Over and Over Again（ISBN=9780883856413） 2025 chm pdf kindle rb azw3 下载 115盘

Iterations, that is transformations that are applied to objects

over and over again, are the subject of this book. Three types of

iteration are considered. The first, smoothing, is the process by

which geometrical shapes can be transformed into standard types.

Functional iteration is a way of describing the phenomenon of

chaos. Finally, iterations involving curves and surfaces play an

important role in computer aided design and graphics. These themes

are explored and developed with no more background required than

high school mathematics and a little calculus. Several basic

strategies are highlighted which unlock solutions to problems

involving iteration, and these are then applied to solve problems

drawn from International Mathematical Olympiads.

1. Transformations and their iteration

2. Arithmetic and geometric means

3. Isoperimetric inequality for triangles

4. Isoperimetric quotient

5. Colored marbles

6. Candy for school children

7. Sugar rather than candy

8. Checkers on a circle

9. Decreasing sets of positive integers

10. Matrix manipulations

11. Nested traingles

12. Morley's theorem and Napoleon's theorem

13. Complex numbers in geometry

14. Birth of an IMO problem

15. Barycentric coordinates

16. Douglas-Neumann theorem

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书籍介绍

Iterations, that is transformations that are applied to objects over and over again, are the subject of this book. Three types of iteration are considered. The first, smoothing, is the process by which geometrical shapes can be transformed into standard types. Functional iteration is a way of describing the phenomenon of chaos. Finally, iterations involving curves and surfaces play an important role in computer aided design and graphics. These themes are explored and developed with no more background required than high school mathematics and a little calculus. Several basic strategies are highlighted which unlock solutions to problems involving iteration, and these are then applied to solve problems drawn from International Mathematical Olympiads.