Geometry of lengths, areas, and volumes
著者
書誌事項
Geometry of lengths, areas, and volumes
（Two-dimensional spaces / James W. Cannon, v. 1）
American Mathematical Society, c2017
- pbk
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注記
Bibliography: p.113-119
内容説明・目次
内容説明
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.
目次
Lengths-The Pythagorean theorem
Consequences of the Pythagorean theorem
Areas
Areas by slicing and scaling
Areas by cut and paste
Areas by counting
Unsolvable problems in Euclidean geometry
Does every set have a size?
Bibliography
「Nielsen BookData」 より
