This is the first of a three volume collection
devoted to the geometry, topology, and curvature of 2-dimensional
spaces. The collection provides a guided tour through a wide range of
topics by one of the twentieth century's masters of geometric
topology. The books are accessible to college and graduate students
and provide perspective and insight to mathematicians at all levels
who are interested in geometry and topology.
The first volume begins with length measurement as dominated by the
Pythagorean Theorem (three proofs) with application to number theory;
areas measured by slicing and scaling, where Archimedes uses the
physical weights and balances to calculate spherical volume and is led
to the invention of calculus; areas by cut and paste, leading to the
Bolyai-Gerwien theorem on squaring polygons; areas by counting,
leading to the theory of continued fractions, the efficient rational
approximation of real numbers, and Minkowski's theorem on convex
bodies; straight-edge and compass constructions, giving complete
proofs, including the transcendence of $e$ and
$\pi$, of the impossibility of squaring the circle,
duplicating the cube, and trisecting the angle; and finally to a
construction of the Hausdorff-Banach-Tarski paradox that shows some
spherical sets are too complicated and cloudy to admit a well-defined
notion of area.
Readership
Undergraduate and graduate students and researchers
interested in topology.