Irrational Angle Research Articles

Noninvertible axial symmetry in QED comes full circle

We revisit the possibility of constructing noninvertible topological defects for the axial symmetry of massless quantum electrodynamics (QED), despite its Adler-Bell-Jackiw anomaly. Dressing the defects with a topological quantum field theory with mixed U(1) and R-valued gauge fields, we are able to describe axial rotations of any rational or irrational angle. We confront our results with the existing proposals, in particular those that concern rational angles. We also provide the symmetry topological field theory that reproduces the action of all such symmetry defects of QED. Finally, we discuss how similar techniques allow the study of condensation defects for a R global symmetry. Published by the American Physical Society 2024

Rational approximation of golden angles: Accelerated reconstructions for radial MRI.

To develop a generic radial sampling scheme that combines the advantages of golden ratio sampling with simplicity of equidistant angular patterns. The irrational angle between consecutive spokes in golden ratio-based sampling schemes enables a flexible retrospective choice of temporal resolution, while preserving good coverage of k-space for each individual bin. Nevertheless, irrational increments prohibit precomputation of the point-spread function (PSF), can lead to numerical problems, and require more complex processing steps. To avoid these problems, a new sampling scheme based on a rational approximation of golden angles (RAGA) is developed. The theoretical properties of RAGA sampling are mathematically derived. Sidelobe-to-peak ratios (SPR) are numerically computed and compared to the corresponding golden ratio sampling schemes. The sampling scheme is implemented in the BART toolbox and in a radial gradient-echo sequence. Feasibility is shown for quantitative imaging in a phantom and a cardiac scan of a healthy volunteer. RAGA sampling can accurately approximate golden ratio sampling and has almost identical PSF and SPR. In contrast to golden ratio sampling, each frame can be reconstructed with the same equidistant trajectory using different sampling masks, and the angle of each acquired spoke can be encoded as a small index, which simplifies processing of the acquired data. RAGA sampling provides the advantages of golden ratio sampling while simplifying data processing, rendering it a valuable tool for dynamic and quantitative MRI.

Atomically precise inorganic helices with a programmable irrational twist.

Helicity in solids often arises from the precise ordering of cooperative intra- and intermolecular interactions unique to natural, organic or molecular systems. This exclusivity limited the realization of helicity and its ensuing properties in dense inorganic solids. Here we report that Ga atoms in GaSeI, a representative III-VI-VII one-dimensional (1D) van der Waals crystal, manifest the rare Boerdijk-Coxeter helix motif. This motif is a non-repeating geometric pattern characterized by 1D face-sharing tetrahedra whose adjacent vertices are rotated by an irrational angle. Using InSeI and GaSeI, we show that the modularity of 1D van der Waals lattices accommodates the systematic twisting of a periodic tetrahelix with a 41 screw axis in InSeI to an infinitely extending Boerdijk-Coxeter helix in GaSeI. GaSeI crystals are non-centrosymmetric, optically active and exfoliable to a single chain. These results present a materials platform towards understanding the origin and physical manifestation of aperiodic helicity in low-dimensional solids.

Tilings of the sphere by congruent quadrilaterals III: Edge combination a 3 b with general angles

Abstract Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This last one classifies the case of a 3 ⁢ b {a^{3}b} -quadrilaterals with some irrational angle: there are a sequence of 1-parameter families of quadrilaterals admitting 2-layer earth map tilings together with their basic flip modifications under extra condition, and 5 sporadic quadrilaterals each admitting a special tiling. A summary of the full classification is presented in the end.

Systems of rank one, explicit Rokhlin towers, and covering numbers

Rotations f_alpha of the one-dimensional torus (equipped with the normalized Lebesgue measure) by an irrational angle alpha are known to be dynamical systems of rank one. This is equivalent to the property that the covering number F^*(f_alpha ) of the dynamical system is one. In other words, there exists a basis B such that for arbitrarily high h, an arbitrarily large proportion of the unit torus can be covered by the Rokhlin tower (f_alpha ^kB)_{k=0}^{h-1}. Although B can be chosen with diameter smaller than any fixed varepsilon > 0, it is not always possible to take an interval for B but this can only be done when the partial quotients of alpha are unbounded. In the present paper, we ask what maximum proportion of the torus can be covered when B is the union of n_B in {mathbb {N}} disjoint intervals. This question has been answered in the case n_B =1 by Checkhova, and here we address the general situation. If n_B = 2, we give a precise formula for the maximum proportion. Furthermore, we show that for fixed alpha , the maximum proportion converges to 1 when n_B rightarrow infty . Explicit lower bounds can be given if alpha has constant partial quotients. Our approach is inspired by the construction involved in the proof of the Rokhlin lemma and furthermore makes use of the three gap theorem.

Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and small complexity, were found in a computer-assisted search. They are surprising since the rotations are given by rational matrices, and the proof of the open set condition usually requires integer data. We develop a classification of self-similar sets by symmetry class and algebraic numbers. Examples are given for various quadratic number fields.

A fast, noniterative approach for accelerated high-temporal resolution cine-CMR using dynamically interleaved streak removal in the power-spectral encoded domain with low-pass filtering (DISPEL) and modulo-prime spokes (MoPS).

To introduce a pair of accelerated non-Cartesian acquisition principles that when combined, exploit the periodicity of k-space acquisition, and thereby enable acquisition of high-temporal cine Cardiac Magnetic Resonance (CMR). The mathematical formulation of a noniterative, undersampled non-Cartesian cine acquisition and reconstruction is presented. First, a low-pass filtering step that exploits streaking artifact redundancy is provided (i.e., Dynamically Interleaved Streak removal in the Power-spectrum Encoded domain with Low-pass filtering [DISPEL]). Next, an effective radial acquisition for the DISPEL approach that exploits the property of prime numbers is described (i.e., Modulo-Prime Spoke [MoPS]). Both DISPEL and MoPS are examined using numerical simulation of a digital heart phantom to show that high-temporal cine-CMR is feasible without removing physiologic motion vs aperiodic interleaving using Golden Angles. The combined high-temporal cine approach is next examined in 11 healthy subjects for a time-volume curve assessment of left ventricular systolic and diastolic performance vs conventional Cartesian cine-CMR reference. The DISPEL method was first shown using simulation under different streak cycles to allow separation of undersampled radial streaking artifacts from physiologic motion with a sufficiently frequent streak-cycle interval. Radial interleaving with MoPS is next shown to allow interleaves with pseudo-Golden-Angle variants, and be more compatible with DISPEL against irrational and nonperiodic rotation angles, including the Golden-Angle-derived rotations. In the in vivo data, the proposed method showed no statistical difference in the systolic performance, while diastolic parameters sensitive to the cine's temporal resolution were statistically significant (P < 0.05 vs Cartesian cine). We demonstrate a high-temporal resolution cine-CMR using DISPEL and MoPS, whose streaking artifact was separated from physiologic motion.

Row-matching in pyramidal Mg2Sn precipitates in Mg–Sn–Zn alloys

Two orientation relationships (ORs) and morphologies with well-defined facets of Mg2Sn precipitates in a Mg–Sn–Zn alloy were recently reported. However, the irrational angles of these two ORs were described without explanation. This study reveals that a unique matching-row-on-terrace structure exists in a dominant facet corresponding to either of the observed ORs. The calculated ORs, the orientations of three facets associated with each OR and the terrace-ledge structures in the dominant facets all match well with observations. The present method of row-matching analysis is useful for understanding high-index faceted interfaces and their associated irrational ORs.

Rational Nonaxis Points on the Unit Circle Have Irrational Angles

(2016). Rational Nonaxis Points on the Unit Circle Have Irrational Angles. The American Mathematical Monthly: Vol. 123, No. 5, pp. 490-490.

Optimally Topologically Transitive Orbits in Discrete Dynamical Systems

Every orbit of a rigid rotation of a circle by a fixed irrational angle is dense. However, the apparent uniformity of the distribution of iterates after a finite number of iterations appears strikingly different for various choices of a rotation angle. Motivated by this observation, we introduce a scalar function on the orbits of a discrete dynamical system defined on a bounded metric space, called the linear limit density, which we interpret as a measure of an orbit's approach to density. Utilizing the three-distance theorem, we compute the exact value of the linear limit density of orbits of rigid rotations by irrational rotation angles with period-1 continued fraction expansions. We further show that any discrete dynamical system defined by an orientation-preserving diffeomorphism of the circle has an orbit with a larger linear limit density than any orbit of the rigid rotation by the golden number. Bernoulli shift maps acting on sequences over a finite alphabet provide another illustrative class of dynamical systems with dense orbits. Our study of the efficiency of an orbit's approach to density leads us to demonstrate the existence of a class of infinite sequences with finite linear limit density constructed by recursively extending finite de Bruijn sequences.

On the partitions with Sturmian-like refinements

In the dynamics of a rotation of the unit circle by an irrational angle $\alpha\in(0,1)$, we study the evolution of partitions whose atoms are finite unions of left-closed right-open intervals with endpoints lying on the past trajectory of the point $0$.Unlike the standard framework, we focus on partitions whose atoms are disconnected sets.We show that the refinements of these partitions eventually coincide with the refinements of a preimage of the Sturmian partition, which consists of two intervals $[0,1-\alpha)$ and $[1-\alpha,1)$.In particular, the refinements of the partitions eventually consist of connected sets, i.e., intervals. We reformulate this result in terms of Sturmian subshifts: we show that for every non-trivial factor mapping from a one-sided Sturmian subshift, satisfying a mild technical assumption, the sliding block code of sufficiently large length induced by the mapping is injective.

A Small Surrogate for the Golden Angle in Time-Resolved Radial MRI Based on Generalized Fibonacci Sequences.

In golden angle radial magnetic resonance imaging a constant azimuthal radial profile spacing of 111.246...(°) guarantees a nearly uniform azimuthal profile distribution in k-space for an arbitrary number of radial profiles. Even though this profile order is advantageous for various real-time imaging methods, in combination with balanced steady-state free precession (SSFP) sequences the large azimuthal angle increment may lead to strong image artifacts, due to the varying eddy currents introduced by the rapidly switching gradient scheme. Based on a generalized Fibonacci sequence, a new sequence of smaller irrational angles is introduced ( 49.750...(°), 32.039...(°), 27.198...(°), 23.628...(°), ... ). The subsequent profile orders guarantee the same sampling efficiency as the golden angle if at least a minimum number of radial profiles is used for reconstruction. The suggested angular increments are applied for dynamic imaging of the heart and the temporomandibular joint. It is shown that for balanced SSFP sequences, trajectories using the smaller golden angle surrogates strongly reduce the image artifacts, while the free retrospective choice of the reconstruction window width is maintained.

Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model

One-dimensional quasiperiodic systems, such as the Harper model and the Fibonacci quasicrystal, have long been the focus of extensive theoretical and experimental research. Recently, the Harper model was found to be topologically nontrivial. Here, we derive a general model that embodies a continuous deformation between these seemingly unrelated models. We show that this deformation does not close any bulk gaps, and thus prove that these models are in fact topologically equivalent. Remarkably, they are equivalent regardless of whether the quasiperiodicity appears as an on-site or hopping modulation. This proves that these different models share the same boundary phenomena and explains past measurements. We generalize this equivalence to any Fibonacci-like quasicrystal, i.e., a cut and project in any irrational angle.

Analytical light scattering and orbital angular momentum spectra of arbitrary Vogel spirals

In this paper, we present a general analytical model for light scattering by arbitrary Vogel spiral arrays of circular apertures illuminated at normal incidence. This model suffices to unveil the fundamental mathematical structure of their complex Fraunhofer diffraction patterns and enables the engineering of optical beams carrying multiple values of orbital angular momentum (OAM). By performing analytical Fourier-Hankel decomposition of spiral arrays and far field patterns, we rigorously demonstrate the ability to encode specific numerical sequences onto the OAM values of diffracted optical beams. In particular, we show that these OAM values are determined by the rational approximations (i.e., the convergents) of the continued fraction expansions of the irrational angles utilized to generate Vogel spirals. These findings open novel and exciting opportunities for the manipulation of complex OAM spectra using dielectric and plasmonic aperiodic spiral arrays for a number of emerging engineering applications in singular optics, secure communication, optical cryptography, and optical sensing.

Sturmian Sequences and Invertible Substitutions

Combinatorics It is known that a Sturmian sequence S can be defined as a coding of the orbit of rho (called the intercept of S) under a rotation of irrational angle alpha (called the slope). On the other hand, a fixed point of an invertible substitution is Sturmian. Naturally, there are two interrelated questions: (1) Given an invertible substitution, we know that its fixed point is Sturmian. What is the slope and intercept? (2) Which kind of Sturmian sequences can be fixed by certain non-trivial invertible substitutions? In this paper we give a unified treatment to the two questions. We remark that though the results are known, our proof is very elementary and concise.

A Decomposition of the Rotation of Circle

The paper concerns the dynamics-related properties of the rotation map of a circle (rotation of the plane). A self-similar structure of orbits of the rotation map is established. That is, a possibility of decomposition of orbits of a given rotation map into a finite set of orbits of other such maps is proved—it is shown that every orbit of iterates of the rotation of circle on irrational angle, after linear re-scaling of its argument can be represented as a finite set of such orbits situated on another circles. A pointwise self-similarity of classical trigonometric system is established and an application to Fourier expansion, which emphasizes a possibility of shifting of signals with respect to time, is presented. The free mechanical motion is also considered. A special dynamical spectrum of frequencies or speeds, associated with a given uniform circular or rectilinear motion, is defined. We prove that an appropriate fragmentation of time axis yields a decomposition of a given orbit of the free continuous-time motion into a set of such orbits propagating in new time and such decomposition is consistent with the decomposition of the per time unit discrete motion. Particularly, our theorems assert that due to a piecewise-linear transform of spatial and time variables the rectilinear rays change their direction.

On irrational lattice angles

The aim of this paper is to generalize the notions of ordinary and expanded lattice angles and their sums studied in the work [7] by the author to the case of angles with lattice vertices but not necessarily with lattice rays. We find normal forms and extend the definition of lattice sums to a certain special case of such angles.

Rigidity results for quasiperiodic &lt;em&gt;SL(2, R)&lt;/em&gt;-cocycles

In this paper we introduce a new technique that allows us to investigate reducibility properties ofsmooth SL(2, R)-cocycles over irrational rotations of the circle beyond the usual Diophantine conditions on these rotations. For any given irrational angle on the base, we show that if the cocycle hasbounded fibered products and if its fibered rotation number belongs to aset of full measure $\Sigma(\a)$, then the matrix map can be perturbed inthe $C^\infty$ topology to yield a $C^\infty$-reducible cocycle. Moreover,the cocycle itself is almost rotations-reducible in the sense thatit can be conjugated arbitrarily close to a cocycle of rotations. If therotation on the circle is of super-Liouville type, the same results hold ifinstead of having bounded products we only assume that the cocycle is$L^2$-conjugate to a cocycle of rotations. When the base rotation is Diophantine, we show that if the cocycle is $L^2$-conjugate to a cocycle of rotations and if its fibered rotation number belongs to a set of full measure, then it is $C^\infty$-reducible. This extends a result proven in [5]. As an application, given any smooth SL(2, R)-cocycle over a irrational rotation of the circle, we show that it is possible to perturb the matrix map in the $C^\infty$ topology in such a way that the upper Lyapunov exponent becomes strictly positive. The latter result is generalized, based on different techniques, by Avila in [1] to quasiperiodic SL(2, R)-cocycles over higher-dimensional tori. Also, in the course of the paper we give a quantitative version of a theorem by L. H. Eliasson, a proof of which is given in the Appendix. This motivates the introduction of a quite general KAM scheme allowing to treat bigger losses of derivatives for which we prove convergence.

On tight projective designs

It is shown that among all tight designs in \({\mathbb {FP}^n \neq \mathbb {RP}^1}\) , where \({\mathbb {F}}\) is \({\mathbb {R}}\) or \({\mathbb {C}}\) , or \({\mathbb {H}}\) (quaternions), only 5-designs in \({\mathbb {CP}^1}\) (Lyubich, Shatalora Geom Dedicata 86: 169–178, 2001) have irrational angle set. This is the only case of equal ranks of the first and the last irreducible idempotent in the corresponding Bose-Mesner algebra.

Dynamical Systems and Irrational Angle Construction by Paper-Folding

In Variations on a theme in paper folding, Burkard Polster [6] discusses an itera tive construction due to Hilton and Pedersen [3] for approximating any acute rational angle (rational multiple of n radians) to any desired accuracy by folding a strip of paper. Hilton and Pedersen, motivated by the construction of regular polygons and star-polygons, focus on the construction of rational angles. The construction employs a periodic sequence composed of two types of folding moves, guided by number theoretic properties of the angle's fractional representation (in fact, analysis of the construction yields new results in number theory?see [4], [1], and [2]). This note generalizes this construction to any acute angle, rational or irrational. For irrational angles, the number-theoretic prescription must obviously be replaced by something new; we show how to derive the appropriate aperiodic sequence of folding moves, via a connection between the folding construction and a simple dynamical system. Hilton and Pedersen's construction recursively generates a sequence of angles a0, otx, ... by a pair of folding operations on a strip of paper which we term Folds and Switchfolds. The angles ak appear between an edge of the strip and a crease created by folding. The Fold operation generates ak+x by bisecting ak (Figure 1):