arXiv subjects

Athanase Papadopoulos

Publications and source records attributed to Athanase Papadopoulos.

At least 19 recordsLinked to original sources

arXiv2026

On spaces of Euclidean triangles and triangulated Euclidean surfaces

In this paper, we introduce an asymmetric distance function on the space of marked Euclidean triangles of normalised area, and we prove several properties of this metric, which turns out to be (a restriction of) a non-symmetric version of the classical Thompson distance. We give a description of the geodesics of this metric, we show that it is Finsler, and we give a formula for its infinitesimal Finsler structure. We then introduce and study a Finsler metric of the space of singular Euclidean structures on a surface adapted to an underlying fixed triangulation, and we also study its geodesics and its Finsler infinitesimal structure. We then develop a theory of completeness and completion of asymmetric metrics which is adapted to our setting, and we use this theory in the study of the completeness of the metric we introduced on the space of triangles. In doing so, we establish a bridgebetween one aspect of Thurston's theory of metrics on spaces of surfaces and Thompson's metrics. The final version of this paper will appear in Monatshefte f{ü}r Mathematik

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arXiv2026

On the construction of geographical maps: Lagrange, Chebyshev, Darboux and Milnor

Lagrange, Chebyshev, and Darboux, in 1779, 1856, and 1911, respectively, wrote articles all bearing the same title, \emph{On the Construction of Geographical Maps}. In 1969, Milnor wrote a paper in which he refers to Chebyshev's paper, of which he provides a new formulation and proof. In the present article, we review the results of all these papers, explaining the main ideas they contain and pointing out connections between them. We give complete proofs of the statements by Darboux and Milnor, both of which aim to make explicit and provide a proof of Chebyshev's result, but whose contents are different. Although Chebyshev did not state explicitly what the word ``best'' means, his conclusion, like that of Darboux and of Milnor, is that a best geographical map is characterised by the fact that its conformal factor is constant on the boundary of the region represented. Our statement and proof of Milnor's theorem work in a more general setting than the one he gives. The final version of this paper will appear in the Handbook of Mathematics in the Arts and Sciences (second edition), ed. Bharath Sriraman, Springer, 2027.

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arXiv2026

Convex structures of the unit tangent spheres in Teichm{ü}ller space

We analyse the convex structure of the Finsler infinitesimal balls of the Thurston metric on Teichm{ü}ller space. We analyse the convex structure of the Finsler unit ball in the tangent space at each point of Teichm\''uller space of a closed surface of genus $\geq 2$ equipped with Thurston's metric. We obtain a characterisation of faces, exposed faces and extreme points of such a unit sphere. In particular, we prove that every face has a unique naturally associated chain-recurrent geodesic lamination such that this face consists of the unit tangent vectors which are linear combinations of stretch vectors along maximal chain-recurrent geodesic laminations containing the given one. We show that a face is exposed if and only if its associated chain-recurrent geodesic lamination is the support of a measured lamination. Furthermore, we show that a point on a tangent unit sphere is an extreme point if and only if it is a stretch vector along some maximal chain-recurrent geodesic lamination. The last result gives an affirmative answer to a conjecture whose answer was known positively in the case where the surface is either the once-punctured torus or the 4-punctured sphere. Our main results also provide an alternative approach to the topological part of the infinitesimal rigidity result concerning Thurston's metric and the equivariance property of stretch vectors.

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arXiv2026

Convergence-Symmetric Metric Spaces

We study some basic properties of spaces which satisfy the usual axioms of a metric space but without the symmetry axiom. We realised that such a study is needed in view of the relatively recent appearance of several papers on natural asymmetric metrics to which the available theories do not apply. One prominent example is Thurston's metric on the Teichm{ü}ller spaces of hyperbolic surfaces of finite type, introduced by Thurston in 1985, with several variants and generalisations. Another asymmetric metric is the earthquake metric, also introduced by Thurston and about which several basic questions remain open. Other asymmetric metrics we consider here include the Funk metric, the Apollonian metric and the left Hausdorff metric. We are particularly interested in questions of completeness and completion and in associated notions of boundary at infinity for these asymmetric metrics. In this paper, after discussing several examples, we prove results such as a Banach fixed point theorem, an Arzel{à}--Ascoli theorem and a Hopf--Rinow theorem adapted to this asymmetric setting. We introduce a property we call ``convergence-symmetry'' which turns out to be crucial in the study of metric completeness of some asymmetric metrics. This property is stronger than a property which was formulated by Herbert Busemann around 1970, and which we call the ``Busemann condition''. Every convergence-symmetric metric space has a unique minimal completion which is convergence-symmetric. This does not hold for spaces satisfying Busemann's condition. Several examples we consider satisfy Busemann's condition but are not convergence-symmetric. We have included throughout the paper a certain number of open questions.

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arXiv2026

The combinatorial structure of the unit tangent spheres and cotangent spheres of Teichm{ü}ller space with Thurston's Finsler metric

We prove several new results on the combinatorial structures of the unit spheres of the norms induced by Thurston's metric on the tangent and cotangent spaces of the Teichm{ü}ller space of a closed surface of negative Euler characteristic. These results include a formula for the dimension of every face of a unit sphere in the tangent space in terms of an invariant of the chain-recurrent lamination representing the face. We then prove that the combinatorial structure of such a unit sphere is independent of the underlying point in Teichm{ü}ller space. Provided the genus of the surface is $\ge$ 2, we show that there is a natural isomorphism between the extended mapping class group of the surface and the group of combinatorial automorphisms of such a unit sphere. In the case of genus 2, we obtain a natural epimorphism between the two groups whose kernel is the class of the hyperelliptic involution. Regarding the unit spheres of Thurston's metric in the cotangent spaces, we obtain a formula describing the codimensions of faces of such a sphere in terms of corresponding projective measured laminations. We then give a necessary and sufficient condition for a face to be exposed, and of a face to correspond to a projectively weighted multi-curve. Some of the results obtained answer open questions.

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arXiv2026

Spherical trigonometry before the modern era:The treatise of Nasir al-Din al-Tusi

This is an overview of Nasir al-Din al-Tusi's Treatise of the quadrilateral, an invaluable 13th century document on spherical geometry which was translated into French in 1891. The title we are using here is the one given by the translator (Alexandre Carath{é}odory). A title which is closer to the original Arabic is ''Disclosing the secrets of the secant figure.'' The term ''secant figure'', to which the title refers, is the so-called ''complete (spherical) quadrilateral'', that is, the figure that underlies what we call today Menelaus' Theorem. This theorem gives a formula that was extensively used by astronomers in their computations and the establishment of their tables since the first century AD, notably by Ptolemy, in the absence of the spherical trigonometric formulae that were discovered later. Nasir's treatise contains much more than Menelaus' theorem, since we find there a complete system of spherical trigonometric formulae, with complete proofs. The treatise includes at the same time invaluable historical information on the discovery of the trigonometric formulae by the Arab mathematicians of the Middle-Ages and the transformation of the field of spherical trigonometry that this discovery led to. The final version of this paper will appear in the book Spherical geometry in the eighteenth century, I: Euler, Lagrange and Lambert, edited by Renzo Caddeo and Athanase Papadopoulos, Springer, 2026.

math.HO↗

arXiv2026

Notes on angles and solid angles, in relation with Euler's memoir De mensura angulorum solidorum

We provide some historical context to the study of solid angles carried out by Euler in his memoir \emph{De mensura angulorum solidorum} (On the measure of solid angles). We extend our study to the general notion of angle (not only solid). While doing so, we explore some works by Ancient Greek mathematicians and others by Arabs mathematicians of the Middle-Ages as well as some later Western authors from the Renaissance. In particular, we review the Pythagorean anthyphairetical perspective on angles which establishes the basis of the important relation between the mathematical notion of angle and the philosophical concept of finitization of the Infinite. In doing so, we shall show that questions addressed by Euler lead us to questions raised about 2500 years ago. At the same time, we highlight the fact that mathematics in those times is also today's mathematics. The reader can also see in this study the intermingling between mathematics and philosophy. This paper will appear in the book \emph{Spherical geometry in the Eighteenth Century, I: Euler, Lagrange and Lambert}, ed. R. Caddeo and A. Papadopoulos, Springer, 2026.

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arXiv2026

Johann Heinrich Lambert's memoir "Theorie der Parallellinien": A review with commentary

We review the memoir \emph{heorie der Parallellinien} by Johann Heinrich Lambert, written in 1766. Lambert, a victim of the prejudices of his time, conceived this memoir as an attempt to prove the so-called parallel postulate of Euclid's \emph{Elements}, and consequently, the non-existence of the geometry that we now call hyperbolic geometry. In fact, by developing the foundations of a geometry obtained by replacing the parallel postulate with its negation while keeping Euclid's other postulates unchanged, Lambert was hoping to arrive at a contradiction. Of course, he failed in his endeavor, but these attempts at proving the parallel postulate implicitly contain, without Lambert having foreseen it, fundamental results of hyperbolic geometry, the discovery of which, by Lobachevsky, Bolyai and Gauss, was not to take place until the following century. Thus, Lambert's memoir (which he did not intend to publish but which was eventually published in 1895) constitutes one of the founding texts of non-Euclidean geometry. Spherical geometry is one of the three geometries of constant curvature, the other two being Euclidean geometry and hyperbolic geometry. In this sense, along with hyperbolic geometry, spherical geometry constitutes one of the two non-Euclidean geometries. In fact, Lambert, like Lobachevsky and others after him, understood the deep relationships between the three geometries: Euclidean, spherical, and hyperbolic, in particular the formal and the more profound analogies between the trigonometric formulae, the properties of birectangular isosceles quadrilaterals and of trirectangular quadrilaterals, the monotonicity properties (which can be formulated in terms of convexity properties) which hold in opposite senses in spherical and hyperbolic geometry which at some points he calls a sphere of imaginary radius. It is for these reasons that we decided to include in this volume, dedicated to spherical geometry, a chapter on this important memoir by Lambert, trying to highlight its most important ideas. This paper will appear as a chapter in the book ``Spherical Geometry in the Eighteenth Century I: Euler, Lagrange and Lambert'', ed. R. Caddeo and A. Papadopoulos, Springer Nature Switzerland, 2026.

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arXiv2026

The horocyclic metric on Teichm{ü}ller spaces

In his paper Minimal stretch maps between hyperbolic surfaces, William Thurston defined a norm on the tangent space to Teichm{ü}ller space of a hyperbolic surface, which he called the earthquake norm. This norm is obtained by assigning a length to a tangent vector after such a vector is considered as an infinitesimal earthquake deformation of the surface. This induces a Finsler metric on the Teichm{ü}ller space, called the earthquake metric. This theory was recently investigated by Huang, Ohshika, Pan and Papadopoulos. In the present paper, we study this metric from the conformal viewpoint and we adapt Thurston's theory to the case of Riemann surfaces of arbitrary genus with marked points. A complex version of the Legendre transform defined for Finsler manifolds gives an analogue of the Wolpert duality for the Weil-Petersson symplectic form, which establishes a complete analogue of Thurston's theory of the earthquake norm in the conformal setting. This paper will appear in the Annales de l'Institut Fourier

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arXiv2026

Simone Weil, Andr{é} Weil, Bourbaki and Pythagorean mathematics

Simone Weil is one of the most prominent 20th century French philosophers. She is the sister of Andr{é} Weil, the renowned mathematician, the father of modern algebraic geometry and the initiator of the Bourbaki group. Simone and Andr{é} Weil shared a love for literature, mathematics, science and philosophy. My aim in this article is to convey, based on their writings and their correspondence, the idea that Pythagoreanism was a central element of their thought. I will put this into context, talking first about the life and work of each of them, showing how much they were linked by essential common ideas, even though their life paths were very different, and how, ultimately, Pythagorean mathematics and philosophy became naturally part of their respective intellectual worlds. The article is the written version of a lecture I gave in October 2025, at the conference ``The Life and Contribution of Pythagoras to Mathematics, Sciences, and Philosophy'' that took place on October 3-4, 2025 at the Cyprus University of Technology in Limassol.

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arXiv2026

On the Lambert conformal conical projection and the general map of the Russian Empire

The problem of drawing geographical maps is the one of mapping a subset of the sphere, representing a country or some other region on the surface of the Earth, into the Euclidean plane, minimising certain distortion properties that are specified in advance. It is known that from the purely mathematical point of view, this is an extremely difficult problem. One of Leonhard Euler's duties during his first stay at the Imperial Academy of Sciences of Saint Petersburg (1727-1741) was to help establishing maps of the Russian Empire. He worked on this project under the direction of the famous French geographer Joseph-Nicolas Delisle, who was the head of the astronomy and geography departments of the Academy. The general map of the Russian Empire, together with several maps of its particular regions were published under Euler's direction in the so-called Russian Atlas in 1745. In his later memoir ``De proiectione geographica De Lisliana in mappa generali imperii russici usitata'', written in 1777, Euler developed the mathematical theory of the method used by Delisle on a heuristic basis, which he himself used for drawing the general map of the Russian Empire. This method usually carries now the name Delisle--Euler map. In a previous paper, the first two authors of the present paper compared the Delisle--Euler map with several other maps of the conical type, with respect to various mathematical distorsion properties. They showed that this map is the best one from all the points of view considered, when it is applied to the drawing of the Russian Empire. In the present paper, we compare the Euler--Delisle map with a map which was not considered in the paper mentioned, namely, the so-called Lambert conformal conical projection, applied to the same region of the Earth. We show that the latter is better in several respects than all the other maps considered in the previous paper, including the Delisle--Euler map.

math.DG↗

arXiv2025

Galilei and Huygens: Music and science

Vincenzo Galilei and Constantijn Huygens were both humanists and eminent musicians, the former from the late Renaissance and the latter from the early Modern era. Their respective sons, Galileo and Christiaan, were scientists whose importance cannot be overestimated. My aim in this chapter is to set the scene for a parallel presentation of the legacy of the Galilei on the one hand, and the Huygens on the other. This will give us an opportunity to talk about mathematics, music and acoustics, but also about science in general, at this time of birth of the Modern era.

math.HO↗

arXiv2025

Andr{é} and Simone Weil: Mathematics, social activism and Indian culture

This is an essay on the relation of Andr{é} and Simone Weil with Indian culture and Sanskrit literature, especially the Bhagavad G{ī}t{ā}, a Hindu scripture which they knew well, which they quoted extensively, and which guided them in making important life decisions. In addressing this question, we will also talk about the life paths of the two Weils, and more specifically about certain aspects that relate to their deep convictions.

math.HO↗

arXiv2025

Euler's work on spherical geometry: An overview with comments

We review Euler's work on spherical geometry. After an introduction concerning the general place that trigonometric formulae occupy in geometry, we start by the two memoirs of Euler on spherical trigonometry, in which he establishes the trigonometric formulae using different methods, namely, the calculus of variations in the first memoir, and classical methods of solid geometry in the other. In another memoir, Euler gives several formulae for the area of a spherical triangle in terms of its side lengths (these are ``spherical Heron formulae''). He uses this in the computation of numerical values of the solid angles of the five regular polyhedra, which is his goal in his memoir. We then review memoirs in which Euler systematically starts by establishing a theorem or a construction in Euclidean geometry and then proves an analogue in spherical geometry. We point out relations between Euler's memoirs on spherical trigonometry and works he did in astronomy, on the problem of drawing geographical maps, and in geomagnetism. We also review some other works of Euler involving spheres, including a memoir on the three-dimensional Apollonius problem and others concerning algebraic curves on the sphere. Even though these works are not properly on spherical geometry, they show Euler's interests in various questions related to spheres and we think that they are worth highlighting in such an overview. Beyond spherical geometry, the reader is invited to discover in this article an important facet of the work of the great Leonhard Euler. This article will appear as a chapter in the book ``Spherical geometry in the eighteenth century, I: Euler, Lagrange and Lambert'', Springer, 2026.

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arXiv2025

The mathematics of periodic anthyphairesis as a basis for the full understanding of Plato's philosophy

Even though Plato's philosophy in ancient times was always closely associated with mathematics, modern Platonic scholarship, during the last five centuries, has moved steadily toward de-mathematization. The present work aims to outline a radical re-interpretation of Plato's philosophy, according to which the Platonic Idea, that is, the intelligible Being, has the structure of the philosophical analogue of a geometric dyad in a philosophic anthyphaeresis -- the precursor of modern continued fractions -- which was studied by the Pythagoreans, Theodorus and Theaetetus in relation with the discoveries of quadratic incommensurabilities. This mathematical structure is clearly visible in the Platonic method of Division and Collection, equivalently Name and Logos, equivalently True Opinion plus Logos, in the dialogues Theaetetus, Sophist, Statesman, Meno, and Parmenides. Equipped with this structure of an intelligible Being, we provide definitive answers to fundamental questions, that were not be resolved by Platonists, concerning the following topics: the dialectic numbers, which are based on the anthyphairetic periodicity and the plus one rule, stating that the dialectic number of terms of a sequence is the (number of) ratios of successive terms plus one (stated in the Parmenides 148d-149d); the description of the intelligible being as an Indivisible Line, a statement bordering on the contradictory; the also seemingly contradictory Sophist 's statement that ``the not-Being is a Being'', based on the equalization of the two elements of the dyad defining an intelligible Being; the more general self-similar Oneness of an intelligible Being, based on the equalization of all parts generated by the anthyphairetic division of an intelligible Being; and finally the Third Man Argument in the Introduction to the Parmenides, appearing as a threat for Plato's theory, but essentially innocuous because of the self-similar Oneness. The third part of our study aims to prove that, contrary to the presently dominant interpretation of Zeno's arguments and paradoxes as being devoid of mathematical content, the analysis of Zeno's presence in the Parmenides, Sophist (via the Eleatic Stranger), and Zeno's verbatim Fragments preserved by Simplicius, show that Plato's intelligible Beings essentially coincide with Zeno's true Beings, and hence that Zeno's philosophical thought was already anthyphairetic, and hence heavily influenced by the Pythagorean's Mathematics. These findings run against Burkert's claim that ``ontology is prior to mathematics''. Modern Platonists have never obtained a clear description of the structure of an intelligible Idea in terms of the mathematics of periodic anthyphairesis, and thus were not able to answer fundamental questions, nor to realize the close connection of Zeno's intelligible beings with Zeno's true Beings.

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arXiv2025

On families of Finsler metrics

In this paper, we answer some natural questions on symmetrisation and more general combinations of Finsler metrics, with a view towards applications to Funk and Hilbert geometries and to metrics on Teichm{ü}ller spaces. For a general non-symmetric Finsler metric on a smooth manifold, we introduce two different families of metrics, containing as special cases the arithmetic and the max symmetrisations respectively of the distance functions associated with these Finsler metrics. We are interested in various natural questions concerning metrics in such a family, regarding its geodesics, its completeness, conditions under which such a metric is Finsler, the shape of its unit ball in the case where it is Finsler, etc. We address such questions in particular in the setting of Funk and Hilbert geometries, and in that of the Teichm{ü}ller spaces of several kinds of surfaces, equipped with Thurstonlike asymmetric metrics.

math.DG↗

arXiv2025

On the Marinus--Ptolemy and Delisle--Euler conical maps

We examine connections between the mathematics behind methods of drawing geographical maps due, on the one hand to Marinos and Ptolemy (1st-2nd c. CE) and on the other hand to Delisle and Euler (18th century). A recent work by the first two authors of this article shows that methods of Delisle and Euler for drawing geographical maps, which are improvements of methods of Marinus and Ptolemy, are best among a collection of geographical maps we term ``conical''. This is an instance where after practitioners and craftsmen (here, geographers) have used a certain tool during several centuries, mathematicians prove that this tool is indeed optimal. Many connections among geography, astronomy and geometry are highlighted. The fact that the Marinos--Ptolemy and the Delisle--Euler methods of drawing geographical maps share many non-trivial properties is an important instance of historical continuity in mathematics.

math.HO↗

arXiv2024

Geometry in the twentieth century: A return to Euclid -- The work of Herbert Busemann

This is a point of view of the work of Herbert Busemann (1905-1994), seen as a return to the geometry of Ancient Greece. The importance of this work, its recognition and its relation with other works are discussed. The final version of this paper will appear in the Handbook of the History and Philosophy of Mathematical Practice, ed. Bharath Sriraman, Springer, 2024.

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