Book contents
- Frontmatter
- Contents
- Mathematics: user's manual
- Appetizers
- 1 Space and geometry
- 2 Motions on the plane
- 3 The many symmetries of planar objects
- 4 The many objects with planar symmetries
- 5 Reflections on the mirror
- 6 A raw material
- 7 Stretching the plane
- 8 Aural wallpaper
- 9 The dawn of perspective
- 10 A repertoire of drawing systems
- 11 The vicissitudes of perspective
- 12 The vicissitudes of geometry
- 13 Symmetries in non-Euclidean geometries
- 14 The shape of the universe
- Appendix: Rule-driven creation
- References
- Acknowledgements
- Index of symbols
- Index of names
- Index of concepts
2 - Motions on the plane
Published online by Cambridge University Press: 05 May 2013
- Frontmatter
- Contents
- Mathematics: user's manual
- Appetizers
- 1 Space and geometry
- 2 Motions on the plane
- 3 The many symmetries of planar objects
- 4 The many objects with planar symmetries
- 5 Reflections on the mirror
- 6 A raw material
- 7 Stretching the plane
- 8 Aural wallpaper
- 9 The dawn of perspective
- 10 A repertoire of drawing systems
- 11 The vicissitudes of perspective
- 12 The vicissitudes of geometry
- 13 Symmetries in non-Euclidean geometries
- 14 The shape of the universe
- Appendix: Rule-driven creation
- References
- Acknowledgements
- Index of symbols
- Index of names
- Index of concepts
Summary
How could we certify that two figures on the plane have the same shape? An approach that seems to be simple is to move one figure to place it on top of the other and, once this is done, check that they match, i.e. that their overlapping is perfect. In a bricolage implementation of this idea, one of the figures would be drawn on a fixed, say white, board and the other on a transparency. In this implementation, “to move one figure” is equivalent to moving the transparency.
If the transparency is initially placed on top of the white board, to each point of the former there corresponds a point on the board, namely the point “below”. After moving it to attempt a matching, each point on the transparency corresponds to another point on the board. This correspondence induces, in turn, a correspondence between points on the board. Indeed, for every point Pb on the board we consider the point Pt above it on the transparency. Then we move the transparency and consider the new point P′b which is now below Pt. A way (admittedly convoluted looking at first) to describe the movement itself would consist of describing the transformation that associates to each point Pb on the board the corresponding P′b. This chapter pursues this idea. In doing so, we will eventually prove that there are, essentially, just a few ways to move figures on the plane.
Translations
Arguably the simplest of the transformations on the plane, a translation maps every point on the plane to the only point at a given distance in a given direction.
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- Information
- Manifold MirrorsThe Crossing Paths of the Arts and Mathematics, pp. 27 - 38Publisher: Cambridge University PressPrint publication year: 2013