MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

172 Biographical Vignettes
the number of visible rectangles (including squares) is 1,296; and that, on a
similar board with n squares on a side, the number of squares is the sum of
the first n square numbers while the number of rectangles (including squares)
is the sum of the first n cube numbers.” He was President of the American
Mathematical Society and Editor of American Journal of Mathematics
(where he finally published Morley’s Theorem). He was also an exceptional
chess player, having defeated fellow Mathematician Emmanuel Lasker while
the latter was still reigning World Champion! His three sons became Rhodes
Scholars: Christopher became a famous novelist, Felix became Editor of The
Washington Post and also President of Haverford College, and Frank became
director of the publishing firm Faber and Faber but was also a Mathematician
who published Inversive Geometry with his father in 1933. He died in
Baltimore, aged 77.
Vignette 30 (Hermann Minkowski: 1864-1909).
Hermann Minkowski was born of German parents in Alexotas, a suburb
of Kaunas, Lithuania which was then part of the Russian Empire [144]. The
family returned to Germany and settled in Königsberg when he was eight years
old. He received his higher education at the University of Königsberg where he
became a lifelong friend of David Hilbert, his fellow student, and Adolf Hurwitz,
his slightly older teacher. In 1883, while still a student at Königsberg,
he was awarded the Mathematics Prize from the French Academy of Sciences
for his manuscript on the theory of quadratic forms. His 1885 doctoral thesis
at Königsberg was a continuation of this prize winning work. In 1887 he
moved to the University of Bonn where he taught until 1894, then he returned
to Königsberg for two years before becoming a colleague of Hurwitz at ETH,
Zurich in 1896 where Einstein was his student. In 1896, he presented his Geometry
of Numbers, a geometrical method for solving problems in number
theory. In 1902, he joined the Mathematics Department of the University of
Göttingen where he was reunited with Hilbert (who had arranged to have the
chair created specifically for Minkowski) and he stayed there for the rest of
his life. It is of great historical interest that it was in fact Minkowski who
suggested to Hilbert the subject of his famous 1900 lecture in Paris on “the
Hilbert Problems” [332]. In 1907, he realized that Einstein’s Special Theory of
Relativity could best be understood in a non-Euclidean four-dimensional space
now called Minkowski spacetime in which time and space are not separate entities
but instead are intermingled. This space-time continuum provided the
framework for all later mathematical work in this area, including Einstein’s
General Theory of Relativity. In 1907, he published his Diophantische Approximationen
which gave an elementary account of his work on the geometry
of numbers and of its application to Diophantine approximation and algebraic
numbers. His subsequent work on the geometry of numbers led him to investigate
convex bodies and packing problems. His Geometrie der Zahlen was