MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Biographical Vignettes 167
study under Steiner, Jacobi, Dirichlet and Eisenstein. In 1849, he returned
to Göttingen and submitted his thesis, supervised by Gauss, in 1851. This
thesis applied topological methods to complex function theory and introduced
Riemann surfaces to study the geometric properties of analytic functions, conformal
mappings and the connectivity of surfaces. (A fundamental theorem on
Riemann surfaces appears in Property 41 of Chapter 2.) In order to become
a Lecturer, he had to work on his Habilitation. In addition to another thesis
(on trigonometric series including a study of Riemann integrability), this required
a public lecture which Gauss chose to be on geometry. The resulting
On the hypotheses that lie at the foundations of geometry of 1854 is considered
a classic of Mathematics. In it, he gave the definition of n-dimensional
Riemannian space and introduced the Riemannian curvature tensor. For the
case of a surface, this reduces to a scalar, the constant non-zero cases corresponding
to the known non-Euclidean geometries. He showed that, in four
dimensions, a collection of ten numbers at each point describe the properties
of a manifold, i.e. a Riemannian metric, no matter how distorted. This provided
the mathematical framework for Einstein’s General Theory of Relativity
sixty years later. This allowed him to begin lecturing at Göttingen, but he
was not appointed Professor until 1857. In 1857, he published another of his
masterpieces, Theory of abelian functions which further developed the idea
of Riemann surfaces and their topological properties. In 1859, he succeeded
Dirichlet as Chair of Mathematics at Göttingen and was elected to the Berlin
Academy of Sciences. A newly elected member was expected to report on
their most recent research and Riemann sent them On the number of primes
less than a given magnitude. This great masterpiece, his only paper on number
theory, introduced the Riemann zeta function and presented a number of
conjectures concerning it, most notably the Riemann Hypothesis, the greatest
unsolved problem in Mathematics [76] (Hilbert’s Eighth Problem [332] and one
of the $1M Millenium Prize Problems [78]) ! It conjectures that, except for
a few trivial exceptions, the roots of the zeta function all have a real part of
1/2 in the complex plane. The Riemann Hypothesis implies results about the
distribution of prime numbers that are in some ways as good as possible. His
work on monodromy and the hypergeometric function in the complex domain
established a basic way of working with functions by consideration of only
their singularities. He died from tuberculosis, aged 39, in Salasca, Italy, where
he was seeking the health benefits of the warmer climate. Source material for
Riemann is available in [23, 42, 287].
Vignette 24 (James Clerk Maxwell: 1831-1879).
James Clerk Maxwell, physicist and Mathematician, was born in Edinburgh,
Scotland [43, 215, 307]. He attended the prestigious Edinburgh Academy
and, at age 14, wrote a paper on ovals where he generalized the definition of an