Hyperbolic Plane Research Articles

Using the Intermediate Value Theorem to Circumscribe Hyperbolic Triangles

Summary We prove that every triangle in the hyperbolic plane can be circumscribed by an ellipse. The Intermediate Value Theorem is a key ingredient in the proof.

Octonionic Planes and Real Forms of $G_2$, $F_4$ and $E_6$

In this work we present a useful way to introduce the octonionic projective and hyperbolic plane $\mathbb{O}P^{2}$ through the use of Veronese vectors. Then we focus on their relation with the exceptional Jordan algebra $\mathfrak{J}_{3}^{\mathbb{O}}$ and show that the Veronese vectors are the rank-one elements of the algebra. We then study groups of motions over the octonionic plane recovering all real forms of $\text{G}_{2}$, $\text{F}_{4}$ and $\text{E}_{6}$ groups and finally give a classification of all octonionic and split-octonionic planes as symmetric spaces.

Growth Competitions on Spherically Symmetric Riemannian Manifolds

Abstract We propose a model for a growth competition between two subsets of a Riemannian manifold. The sets grow at two different rates, avoiding each other. It is shown that if the competition takes place on a surface which is rotationally symmetric about the starting point of the slower set, then if the surface is conformally equivalent to the Euclidean plane, the slower set remains in a bounded region, while if the surface is nonpositively curved and conformally equivalent to the hyperbolic plane, both sets may keep growing indefinitely.

On the distribution of lattice points on hyperbolic circles

We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane $\mathbb{H}$. The angles of lattice points arising from the orbit of the modular group $PSL_{2}(\mathbb{Z})$, and lying on hyperbolic circles, are shown to be equidistributed for generic radii. However, the angles fail to equidistribute on a thin set of exceptional radii, even in the presence of growing multiplicity. Surprisingly, the distribution of angles on hyperbolic circles turns out to be related to the angular distribution of $\mathbb{Z}^2$-lattice points (with certain parity conditions) lying on circles in $\mathbb{R}^2$, along a thin subsequence of radii. A notable difference is that measures in the hyperbolic setting can break symmetry - on very thin subsequences they are not invariant under rotation by $\frac{\pi}{2}$, unlike the Euclidean setting where all measures have this invariance property.

The modularity of random graphs on the hyperbolic plane

Abstract Modularity is a quantity which has been introduced in the context of complex networks in order to quantify how close a network is to an ideal modular network in which the nodes form small interconnected communities that are joined together with relatively few edges. In this article, we consider this quantity on a probabilistic model of complex networks introduced by Krioukov et al. (2010, Phys. Rev. E, 82, 036106). This model views a complex network as an expression of hidden popularity hierarchies (i.e. nodes higher up in the hierarchies have more global reach), encapsulated by an underlying hyperbolic space. For certain parameters, this model was proved to have typical features that are observed in complex networks such as power law degree distribution, bounded average degree, clustering coefficient that is asymptotically bounded away from zero and ultra-small typical distances. In the present work, we investigate its modularity and we show that, in this regime, it converges to one in probability.

Vortex Filament Equation for a Regular Polygon in the Hyperbolic Plane

The aim of this paper is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow up exponentially, which makes the problem more challenging from a numerical point of view. However, using a finite difference scheme in space combined with a fourth-order Runge–Kutta method in time and fixed boundary conditions, we show that the numerical solution is in complete agreement with the one obtained by means of algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with the evolution in the Euclidean case.

Finiteness in polygonal billiards on hyperbolic plane

\textsc{J. Hadamard} studied the geometric properties of geodesic flows on surfaces of negative curvature, thus initiating "Symbolic Dynamics". In this article, we follow the same geometric approach to study the geodesic trajectories of billiards in "rational polygons" on the hyperbolic plane. We particularly show that the billiard dynamics resulting thus are just 'Subshifts of Finite Type' or their dense subsets. We further show that 'Subshifts of Finite Type' play a central role in subshift dynamics and while discussing the topological structure of the space of all subshifts, we demonstrate that they approximate any shift dynamics.

Scattering in the Poincaré disk and in the Poincaré upper half-plane

We investigate the scattering of a plane wave in the hyperbolic plane. We formulate the problem in terms of the Lippmann-Schwinger equation and solve it exactly for barriers modeled as Dirac delta functions running along: (i) N − horizontal lines in the Poincaré upper half-plane; (ii) N − concentric circles centered at the origin; and, (iii) a hypercircle in the Poincaré disk.

Duality Notions in Real Projective Plane

Summary In this article, we check with the Mizar system [1], [2], the converse of Desargues’ theorem and the converse of Pappus’ theorem of the real projective plane. It is well known that in the projective plane, the notions of points and lines are dual [11], [9], [15], [8]. Some results (analytical, synthetic, combinatorial) of projective geometry are already present in some libraries Lean/Hott [5], Isabelle/Hol [3], Coq [13], [14], [4], Agda [6], . . . . Proofs of dual statements by proof assistants have already been proposed, using an axiomatic method (for example see in [13] - the section on duality: “[...] For every theorem we prove, we can easily derive its dual using our function swap [...]2”). In our formalisation, we use an analytical rather than a synthetic approach using the definitions of Leończuk and Prażmowski of the projective plane [12]. Our motivation is to show that it is possible by developing dual definitions to find proofs of dual theorems in a few lines of code. In the first part, rather technical, we introduce definitions that allow us to construct the duality between the points of the real projective plane and the lines associated to this projective plane. In the second part, we give a natural definition of line concurrency and prove that this definition is dual to the definition of alignment. Finally, we apply these results to find, in a few lines, the dual properties and theorems of those defined in the article [12] (transitive, Vebleian, at_least_3rank, Fanoian, Desarguesian, 2-dimensional). We hope that this methodology will allow us to continued more quickly the proof started in [7] that the Beltrami-Klein plane is a model satisfying the axioms of the hyperbolic plane (in the sense of Tarski geometry [10]).

Borcherds products of half-integral weight

We give a necessary and sufficient criterion for the existence of Borcherds products of half-integral weight associated to even lattices that split two hyperbolic planes. In particular we prove that half-integral weight Borcherds products exist for lattices of arbitrary rank.

From Euclidean triangles to the hyperbolic plane

We introduce a model of the hyperbolic plane which arises from Thurston’s ideas on best Lipschitz maps between surfaces. More precisely, we apply Thurston’s theory to the space of Euclidean triangles. We prove that the Lipschitz constant of the unique affine map between two Euclidean triangles induces a metric on the set of isometry classes of the triangles with area 12. We then prove that this space of triangles together with 2 times its natural metric is isometric to the hyperbolic plane. Our construction is simple and natural and it makes relations between several geometrical ideas.

Diameter, width and thickness in the hyperbolic plane

In hyperbolic geometry there are several concepts to measure the breadth or width of a convex set. In the first part of the paper we collect them and compare their properties. Than we introduce a new concept to measure the width and thickness of a convex body. Correspondingly, we define three classes of bodies, bodies of constant with, bodies of constant diameter and bodies having the constant shadow property, respectively. We prove that the property of constant diameter follows to the fulfilment of constant shadow property, and both of them are stronger as the property of constant width. In the last part of this paper, we introduce the thickness of a constant body and prove a variant of Blaschke’s theorem on the larger circle inscribed to a plane-convex body of given thickness and diameter.

Aperiodic order and spherical diffraction, III: The shadow transform and the diffraction formula

We define spherical diffraction measures for a wide class of weighted point sets in commutative spaces, i.e. proper homogeneous spaces associated with Gelfand pairs. In the case of the hyperbolic plane we can interpret the spherical diffraction measure as the Mellin transform of the auto-correlation distribution. We show that uniform regular model sets in commutative spaces have a pure point spherical diffraction measure. The atoms of this measure are located at the spherical automorphic spectrum of the underlying lattice, and the diffraction coefficients can be characterized abstractly in terms of the so-called shadow transform of the characteristic functions of the window. In the case of the Heisenberg group we can give explicit formulas for these diffraction coefficients in terms of Bessel and Laguerre functions.

K–isotoxal tilings from [pn ] tilings

A tiling is isotoxal if its edges form orbits or transitivity classes under the action of its symmetry group. In this article, a method is presented that facilitates the systematic derivation of planar edge-to-edge isotoxal tilings from isohedral tilings. Two well-known subgroups of triangle groups will be used to create and determine classes of isotoxal tilings in the Euclidean, hyperbolic and spherical planes which will be described in terms of their symmetry groups and symbols. The symmetry properties of isotoxal tilings make these appropriate tools to create geometrically influenced artwork such as Escher-like patterns or aesthetically pleasing designs in the three classical geometries.

Real Hypersurfaces in ℂP 2 and ℂH 2 with Cyclic Parallel ∗-Ricci Tensor

In this paper, we prove that the ∗-Ricci tensor of a real hypersurface in complex projective plane ℂP 2 or complex hyperbolic plane ℂH 2 is cyclic parallel if and only if the hypersurface is of type (A). We find some three-dimensional real hypersurfaces having non-vanishing and non-parallel ∗-Ricci tensors which are cyclic parallel.

Convergence of elastic flows of curves into manifolds

For a given p∈[2,+∞), we define the p-elastic energy E of a closed curve γ:S1→M immersed in a complete Riemannian manifold (M,g) as the sum of the length of the curve and the Lp-norm of its curvature (with respect to the length measure). We are interested in the convergence of the (Lp,Lp′)-gradient flow of these energies to critical points. By means of parabolic estimates, it is usually possible to prove sub-convergence of the flow, that is, convergence to critical points up to reparametrizations and, more importantly, up to isometry of the ambient. Assuming that the flow sub-converges, we are interested in proving the smooth convergence of the flow, that is, the existence of the full limit of the evolving flow.We first give an overview of the general strategy one can apply for proving such a statement. The crucial step is the application of a Łojasiewicz–Simon gradient inequality, of which we present a versatile version. Then we apply such strategy to the flow of E of curves into manifolds, proving the desired improvement of sub-convergence to full smooth convergence of the flow to critical points. As corollaries, we obtain the smooth convergence of the flow for p=2 in the Euclidean space Rn, in the hyperbolic plane H2, and in the two-dimensional sphere S2. In particular, the result implies that such flow in Rn or H2 remains in a bounded region of the space for any time.

Poincaré and his polarization sphere.

In a text (1892) on light, Jules Henri Poincaré introduced a geometrical device for tracking the polarization of state of light interacting with matter. Poincaré first mapped all polarization ellipses onto the surface of sphere and changes of state were represented as rotations from one point to another about prescribed axes depending on linear optical properties of the medium. Here, we consider how Poincaré, the mathematician, invented his sphere. Poincaré's professional activities in the service of geodesy appear at first glance to provide a borrowed geometry for his one-to-one mapping of polarization ellipses to global lines of latitude and longitude. However, this association falls apart in the face of a close reading of Poincaré's biography and his influences, especially the research interests of his teachers from the École Polytechnique and the Écoles de Mines. The work of Tissot and Mallard on distortion ellipses in cartography and the etiology of optical activity in crystals, respectively, together with Poincaré's own study of the qualitative theory of differential equations, provide the iconography of, and motivation for, the sphere. Whether Poincaré's mentors were unwitting partners in the invention of the sphere-it is impossible to be sure-Poincaré's otherworldly geometric sensibilities carried him through isometries (rotations) on hyperbolic planes and beyond. The apparent ingenuity behind Poincaré's sphere is diminished in comparison to his fulsome achievements. Moreover, Poincaré's considerations of the psychology of invention further emphasize that sometimes great ideas arrive to those fortunate to receive them by mechanisms that resist interpretation.

Paper, Plaster, Strings: Exploratory Material Mathematical Models between the 1860s and 1930s

AbstractDoes the materiality of a three-dimensional model have an effect on how this model operates in an exploratory way, how it prompts discovery of new mathematical results? Material mathematical models were produced and used during the second half of the nineteenth century, visualizing mathematical objects, such as curves and surfaces—and these were produced from a variety of materials: paper, cardboard, plaster, strings, wood. However, the question, whether their materiality influenced the status of these models—considered as exploratory, technical, or representational—was hardly touched upon. This article aims to approach this question by investigating two case studies: Beltrami’s paper models vs. Dyck’s plaster ones of the hyperbolic plane; and Chisini’s string models of braids vs. Artin’s and Moishezon’s algebraization of these braids. These two case studies indicate that materiality might have a decisive role in how the model was taken into account mathematically: either as an exploratory or rather as a technical or pedagogical object.

Beltrami equation for the harmonic diffeomorphisms between surfaces

We study harmonic maps between surfaces, that are solutions to a nonlinear elliptic PDE. In Refs. Minsky (1992); Wolf (1989) it was proved that harmonic diffeomorphisms, with nonvanishing Hopf differential, satisfy a Beltrami equation of a certain type: the imaginary part of the logarithm of the Beltrami coefficient coincides with the imaginary part of the logarithm of the Hopf differential. Therefore, it is a harmonic function. The real part of the logarithm of the Beltrami coefficient satisfies an elliptic nonlinear differential equation, which in the case of constant curvature is the elliptic sinh-Gordon equation.In this paper we also prove the converse: if the imaginary part of the logarithm of the Beltrami coefficient is a harmonic function, then the target surface can be equipped with a metric, conformal to the original one, and the solution of the Beltrami equation is a harmonic map. Therefore, solving a certain Beltrami equation is equivalent to solving the harmonic map problem. Harmonic maps to a constant curvature surface are therefore classified by the classification of the solutions of the elliptic sinh-Gordon equation.The general problem of solving the sinh-Gordon equation is a nonlinear problem and is still open. Different well-known harmonic maps to the hyperbolic plane are proved to be related to the one-soliton solutions of the elliptic sinh-Gordon equation. Moreover, an example is proposed which does not belong to the one-soliton solution of the elliptic sinh-Gordon equation. Solutions are calculated for the constant curvature case in a unified way, for positive, negative and zero curvature of the target surface.

The classification of free algebras of orthogonal modular forms

We prove a necessary and sufficient condition for the graded algebra of automorphic forms on a symmetric domain of type IV being free. From the necessary condition, we derive a classification result. Let $M$ be an even lattice of signature $(2,n)$ splitting two hyperbolic planes. Suppose $\Gamma$ is a subgroup of the integral orthogonal group of $M$ containing the discriminant kernel. It is proved that there are exactly 26 groups $\Gamma$ such that the space of modular forms for $\Gamma$ is a free algebra. Using the sufficient condition, we recover some well-known results.