Hyperbolic Geometry Research Articles

Light rings in static and extremal black holes

In this work we establish some results concerning the existence of external light rings in extremal black hole spacetimes through the Newman-Penrose formalism. Specifically, assuming flat or (anti)--de Sitter asymptotics, staticity and the null energy condition, we show that a sufficient condition for the existence of external light rings is $R<2{K}_{G}$, where $R$, the curvature scalar of the spacetime and ${K}_{G}$, the Gaussian curvature of a spacelike two-surface, are both evaluated at the outermost event horizon, which can be endowed with spherical, hyperbolic or planar geometry. Our results are valid for any metric gravity theory where photons follow null geodesics.

Generation of Escher‐like spiral drawings in a modified hyperbolic space

Dutch graphic artist M.C. Escher created many famous drawings with a deep mathematical background based on wallpaper symmetry, hyperbolic geometry, spirals, and regular polyhedra. However, he did not attempt any spiral drawings in hyperbolic space. In this paper, we consider a modified hyperbolic geometry by removing the condition that a geodesic is orthogonal to the unit circle in the Poincaré model. We show that spiral symmetry and the similarity property exist in this modified geometry so that the creation of uncommon hyperbolic spiral drawings is possible. To this end, we first establish the theoretical foundation for the proposed method by deriving a contraction mapping and a rotation for constructing modified hyperbolic spiral tilings (MHSTs) and introduce symmetry groups to analyze the structure of MHSTs. Then, to embed a pre‐designed wallpaper template into the tiles, we derive a one‐to‐one mapping between a tile of MHST and a rectangle. Finally, we specify some technical implementation details and give a gallery of the resulting MHST drawings. Using existing wallpaper templates, the proposed method is able to generate a great variety of exotic Escher‐like drawings.

Topological Quantum Computation is Hyperbolic

We show that a topological quantum computer based on the evaluation of a Witten–Reshetikhin–Turaev TQFT invariant of knots can always be arranged so that the knot diagrams with which one computes are diagrams of hyperbolic knots. The diagrams can even be arranged to have additional nice properties, such as being alternating with minimal crossing number. Moreover, the reduction is polynomially uniform in the self-braiding exponent of the coloring object. Various complexity-theoretic hardness results regarding the calculation of quantum invariants of knots follow as corollaries. In particular, we argue that the hyperbolic geometry of knots is unlikely to be useful for topological quantum computation.

The Flow-geodesic Curvature and the Flow-evolute of Hyperbolic Plane Curves

We introduce a new type of curvature function and associated evolute curve for a given curve in the hyperboloid model of plane hyperbolic geometry. A special attention is devoted to the examples, particularly to a horocycle provided by the null Lorentzian rotation.

Fullerene and nanotube models in Bolyai - Lobachevsky hyperbolic geometry H3 on the 200th anniversary of its discovery

The Archimedean solid (5, 6, 6), where regular pentagon, hexagon and hexagon surround each vertex, so altogether 60 vertices (with carbon atoms for C60 fullerene). 12 pentagons and 20 hexagons bound this football polyhedron,

Dynamic Scale-free Graph Embedding via Self-attention

Graph neural networks (GNNs) have recently become increasingly popular due to their ability to learn node representations in complex graphs. Existing graph representation learning methods mainly target static graphs in Euclidean space, whereas many graphs in practical applications are dynamic and evolve continuously over time. Recent work has demonstrated that real-world graphs exhibit hierarchical properties. Unfortunately, many methods typically do not account for these latent hierarchical structures. In this work, we propose a dynamic network in hyperbolic space via self-attention, referred to as DynHAT, which leverages both the hyperbolic geometry and attention mechanism to learn node representations. More specifically, DynHAT captures hierarchical information by mapping the structural graph onto hyperbolic space, and time-varying dynamic evolution by flexibly weighting historical representations. Through extensive experiments on three real-world datasets, we show the superiority of our model in embedding dynamic graphs in hyperbolic space and competing methods in a link prediction task. In addition, our results show that embedding dynamic graphs in hyperbolic space has competitive performance when necessitating low dimensions.

Large nearest neighbour balls in hyperbolic stochastic geometry

Consider a stationary Poisson process in a d-dimensional hyperbolic space. For R>0 define the point process xi _R^{(k)} of exceedance heights over a suitable threshold of the hyperbolic volumes of kth nearest neighbour balls centred around the points of the Poisson process within a hyperbolic ball of radius R centred at a fixed point. The point process xi _R^{(k)} is compared to an inhomogeneous Poisson process on the real line with intensity function e^{-u} and point process convergence in the Kantorovich-Rubinstein distance is shown. From this, a quantitative limit theorem for the hyperbolic maximum kth nearest neighbour ball with a limiting Gumbel distribution is derived.

Topological Transitions Enhance the Performance of Twisted Thermophotovoltaic Systems

Twisted bilayer two-dimensional electronic systems give rise to many exotic phenomena and unveil a new frontier for the study of quantum materials. In photonics, twisted two-dimensional systems coupled via near-field interactions offer a platform to study localization and lasing. Here, we theoretically propose that twisting can be an unprecedented tool to tune the performance of near-field thermophotovoltaic systems. Remarkably, through twisting-induced photonic topological transitions, we achieve significant tuning of the thermophotovoltaic energy efficiency and power. The underlying mechanism is related to the change of the photonic isofrequency contours from elliptical to hyperbolic geometries in a setup where the hexagonal-boron-nitride metasurface serves as the heat source and the indium-antimonide $p$-$n$ junction serves as the cell. We find that a notably high energy efficiency, nearly 53 of the Carnot efficiency, can be achieved in our thermophotovoltaic system, while the output power can reach up to $1.1\ifmmode\times\else\texttimes\fi{}{10}^{4}\phantom{\rule{0.2em}{0ex}}\mathrm{W}/{\mathrm{m}}^{2}$ without requiring a large temperature difference between the source and the cell. Our results indicate the promising future of twisted near-field thermophotovoltaics and paves the way towards tunable, high-performance thermophotovoltaics and infrared detection.

Hyperbolic linear canonical transforms of quaternion signals and uncertainty

This paper is concerned with Linear Canonical Transforms (LCTs) associated with two-dimensional quaternion-valued signals defined in an open rectangle of the Euclidean plane endowed with a hyperbolic measure, which we call Quaternion Hyperbolic Linear Canonical Transforms (QHLCTs). These transforms are defined by replacing the Euclidean plane wave with a corresponding hyperbolic relativistic plane wave in one dimension multiplied by quadratic modulations in both the hyperbolic spatial and frequency domains, giving the hyperbolic counterpart of the Euclidean LCTs. We prove the fundamental properties of the partial QHLCTs and the right-sided QHLCT by employing hyperbolic geometry tools and establish main results such as the Riemann–Lebesgue Lemma, the Plancherel and Parseval Theorems, and inversion formulas. The analysis is carried out in terms of novel hyperbolic derivative and hyperbolic primitive concepts, which lead to the differentiation and integration properties of the QHLCTs. The results are applied to establish two quaternionic versions of the Heisenberg uncertainty principle for the right-sided QHLCT. These uncertainty principles prescribe a lower bound on the product of the effective widths of quaternion-valued signals in the hyperbolic spatial and frequency domains. It is shown that only hyperbolic Gaussian quaternion functions minimize the uncertainty relations.

The Geodesics for Poincaré’s Half-Plane: A Nonstandard Derivation

Constants of motion in Mechanics are usually inferred from groups of symmetry transformations of the given system, as, for example, a Lagrangian function that is time-invariant implies the conservation of energy. Here we wish to show that useful properties of a mechanical system can sometimes be deduced from a family of Noether-like transformations that are not inspired by any symmetry whatsoever. The sample system we concentrate on is the Lagrangian interpretation of Poincaré’s half plane of hyperbolic geometry, and the properties we will derive in a new way are the shape and the time parameterization of its geodesics.

A Fourfold Gold(I)-Aryl Macrocycle with Hyperbolic Geometry and its Reductive Elimination to a Carbon Nanoring Host.

An ethylene glycol-decorated [6]cyclo-meta-phenylene (CMP) macrocycle was synthesized and utilized as a subunit to construct a fourfold AuI 2 -aryl metallacycle with an overall square arrangement. The corners consist of rigid dinuclear gold(I) complexes previously known to form only triangular metallacycles. The interplay between the conformational flexibility of the [6]CMP macrocycle and the rigid dinuclear gold(I) moieties enable the square geometry, as revealed by single-crystal X-ray diffraction. The formation of the gold complex shows size-selectivity compared to an alternative route using platinum(II) corner motifs. Upon reductive elimination, an all-organic ether-decorated carbon nanoring was obtained. Investigation as a host for the complexation of large guest molecules with a suitable convex π-surfaces was accomplished using isothermal NMR binding titrations. Association constants for [6]cycloparaphenylene ([6]CPP), [7]CPP, C60 , and C70 were determined.

Ein vierfacher Gold(I)−Aryl‐Makrozyklus mit hyperbolischer Geometrie und dessen reduktive Eliminierung zu einem Kohlenstoffnanoring‐Wirt

AbstractEin Ethylenglykol‐dekorierter [6]Cyclo‐meta‐phenylen(CMP)‐Makrozyklus wurde hergestellt und als Untereinheit für den Aufbau eines vierfachen AuI2−Aryl‐Metallazyklus mit einer quadratischen Überstruktur genutzt. Die Ecken bestehen aus dinuklearen Gold(I)‐Komplexen, von denen bisher nur dreieckige Metallazyklen bekannt waren. Das Zusammenspiel von flexibler Konformation des [6]CMP‐Makrozyklus und der starren dinuklearen Gold(I)‐Einheit ermöglicht die quadratische Geometrie, die durch Kristallstrukturanalyse charakterisiert wurde. Die Bildung des Goldkomplex läuft im Vergleich zu einer alternativen Routen über Platin(II)‐Ecken größenselektiv ab. Durch reduktive Eliminierung wird ein vollständig organischer Ether‐dekorierter Kohlenstoffnanoring erhalten. Dieser wurde als Wirt zur Komplexierung von großen Gastmolekülen mit geeigneter konvexer π‐Oberfläche mittels isothermer NMR‐Bindungstitration untersucht. Die Assoziationskonstanten für [6]Cycloparaphenylen ([6]CPP), [7]CPP, C60 und C70 wurden bestimmt.

Ptolemy’s Theorem in the Relativistic Model of Analytic Hyperbolic Geometry

Ptolemy’s Theorem in Euclidean geometry, named after the Greek astronomer and mathematician Ptolemy, is well-known. By means of the relativistic model of hyperbolic geometry, we translate Ptolemy’s Theorem from Euclidean geometry into the hyperbolic geometry of Lobachevsky and Bolyai. The relativistic model of hyperbolic geometry is based on the Einstein addition of relativistically admissible velocities and, as such, it coincides with the well-known Beltrami–Klein ball model of hyperbolic geometry. The translation of Ptolemy’s Theorem from Euclidean geometry into hyperbolic geometry is achieved by means of hyperbolic trigonometry, called gyrotrigonometry, to which the relativistic model of analytic hyperbolic geometry gives rise.

Kähler–Einstein metrics near an isolated log-canonical singularity

Abstract We construct Kähler–Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity. Such metrics are complete near the singularity if the underlying space has complex dimension 2. We also establish a stability result for Kähler–Einstein metrics near certain types of isolated log canonical singularity. As application, for complex dimension 2 log canonical singularity, we show that any complete Kähler–Einstein metric of negative scalar curvature near an isolated log canonical (non-log terminal) singularity is smoothly asymptotically close to model Kähler–Einstein metrics from hyperbolic geometry.

Simpler foundations for the hyperbolic plane

Abstract H. L. Skala (1992) gave the first elegant first-order axiom system for hyperbolic geometry by replacing Menger’s axiom involving projectivities with the theorems of Pappus and Desargues for the hyperbolic plane. In so doing, Skala showed that hyperbolic geometry is incidence geometry. We improve upon Skala’s formulation by doing away with Pappus and Desargues altogether, by substituting for them two simpler axioms.

Caroline Series and Hyperbolic Geometry

Caroline Series and Hyperbolic Geometry

Measurement of individual color space using a luminous vector field.

This study is intended to measure the geometry of the observer's color space when viewing a computer screen and to define individual variations from these data. A CIE photometric standard observer assumes that the eye's spectral efficiency function is constant, and photometry measurements correspond to vectors with fixed directions. By definition, the standard observer decomposes color space into planar surfaces of constant luminance. Using heterochromatic photometry with a minimum motion stimulus, we systematically measure the direction of luminous vectors for many observers and many color points. During the measurement process, the background and stimulus modulation averages are fixed to the given points to ensure that the observer is in a fixed adaptation mode. Our measurements result in a vector field or set of vectors (x,v), where x is the point's color space position, and v is the observer's luminosity vector. To estimate surfaces from vector fields, two mathematical hypotheses were used: (1)that surfaces are quadratic or, equivalently, that the vector field model is affine, and (2)that the metric of surfaces is proportional to a visual origin. Across 24 observers, we found that vector fields are convergent and the corresponding surfaces are hyperbolic. The equation of the surface in the display's color space coordinate system, and in particular the axis of symmetry, varied systematically from individual to individual. A hyperbolic geometry is compatible with studies that emphasize a modification of the photometric vector with changing adaptations.

HyGGE: Hyperbolic graph attention network for reasoning over knowledge graphs

Recently, hyperbolic embedding has successfully demonstrated its superiority over Euclidean analogues in representing hierarchical data. As the scale-free network that usually exhibits rich hierarchical structures, knowledge graphs naturally become a field where hyperbolic embedding shows its talents. Furthermore, hyperbolic embedding is also an expected solution to comprehensively reproduce the semantic features and underlying structures of KGs in the embedding space, which will significantly optimize the interpretability and performance of embedding models. However, most of the several existing hyperbolic studies only individually learn semantic information indicated by triples individually, making the embedding space relatively one-sided and simplified. In addition, many issues that limit reasoning performance are still ignored and unresolved in the context of hyperbolic geometry, like the response to complex relations and relation patterns. Motivated by these concerns, we propose the hyperbolic embedding model for KG reasoning, HyGGE. It is based on an innovative hyperbolic graph attention network. Furthermore, the response to complex relations, which is a well-known problem that constrains reasoning performance is also discussed in HyGGE. On the one hand, the focus on neighborhood structures and relation features makes up for the singularity that the embedding space is completely induced by triples individually, thereby optimizing the expressiveness of the embedding space. On the other hand, they cooperate with the effect of hyperbolic geometry to capture hierarchical features contained in local structures, and thus giving the hyperbolic embedding a fuller play to its advantages. Extensive experiments have validated the effectiveness and advantages of HyGGE.

Categorical representation learning and RG flow operators for algorithmic classifiers

Following the earlier formalism of the categorical representation learning, we discuss the construction of the ‘RG-flow-based categorifier’. Borrowing ideas from the theory of renormalization group (RG) flows in quantum field theory, holographic duality, and hyperbolic geometry and combining them with neural ordinary differential equation techniques, we construct a new algorithmic natural language processing architecture, called the RG-flow categorifier or for short the RG categorifier, which is capable of data classification and generation in all layers. We apply our algorithmic platform to biomedical data sets and show its performance in the field of sequence-to-function mapping. In particular, we apply the RG categorifier to particular genomic sequences of flu viruses and show how our technology is capable of extracting the information from given genomic sequences, finding their hidden symmetries and dominant features, classifying them, and using the trained data to make a stochastic prediction of new plausible generated sequences associated with a new set of viruses which could avoid the human immune system.

Extrapolation over temporal knowledge graph via hyperbolic embedding

AbstractPredicting potential facts in the future, Temporal Knowledge Graph (TKG) extrapolation remains challenging because of the deep dependence between the temporal association and semantic patterns of facts. Intuitively, facts (events) that happened at different timestamps have different influences on future events, which can be attributed to a hierarchy among not only facts but also relevant entities. Therefore, it is crucial to pay more attention to important entities and events when forecasting the future. However, most existing methods focus on reasoning over temporally evolving facts or mining evolutional patterns from known facts, which may be affected by the diversity and variability of the evolution, and they might fail to attach importance to facts that matter. Hyperbolic geometry was proved to be effective in capturing hierarchical patterns among data, which is considered to be a solution for modelling hierarchical relations among facts. To this end, we propose ReTIN, a novel model integrating real‐time influence of historical facts for TKG reasoning based on hyperbolic geometry, which provides low‐dimensional embeddings to capture latent hierarchical structures and other rich semantic patterns of the existing TKG. Considering both real‐time and global features of TKG boosts the adaptation of ReTIN to the ever‐changing dynamics and inherent constraints. Extensive experiments on benchmarks demonstrate the superiority of ReTIN over various baselines. The ablation study further supports the value of exploiting temporal information.