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
7 - Stretching 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
The drawing on the left of Figure 7.1 shows a fish (known as Argyropelecus olfersi among zoologists) over a grid with square units. The drawing on the right shows another fish (known as Sternoptyx diaphana), also over a grid whose cells are now parallelograms.
Their original purpose was to illustrate a way of comparing the forms of these two species which would be simpler and more precise than that favoured by morphologists at the beginning of the twentieth century. For the morphologist, Thompson writes (1961: 274), “when comparing one organism with another, describes the differences between them point by point, and ‘character’ by ‘character’” even though “he is from time to time constrained to admit the existence of ‘correlation’ between characters […].” Thompson is thus finding fault in the “local” nature of the morphologist's comparisons, the fact that they rely on an accumulation of details lacking a “global”, all encompassing, capacity of explanation. In contrast to this approach, Thompson suggests that a correlation in the forms of the two species above can be found of such a particular nature that it would explain all the “point by point differences”. More precisely, he maintains that the form on the right is the image of that on the left by a transformation of the plane.
A cursory glance at Figure 7.1 shows, however, that the required transformation is not an isometry: squares cannot become (non-square) parallelograms under the action of an isometry.
- Type
- Chapter
- Information
- Manifold MirrorsThe Crossing Paths of the Arts and Mathematics, pp. 158 - 187Publisher: Cambridge University PressPrint publication year: 2013