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
3 - The many symmetries of planar objects
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 composition of a temple is based on symmetry, whose principles architects should take the greatest care to master. Symmetry derives from proportion […]. Proportion is the mutual calibration of each element of the work and of the whole, from which the proportional system is achieved. No temple can have any compositional system without symmetry and proportion, unless, as it were, it has an exact system of correspondence to the likeness of a well-formed human being.
Vitruvius (1999: III.1.1)What immortal hand or eye
Could frame thy fearful symmetry?
W. Blake (1970: Tiger 4)The first epigraph opening this chapter is taken from Vitruvius' De Architectura, a compendium of architectural practice from the first century bce with enduring influence in Western architecture. Vitruvius states that architects “should take the greatest care to master” symmetry while designing a temple. A precise definition of symmetry is, however, missing from this text. We are instead told that the respect of symmetry passes through observing some proportions between the constituent parts of what is designed. The instance chosen by Vitruvius to exemplify this observance of proportions was the human body, and the proportions respected by nature, in the case of “a well-formed human being”, have the form of simple ratios between lengths in the body, as in “the face, from the chin to the top of the forehead […] should be one-tenth [of the total height of the body]”. This set of canonical proportions was the source of one of the most celebrated drawings of Leonardo da Vinci.
- Type
- Chapter
- Information
- Manifold MirrorsThe Crossing Paths of the Arts and Mathematics, pp. 39 - 82Publisher: Cambridge University PressPrint publication year: 2013