Book contents
- Frontmatter
- Contents
- Acknowledgements
- Part I Mathematical Recreations and Abstract Games
- Part II Mathematics: game-like, scientific and perceptual
- 6 Game-like mathematics
- 7 Euclid and the rules of his geometrical game
- 8 New concepts and new objects
- 9 Convergent and divergent series
- 10 Mathematics becomes game-like
- 11 Mathematics as science
- 12 Numbers and sequences
- 13 Computers and mathematics
- 14 Mathematics and the sciences
- 15 Minimum paths: elegant simplicity
- 16 The foundations: perception, imagination, insight
- 17 Structure
- 18 Hidden structure, common structure
- 19 Mathematics and beauty
- 20 Origins: formality in the everyday world
- References
- Index
10 - Mathematics becomes game-like
Published online by Cambridge University Press: 05 November 2012
- Frontmatter
- Contents
- Acknowledgements
- Part I Mathematical Recreations and Abstract Games
- Part II Mathematics: game-like, scientific and perceptual
- 6 Game-like mathematics
- 7 Euclid and the rules of his geometrical game
- 8 New concepts and new objects
- 9 Convergent and divergent series
- 10 Mathematics becomes game-like
- 11 Mathematics as science
- 12 Numbers and sequences
- 13 Computers and mathematics
- 14 Mathematics and the sciences
- 15 Minimum paths: elegant simplicity
- 16 The foundations: perception, imagination, insight
- 17 Structure
- 18 Hidden structure, common structure
- 19 Mathematics and beauty
- 20 Origins: formality in the everyday world
- References
- Index
Summary
Divergent series are an example of an idea so novel and so difficult to understand that early work on divergent series was a mixture of uncertain manipulation, brilliant intuition, apparent failure, genuine errors, and incomprehensible success. It was only after many brilliant mathematicians had struggled with the idea that the dark and obscure landscape of divergent series was illuminated, and mathematicians finally decided what a divergent series ‘really is’ and how they could safely be handled.
Euler's relation for polyhedra
A similar confusion arose over another of Euler's interests: polyhedra. He noticed what Descartes had also seen earlier that for the simplest polyhedra, the number of vertices plus the number of faces exceeded the number of edges by 2: V + F = E + 2.
Thus, in Figure 10.1 a cube has 8 vertices, 6 faces and 12 edges, and 6 + 8 = 14 + 2. The octahedron on the right has 6 vertices, 8 faces and also 12 edges, while the irregular polyhedron below which is a square pyramid face-to-face with a cube, has 9 vertices, 9 faces and 16 edges, and 9 + 9 = 16 + 2.
- Type
- Chapter
- Information
- Games and MathematicsSubtle Connections, pp. 115 - 121Publisher: Cambridge University PressPrint publication year: 2012