , ,...,n − (2 x | ) ... (2 xy | ... ( m z |
0) ( n j m z | ) | j =0 , ,...n − (2 x m z | ) ... (2 xy m z | ) ... ( n j | | j =0 , ,...n − (2 x | ... (2 xy | ) ... ( m z | )( n j m z | ) | j =0 , ,...n − (2 x m z | ) ... (2 xy m z | ) ... ( n j | | j =0 , ,...n − (2 x | ) ... (2 xy | ) ... ( m z | )( n j m z | ) | j =0 , ,...n − (2 x m z | ... (2 xy m z | ... ( n j | | j =0 , ,...,n − ( n j | ) | j =1 , ,...,n − (2 x | ... (2 xy | ) ... ( m z | ) ( n j m z | ) | j =0 , ,...n − (2 x m z | ) ... (2 xy m z | ... ( n j | | j =0 , ,...,n − ( n j | ) | j =1 , ,...,n − (2 x | ) ... (2 xy | ... ( m z | ) ( n j m z | | j =0 , ,...n − (2 x m z | ... (2 xy m z | ) ... n m2 (odd n) ( n j | | j =0 , ,...,n − (2 x | ... ( n j m z | | j =0 , ,...,n − x m z | ... ( n j | | j =0 , ,...,n − (2 x | ) ... ( n j m z | | j =0 , ,...n − (2 x m z | ) ... ( n j | | j =0 , ,...,n − (2 x | ... ( n j m z | ) | j =0 , ,...n − (2 x m z | ) ... ( n j | | j =0 , ,...,n − (2 x | ) ... ( n j m z | ) | j =0 , ,...n − (2 x m z | ...D nd n
2m (even n) (2 n j | | j =0 , ,... n ( m x | ... (2 xy | ... (2 n j | | j =0 , ,... n ( m x | ) ... (2 xy | ) ... (2 n j | | j =0 , ,... n ( m x | ... (2 xy | ) ... (2 n j | | j =0 , ,... n ( m x | ) ... (2 xy | ...n m (odd n) (2 n j | | j =0 , ,... n − ( m x | ... (2 xy | ... (2 n j | | j =0 , ,... n − ( m x | ) ... (2 xy | ) ... (2 n j | | j =0 , ,... n − ( m x | ... (2 xy | ) ... (2 n j | | j =0 , ,... n − ( m x | ) ... (2 xy | ...C ∞ ∞ (1 |
0) ( ∞| ... (1 |
0) ( ∞| φ π ) ... (1 |
0) ( ∞| − φ π ) ...C ∞ v ∞ mm (1 |
0) ( ∞| ... ( ∞| ... ( m x | ... (1 |
0) ( ∞| ... ( ∞| ... ( m x | ) ...C ∞ h ∞ /m (1 |
0) ( ∞| ... ( ∞| ... (1 |
0) ( ∞| φ π ) ... ( ∞| − φ π ) ... (1 |
0) ( ∞| − φ π ) ... ( ∞| φ π ) ... (1 |
0) ( ∞| φ π ) ... ( ∞| φ π ) ... (1 |
0) ( ∞| − φ π ) ... ( ∞| − φ π ) ...D ∞ ∞ |
0) ( ∞| ... (2 x | ... (1 |
0) ( ∞| ... (2 x | ) ...D ∞ h ∞ /mmm (1 |
0) ( ∞| ... ( m x | ... (1 |
0) ( ∞| ... (2 x | ... (1 |
0) ( ∞| ... ( m x | ) ... (1 |
0) ( ∞| ... (2 x | ) ... (1 |
0) ( ∞| ... ( m x | ... (1 | ) ( ∞| ) ... (2 x | ) ... (1 |
0) ( ∞| ... ( m x | ) ... (1 | ) ( ∞| ) ... (2 x | ...K ∞∞ (1 |
0) ( ∞| ...K h ∞∞ m (1 |
0) ( ∞| ... ( ∞| ... (1 | |
0) ( ∞| ... ( ∞| ) ... (1 | )16 Acknowledgements
The authors thank Latham Boyle and Kendrick Smith for sharing their noteson platonic orbits and the character table method to obtain spatio-temporalgroups. The authors also thank Jeremy Karl Cockcroft for giving permission tomodify and use the stereographic projections from his online course on powderdiffraction [20].
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