Quasiconformal mappings, from Ptolemy's geography to the work of Teichm\"uller

TL;DR

This paper traces the historical development of quasiconformal mappings from ancient cartography to Teichmüller's foundational work, highlighting key ideas, mathematicians, and their contributions over nearly two millennia.

Contribution

It provides a comprehensive historical survey of quasiconformal mappings, emphasizing overlooked ideas and early developments leading to Teichmüller's influential theories.

Findings

Highlights the role of Tissot's work in early mapping distortions.

Details the contributions of Teichmüller to the theory of Riemann surfaces.

Connects historical cartography with modern quasiconformal theory.

Abstract

The origin of quasiconformal mappings, like that of conformal mappings, can be traced back to old cartography where the basic problem was the search for mappings from the sphere onto the plane with minimal deviation from conformality, subject to certain conditions which were made precise. In this paper, we survey the development of cartography, highlighting the main ideas that are related to quasiconformality. Some of these ideas were completely ignored in the previous historical surveys on quasiconformal mappings. We then survey early quasiconformal theory in the works of Gr\"otzsch, Lavrentieff, Ahlfors and Teichm\"uller, which are the 20th-century founders of the theory. The period we consider starts with Claudius Ptolemy (c. 100--170 A.D.) and ends with Oswald Teichm\"uller (1913--1943). We mention the works of several mathematicians-geographers done in this period, including Euler, Lagrange, Lambert, Gauss, Chebyshev, Darboux and others. We highlight in particular the work of Nicolas-Auguste Tissot (1824--1897), a French mathematician and geographer who (according to our knowledge) was the first to introduce the notion of a mapping which transforms infinitesimal circles into infinitesimal ellipses, studying parameters such as the ratio of the major to the minor axes of such an infinitesimal ellipse, its area divided by the area of the infinitesimal circle of which it is the image, and the inclination of its axis with respect to a fixed axis in the plane. We also give some information about the lives and works of Gr\"otzsch, Lavrentieff, Ahlfors and Teichm\"uller. The latter brought the theory of quasiconformal mappings to a high level of development. He used it in an essential way in his investigations of Riemann surfaces and their moduli and in function theory (in particular, in his work on the Bieberbach conjecture and the type problem). We survey in detail several of his results. We also discuss some aspects of his life and writings, explaining why his papers were not read and why some of his ideas are still unknown even to Teichm\"uller theorists. The final version of this paper will appear in the book "Uniformization, Riemann-Hilbert Correspondence, Calabi-Yau Manifolds, and Picard-Fuchs Equations" (ed. L. Ji and S.-T. Yau), International Press and Higher Education Press (2017).