Threefold Symmetry Research Articles

Engineering the Dipole Orientation and Symmetry Breaking with Mixed-Dimensional Heterostructures.

Engineering of the dipole and the symmetry of materials plays an important role in fundamental research and technical applications. Here, a novel morphological manipulation strategy to engineer the dipole orientation and symmetry of 2D layered materials by integrating them with 1D nanowires (NWs) is reported. This 2D InSe –1D AlGaAs NW heterostructure example shows that the in‐plane dipole moments in InSe can be engineered in the mixed‐dimensional heterostructure to significantly enhance linear and nonlinear optical responses (e.g., photoluminescence, Raman, and second harmonic generation) with an enhancement factor of up to ≈12. Further, the 1D NW can break the threefold rotational symmetry of 2D InSe, leading to a strong optical anisotropy of up to ≈65%. These results of engineering dipole orientation and symmetry breaking with the mixed‐dimensional heterostructures open a new path for photonic and optoelectronic applications.

Phonon-induced rotation of the electronic nematic director in superconducting Bi2Se3

The doped topological insulator ${A}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, with $A={\mathrm{Cu},\mathrm{Sr},\mathrm{Nb}}$, becomes a nematic superconductor below ${T}_{c}\ensuremath{\approx}3$--4 K. The associated electronic nematic director is described by an angle $\ensuremath{\alpha}$ and is experimentally manifested in the elliptical shape of the in-plane critical magnetic field ${H}_{c2}$. Because of the threefold rotational symmetry of the lattice, $\ensuremath{\alpha}$ is expected to align with one of three high-symmetry directions corresponding to the in-plane nearest-neighbor bonds, consistent with a ${Z}_{3}$-Potts nematic transition. Here, we show that the nematic coupling to the acoustic phonons, which makes the nematic correlation length tend to diverge along certain directions only, can fundamentally alter this phenomenology in trigonal lattices. Compared to hexagonal lattices, the former possesses a sixth independent elastic constant ${c}_{14}$ due to the fact that the in-plane shear strain doublet $({\ensuremath{\epsilon}}_{xx}\ensuremath{-}{\ensuremath{\epsilon}}_{yy},\ensuremath{-}2{\ensuremath{\epsilon}}_{xy})$ and the out-of-plane shear strain doublet $(2{\ensuremath{\epsilon}}_{yz},\ensuremath{-}2{\ensuremath{\epsilon}}_{zx})$ transform as the same irreducible representation. We find that, when ${c}_{14}$ overcomes a threshold value, which is expected to be the case in doped ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, the nematic director $\ensuremath{\alpha}$ unlocks from the high-symmetry directions due to the competition between the quadratic phonon-mediated interaction and the cubic nematic anharmonicity. This implies the breaking of the residual in-plane twofold rotational symmetry (${C}_{2x}$), resulting in a triclinic phase. We discuss the implications of these findings for the structure of nematic domains, for the shape of the in-plane ${H}_{c2}$ in ${A}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, and for the presence of nodes inside the superconducting state.

Partially Conserved Motifs Support an Alternating Access Model for the Human Mitochondrial NAD+ Transporter

SLC25A51 resides in the inner mitochondrial membrane and it is required for cellular respiration and the mitochondrial production of ATP. As a member of the mitochondrial carrier family (MCF), it controls NAD+ concentrations in the mitochondrial matrix by directly importing the oxidized dinucleotide. Nevertheless, the mechanism by which SLC25A51 transports NAD+ across the inner mitochondrial membrane is not known, and its primary sequence lacks key residues that have been otherwise attributed to shared mechanisms of MCF transporters. To investigate this discrepancy, we used Molecular Dynamics simulations to characterize both the outward‐open conformation of SLC25A51 where NAD+ is expected to bind and the inward‐open conformation. We tested predictions from the simulations using a genetically‐encoded fluorescent biosensor that can directly monitor changes in free NAD+ concentrations in the mitochondrial matrix.We found that SLC25A51 lacks a three‐fold symmetry of its pore that is a characteristic of many MCF transporters. Instead, the core of SLC25A51 adopts an oblong and flattened shape. We predict this may be critical for the recognition and discrimination of the oxidized dinucleotide from other structurally related molecules, including its reduced form. A similar asymmetry was observed at the salt‐bridge network on the matrix side, produced by the asymmetric formation of a single salt‐bridge between E139 and K236 that linked helices 3 and 5. This ionic interaction was consistently observed throughout the simulations and mutation of either residue ablated SLC25A51 activity. We identified three cardiolipin binding sites on the membrane‐facing exterior of SLC25A51 that permitted cardiolipin to bridge interactions between the helical chains. We showed that these sites impact SLC25A51 function when mutated. Our simulations further predicted an energetically‐comparable single salt‐bridge on the cytoplasmic side of the transporter that could stabilize a complementary matrix‐open state. Mutation of the predicted cytoplasmic salt bridge between K198 and E291, as well as introduction of an ectopic salt bridge to imbalance the potential energetics of transport, diminished SLC25A51 activity. Together, the data indicate that although SLC25A51 lacks may conserved residues compared to other MCF family members, it functions through an alternating access mechanism and its activity depends on cardiolipin.

Oyster (Crassostrea gigas) ferritin can efficiently reduce the damage of Pb2+in vivo by electrostatic attraction

Heavy metal ions pollution can cause damage to human body through food, so the development of a new kind of macromolecular that can remove heavy metal ions damage has a good application prospect. The possibilities of removing heavy metal ions from food system with ferritin were studied in this paper. In this study, oyster ferritin (GF1) can resistant to denaturation induced by Pb2+, Cd2+, Cr3+ and still maintains its basic structure. GF1 can bind more Pb2+, Cd2+, Cr3+ than recombinant human H-chain ferritin (rHuHF), especially Pb2+, and the findings suggest that each GF1 can capture about 51.42 Pb2+ in solution. The hard and soft acids and base also verifies that Pb2+ have stronger binding ability to the key amino acids at the outer end of the three-fold symmetry channel. Cells preprotected by ferritin could resistant to heavy metal ions. And GF1 can reduce the high blood lead in mice and may play a role in alleviating lead poisoning in vivo. All findings demonstrated that GF1 can be used as a novel macromolecule to bind heavy metal ions, and the study can broaden the research scope of ferritin in contaminated food systems.

Dimensional crossover and symmetry transformation of charge density waves in VSe2

Collective phenomena in solids can be sensitive to the dimensionality of the system; a case of special interest is VSe2, which shows a (r7 x r3) charge density wave (CDW) in the single layer with the three-fold symmetry in the normal phase spontaneously broken, in contrast to the (4 x 4) in-plane CDW in the bulk. Angle-resolved photoemission spectroscopy (ARPES) from VSe2 ranging from a single layer to the bulk reveals the evolution of the electronic structure including the Fermi surface contours and the CDW gap. At a thickness of two layers, the ARPES maps are already nearly bulklike, but the transition temperature TC for the (4 x 4) CDW is much higher than the bulk value of 110 K. These results can be understood as a result of dimensional crossover of phonon instability driven by a competition of nesting vectors. Our study provides key insights into the CDW mechanisms and offers a perspective in the search and control of emergent phases in quantum materials.

The Acetyltransferase RibT From Bacillus subtilis Affects in vivo Dynamics of the Multimeric Heavy Riboflavin Synthase Complex.

Flavins are ubiquitous molecules in life as they serve as important enzyme cofactors. In the Gram-positive, soil-dwelling bacterium Bacillus subtilis, four well-characterized gene products (the enzymes RibDG, RibE, RibAB, and RibH) catalyze the biosynthesis of riboflavin (RF) from guanosine-triphosphate (GTP) and ribulose-5-phosphate (R5P). The corresponding genes form an operon together with the gene ribT (ribDG-E-AB-H-T), wherein the function of this terminal gene remained enigmatic. RibT has been structurally characterized as a GCN5-like acetyltransferase (GNAT), however, with unidentified target molecules. Bacterial two-hybrid system revealed interactions between RibT, RibH, and RibE, forming the heavy RF synthase complex. Applying single particle tracking (SPT), we found that confined (sub)diffusion of RibT is largely dependent on interacting RibE and, to a lesser degree, on interacting RibH. By induced expression of otherwise low-expressed ribT from an ectopic locus, we observed a decrease in the subpopulation considered to represent capsids of the heavy RF synthase and an increase in the subpopulation thought to represent pentamers of RibH, pointing to a putative role for RibT in capsid disassembly. Complementarily, either deletion of ribT or mutation of a key residue from RibH (K29) suspected to be the substrate of RibT for acetylation leads to increased levels of subpopulations considered as capsids of RibH-mVenus (RibH-mV) in comparison to wild-type (wt)-like cells. Thus, we provide evidence for an indirect involvement of RibT in RF biosynthesis by a putative capsid disassembling mechanism considered to involve acetylation of RibH residue K29 at the three-fold symmetry axis of 60-mer capsids.

Spirals in Galaxies

Spirals in galaxies have long been thought to be caused by gravitational instability in the stellar component of the disk, but discerning the precise mechanism had proved elusive. Tidal interactions, and perhaps bars, may provoke some spiral responses, but spirals in many galaxies must be self-excited. We survey the relevant observational data and aspects of disk dynamical theory. The origin of the recurring spiral patterns in simulations of isolated disk galaxies has recently become clear, and it is likely that the mechanism is the same in real galaxies, although evidence to confirm this supposition is hard to obtain. As transient spiral activity increases random motion, the patterns must fade over time unless the disk also contains a dissipative gas component. Continuing spiral activity alters the structure of the disks in other ways: reducing metallicity gradients and flattening rotation curves are two of the most significant. The overwhelming majority of spirals in galaxies have two- or three-fold rotational symmetry, indicating that the cool, thin disk component is massive. Spirals in simulations of halo-dominated disks instead manifest many arms and, consequently, do not capture the expected full spiral-driven evolution. We conclude by identifying areas where further work is needed.

Ultrafast Pump–Probe Microscopy on 2D Transition Metal Dichalcogenides

Although microscopic techniques have been used to characterize transition metal dichalcogenides (TMDs), direct observation of charge carrier dynamics distribution in TMDs with diverse shapes remains unexplored. Herein, ultrafast pump–probe microscopy (UPPM) is employed to reveal the carrier dynamics distribution in molybdenum disulfide (MoS2) and tungsten disulfide (WS2) monolayer of four shapes: triangular (t‐MoS2), curved triangular (c‐MoS2), triangular (t‐WS2), and hexagonal (h‐WS2). Monitoring the photon transmission T at 1.55 eV after pumping with a photon energy of 3.1 eV, a negative ΔT/T occurs in t‐MoS2 and c‐MoS2, while a positive ΔT/T is detected in t‐WS2 and h‐WS2 after 3–7 ps time evolution. This distinctive behavior is attributed to deep/shallow defects below the conduction band minimum (CBM) in MoS2 and WS2. Spatial‐independent ΔT/T is observed in t‐MoS2 and t‐WS2, while the ΔT/T in c‐MoS2 has a rapid decay of photoexcited carriers at the vertices and curved edges. Additionally, a threefold symmetry of ΔT/T is revealed in h‐WS2, attributed to the dissimilar occupation of defect states near the h‐WS2 CBM. This work paves the way for examining charge carrier dynamics of various shapes of TMDs and provides a unique microscopic method for studying the charge carrier dynamics in emerging TMDs heterostructures.

Pseudogauge field driven acoustoelectric current in two-dimensional hexagonal Dirac materials

Using a diagrammatic scheme, we study the acoustoelectric effects in two-dimensional (2D) hexagonal Dirac materials due to the sound-induced pseudogauge field. We analyze both uniform and spatially dispersive currents in response to copropagating and counterpropagating sound waves, respectively. In addition to the longitudinal acoustoelectric current, we obtain an exotic transverse charge current flowing perpendicular to the sound propagation direction owing to the interplay of transverse and longitudinal gauge field components ${j}_{T}\ensuremath{\propto}{A}_{L}{A}_{T}^{*}$. In contrast to the almost isotropic directional profile of the longitudinal uniform current, a highly anisotropic transverse component ${j}_{T}\ensuremath{\sim}sin(6\ensuremath{\theta})$ is achieved that stems from the inherited threefold symmetry of the hexagonal lattice. However, both longitudinal and transverse parts of the dispersive current are predicted to be strongly anisotropic $\ensuremath{\sim}{sin}^{2}(3\ensuremath{\theta})$ or ${cos}^{2}(3\ensuremath{\theta})$. We quantitatively estimate the pseudogauge field contribution to the acoustoelectric current that can be probed in future experiments in graphene and other 2D hexagonal Dirac materials.

Chiral phonons in lattices with C4 symmetry

Chiral phonons were initially proposed and further verified experimentally in two-dimensional (2D) hexagonal crystal lattices. Many intriguing features brought about by chiral phonons are attributed to the pseudoangular momenta which are associated with the threefold rotational symmetry of hexagonal lattices. Here, we go beyond the hexagonal crystals and investigate the chiral phonons in systems with fourfold rotational symmetry. We clarify the symmetry requirements for the emergence of chiral phonons in both 2D square lattices and three-dimensional tetragonal lattices. For two dimensions, the realization of ${C}_{4}$ chiral phonons requires the breaking of time-reversal symmetry; while for three dimensions, they can exist on the ${C}_{4}$-invariant path in a chiral tetragonal lattice. These phonons have the advantage that they can be more readily coupled with optical transitions, which facilitates their experimental detection. We demonstrate our idea via model analysis and first-principles calculations of concrete materials, including the MnAs monolayer and the $\ensuremath{\alpha}$-cristobalite. Our work reveals chiral phonons beyond the hexagonal lattices and paves the way for further exploration of chiral phonon physics in square/tetragonal materials and metamaterials.

Synchronization of a genetic oscillator with the cell division cycle

Genetic circuits that control specific cellular functions are never fully insulated against influences of other parts of the cell. For example, they are subject to periodic modulation by the cell cycle through volume growth and gene doubling. To investigate possible effects of the cell cycle on oscillatory gene circuits dynamics, we modelled a simple synthetic genetic oscillator, the repressilator, and studied hallmarks of the resulting nonlinear dynamics. We found that the repressilator coupled to the cell cycle shows typical quasiperiodic motion with discrete Fourier spectra and windows in parameter space with synchronization of the two oscillators, with a devil’s stair case indicating the Arnold tongues of synchronization. In the case of identical parameters for the three genes of the repressilator and simultaneous gene duplication, we identify two classes of synchronization windows, symmetric and asymmetric, depending on whether the trajectories satisfy a discrete three-fold rotation symmetry, corresponding to cyclic permutation of the three genes. Unexpectedly changing the gene doubling time revealed that the width of the Arnold tongues is connected to that three-fold symmetry of the synchronization trajectories: non-simultaneous gene duplication increases the width of asymmetric synchronization regions, for some of them by an order of magnitude. By contrast, there is only a small or even a negative effect on the window size for symmetric synchronization. This observation points to a control mechanism of synchronization via the location of the genes on the chromosome.

Presence and absence of intrinsic magnetism in graphitic carbon nitrides designed through C\u2013N\u2013H building blocks

We use the first principle calculation to investigate the intrinsic magnetism of graphitic carbon nitrides (GCNs). By preserving three-fold symmetry, the GCN building blocks have been built out of different combinations between 6 components which are C atom, N atom, s-triazine, heptazine, heptazine with C atom at the center, and benzimidazole-like component. That results in 20 phases where 11 phases have been previously reported, and 9 phases are newly derived. The partial density of states and charge density have been analyzed through 20 phases to understand the origin of the presence and absence of intrinsic magnetism in GCNs. The intrinsic magnetism will be present not only because the GCNs comprising of radical components but also the pi-conjugated states are not the valence maximum to break the delocalization of unpaired electrons. The building blocks are also employed to study alloys between g-hbox {C}_3hbox {N}_4 and g-hbox {C}_4hbox {N}_3. The magnetization of the alloys has been found to be linearly dependent on a number of C atoms in the unit cell and some magnetic alloys are energetically favorable. Moreover, the intrinsic magnetism in GCNs can be promoted or demoted by passivating with a H atom depending on the passivated positions.

Second-harmonic generation in atomically thin 1T−TiSe2 and its possible origin from charge density wave transitions

Optical second-harmonic generation (SHG) can only occur in noncentrosymmetric crystals in the leading electric-dipole approximation. Transition metal dichalco genides with the $1T$ octahedral coordination is centrosymmetric, hence precluding SHG. Here we report the surprising observation of SHG in atomically thin $1T\text{\ensuremath{-}}\mathrm{Ti}{\mathrm{Se}}_{2}$, a prototypical charge density wave (CDW) material. Its intensity peaks in the trilayer, reaching 2% of that in monolayer $\mathrm{Mo}{\mathrm{S}}_{2}$, a two-dimensional crystal featuring pronounced nonlinear optical effect. The SHG signal exhibits a sixfold polarization dependence characteristic of a lattice with threefold rotational symmetry. It monotonically decreases with increasing temperature and persists at room temperature. Raman spectroscopy demonstrates that the CDW order is robust in atomically thin samples, with the transition temperature slightly lower than in the bulk. The SHG can be explained by the lattice distortion associated with the CDW as well as its fluctuation above the transition temperature. These results challenge the exciton insulator scenario and the chiral nature of the CDW, but support the band Jahn-Teller mechanism. Our work demonstrates SHG as a sensitive probe of the stacking order of the CDW in $1T\text{\ensuremath{-}}\mathrm{Ti}{\mathrm{Se}}_{2}$ and enriches the material base for nonlinear optical effects.

Spin coupling in Ag film growth on Sn/Ge(111)-[formula omitted

Silver has been deposited on the Sn/Ge(111)-(3×3)-R30° surface at room temperature. The Ag growth and resulting surface morphology have been investigated using scanning tunneling microscopy. The first layer of silver forms an interface with domains of two different phases. One structure consists of short atomic rows with three-fold symmetry, oriented in the directions of the 3×3 surface. These rows are separated by a distance equal to 3 and are found to fit a 23×3 unit cell. The other phase is a 3×3 honeycomb structure, oriented in the Ge(111) 1 × 1 directions. Atomic structural models for the two interface phases are proposed, based on two different spin arrangements of the Sn/Ge(111)-3×3 surface. The results highlight the topological coupling of the two interface faces. Both interface structures are preserved with additional silver deposition. The second layer of Ag grows with a bulk-like lattice thickness on top of both interfaces. Low-energy electron diffraction on a mostly two layer Ag film reveals that it consists of domains where Ag grows in different orientations. These domains are rotated 30° with respect to each other, and thus mirror the symmetry directions of the two interfacial phases.

Tunable circular dichroism through absorption in coupled optical modes of twisted triskelia nanostructures

We present a system consisting of two stacked chiral plasmonic nanoelements, so-called triskelia, that exhibits a high degree of circular dichroism. The optical modes arising from the interactions between the two elements are the main responsible for the dichroic signal. Their excitation in the absorption cross section is favored when the circular polarization of the light is opposite to the helicity of the system, so that an intense near-field distribution with 3D character is excited between the two triskelia, which in turn causes the dichroic response. Therefore, the stacking, in itself, provides a simple way to tune both the value of the circular dichroism, up to 60%, and its spectral distribution in the visible and near infrared range. We show how these interaction-driven modes can be controlled by finely tuning the distance and the relative twist angle between the triskelia, yielding maximum values of the dichroism at 20° and 100° for left- and right-handed circularly polarized light, respectively. Despite the three-fold symmetry of the elements, these two situations are not completely equivalent since the interplay between the handedness of the stack and the chirality of each single element breaks the symmetry between clockwise and anticlockwise rotation angles around 0°. This reveals the occurrence of clear helicity-dependent resonances. The proposed structure can be thus finely tuned to tailor the dichroic signal for applications at will, such as highly efficient helicity-sensitive surface spectroscopies or single-photon polarization detectors, among others.

Образование и рост фуллеренов и нанотрубок, имеющих Т-симметрию третьего порядка

According to the periodic system of fullerenes, all the fullerenes can be classified into the groups having different symmetry. It is supposed that the fullerenes of one and the same symmetry have similar properties. Before the appearance of the periodic system in 2017 the fullerenes were chosen for study at a random way that instead of ordering the results only increased information entropy. We have studied possible ways of generation and growing the fullerenes, which refer to the group having three-fold T-symmetry. Beginning with cyclopropane C3H6 producing clusters C6, we have obtained elementary fullerenes C6 as well as mini-fullerenes C12, which in their turn have produced the fullerenes from C18 to C48, perfect and imperfect, as well as nanotubes. The basic perfect fullerenes C18, C24, C30, C36, C42 and C48 have the ordinary three-fold symmetry, the intermediate ones having no such symmetry. Their imperfection is connected with extra ‘interstitial’ or carbon dimers, the dimers playing the role of defects. One can define the imperfect fullerenes with defects as the fullerenes having topological three-fold symmetry. We have calculated their shape and energies using Avogadro package and discussed possible reasons of their dependence on a fullerene size and shape. We have found that the fullerenes can be divided into two groups, alive that can grow, and dead which are impotent. Taking into account the results obtained early, allows us to make predictions that the dead fullerenes C24R, C32R, C40R and C48R of three-, four-, five- and six-fold symmetry have the most chance to be found experimentally with comparison of their isomers.

Design of continuously deformable topological phononic structures and their application to elastic waveguides

We design a topological phononic crystal in a thin plate, aiming at the development of an efficient elastic waveguide based on the concept of band topology in phonon dispersion. We adopt a snowflake-like structure for the crystal unit cell and search for the optimal structure that exhibits the topological phase transition of the three-dimensional phononic crystal by changing the unit cell structure. The bandgap width can be adjusted by varying the length of the branch of the snow side, and a topological phase transition occurs at the unit cell structure with three-fold rotational symmetry. Elastic waveguides based on edge modes appearing at interfaces between crystals with different band topologies are designed, and their transmission efficiency is evaluated numerically and experimentally. The results demonstrate robustness of the elastic wave propagation in thin plates. Moreover, we experimentally estimate a robustness of the topologically protected propagating states against structural inhomogeneities. The results obtained in this study pave the way for practical development of new surface acoustic wave technologies for high-frequency communication devices.

Epitaxial growth of bilayer Bi(110) on two-dimensional ferromagnetic Fe3GeTe2

Heterostructures of two-dimensional (2D) layered materials with selective compositions play an important role in creating novel functionalities. Effective interface coupling between 2D ferromagnet and electronic materials would enable the generation of exotic physical phenomena caused by intrinsic symmetry breaking and proximity effect at interfaces. Here, epitaxial growth of bilayer Bi(110) on 2D ferromagnetic Fe3GeTe2 (FGT) with large magnetic anisotropy has been reported. Bilayer Bi(110) islands are found to extend along fixed lattice directions of FGT. The six preferred orientations could be divided into two groups of three-fold symmetry axes with the difference approximately to 26°. Moreover, dI/dV measurements confirm the existence of interface coupling between bilayer Bi(110) and FGT. A variation of the energy gap at the edges of bilayer Bi(110) is also observed which is modulated by the interface coupling strengths associated with its buckled atomic structure. This system provides a good platform for further study of the exotic electronic properties of epitaxial Bi(110) on 2D ferromagnetic substrate and promotes potential applications in the field of spin devices.

Topological invariants of rotationally symmetric crystals

Recent formal classifications of crystalline topological insulators predict that the combination of time-reversal and rotational symmetry gives rise to topological invariants beyond the ones known for other lattice symmetries. Although the classification proves their existence, it does not indicate a way of calculating the values of those invariants. Here, we show that a specific set of concentric Wilson loops and line invariants yields the values of all topological invariants in two-dimensional systems with pure rotation symmetry in class AII. Examples of this analysis are given for specific models with two-fold and three-fold rotational symmetry. We find new invariants that relate to the presence of higher-order topology and corner charges.

Symmetric projection attractor reconstruction: Embedding in higher dimensions.

Symmetric Projection Attractor Reconstruction (SPAR) provides an intuitive visualization and simple quantification of the morphology and variability of approximately periodic signals. The original method takes a three-dimensional delay coordinate embedding of a signal and subsequently projects this phase space reconstruction to a two-dimensional image with threefold symmetry, providing a bounded visualization of the waveform. We present an extension of the original work to apply delay coordinate embedding in any dimension N≥3 while still deriving a two-dimensional output with some rotational symmetry property that provides a meaningful visualization of the higher dimensional attractor. A generalized result is developed for taking N≥3 delay coordinates from a continuous periodic signal, where we determine invariant subspaces of the phase space that provide a two-dimensional projection with the required rotational symmetry. The result in each subspace is shown to be equivalent to following each pair of coefficients of the trigonometric interpolating polynomial of N evenly spaced points as the signal is translated horizontally. Bounds on the mean and the frequency response of our new coordinates are derived. We demonstrate how this aids our understanding of the attractor properties and its relationship to the underlying waveform. Our generalized result is then extended to real, approximately periodic signals, where we demonstrate that the higher dimensional SPAR method provides information on subtle changes in different parts of the waveform morphology.