MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

170 Biographical Vignettes
Schwarz reflection priniciple. The latter has previously been described in the
context of Fagnano’s Problem (Property 53) and Triangular Billiards (Recreation
20). He died in Berlin, aged 78. Source material for Schwarz is available
in [23].
Vignette 27 (Jules Henri Poincaré: 1854-1912).
Henri Poincaré, described by many as The Last Universalist in Mathematics,
was born into an upper middle class family in Nancy, France [67]. He
was not the only distinguished member of his family. His cousin, Raymond
Poincaré, was several times Prime Minister of France and President of the
French Republic during World War I. In 1862, Henri entered the Lycée in
Nancy (now renamed after him) and spent eleven years there as one of the top
students in every subject. He won first prizes in the concours général, a competition
between the top students from all across France. In 1873, he entered
l’ École Polytechnique, graduating in 1875. After graduation, he continued his
studies at l’ École des Mines after which he spent a short time working as a
mining engineer while completing his doctoral work. In 1879, he received his
doctorate under Charles Hermite at the University of Paris with a thesis on differential
equations where he introduced the qualitative geometric theory which
was to become so influential. He then was appointed to teach mathematical
analysis at the University of Caen. In 1881, he became a Professor at the
University of Paris and also at l’ École Polytechnique, holding both posts for
the rest of his life. The breadth and depth of his mathematical contributions
is truly staggering. He won a mathematical competition based on his work on
the three-body problem which used invariant integrals, introduced homoclinic
points and gave the first mathematical description of chaotic motion. He also
made fundamental contributions to number theory, automorphic functions and
the theory of analytic functions of several complex variables. His work in algebraic
topology was especially noteworthy where he created homotopy theory
and introduced the notion of the fundamental group as well as formulated the
celebrated Poincaré Conjecture which has only recently been settled in the
affirmative by Grigory Perelman [141]. In Applied Mathematics, he made advances
in fluid mechanics, optics, electricity, telegraphy, capillarity, elasticity,
thermodynamics, potential theory, quantum theory, theory of relativity and
celestial mechanics, the latter culminating in his masterpiece Les Méthodes
nouvelles de la mécanique céleste in three volumes published between 1892
and 1899. See Property 84 of Chapter 2 for a description of the Poincaré disk
model of the hyperbolic plane. His name has been enshrined in the Poincaré-
Bendixson Theorem, the Poincaré Group, the Poincaré-Linstedt Method, the
Poincaré Inequality, the Poincaré Metric and the Poincaré Map, to mention
but a few. Poincaré’s popular works included Science and Hypothesis (1901),
The Value of Science (1905) and Science and Method (1908). He was the only