# Symmetry and Beauty in Plato

## Abstract

**:**

## 1. Introduction

## 2. Symmetry, and the use of the word "summetria" by Plato

“in Mathematics, having a common measure; freq. denied of the relation between the diagonal of a square and its side.”

## 3. Beauty in Plato

“I do not mean by beauty of form such beauty as that of animals or pictures, which many would suppose to be my meaning; but, says the argument, understand me to mean straight lines and circles, and the plane or solid figures which are formed out of them by turning-lathes and rulers and measures of angles; for these I affirm to be not only relatively beautiful, like other things, but they are eternally and absolutely beautiful, and they have peculiar pleasure, quite unlike the pleasures of scratching.”

## 4. The Meno argument

Socrates: "And does not this line, reaching from corner to corner, bisect each of these spaces?”

## 5. The Timaeus

“there is no shape more perfect and none more similar to itself.”(33b)

_{4h}symmetry. Theaetetus’s discovery means that if, and only if, the dimensions of such an object are adjusted correctly, then a previously unrecognised regular figure, one which is ’similar to itself’, the regular octahedron, appears. Bipyramids might well have been constructed or visualised at earlier times, but there would have been nothing special about any of them before the concept of regularity had been recognised.

_{d}, O

_{h}and I

_{h}of the regular polyhedra [24,25]. In language which might have been easier to explain to Plato, he has preserved the regularity of the bodies by using his composition of the triangles.

## 6. Final remarks

“There is no evidence that either Plato or Euclid possessed a mathematical formulation of the concept of symmetry; to them the appeal of the regular solids seems to have been primarily aesthetic.”

## 7. Acknowledgements

## References and Notes

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**Figure 1.**Constructions of equilateral triangles; a) An apparently simple species; b) Plato’s specification at Timaeus 54d-e.

**Figure 2.**A model octahedron constructed from 1b equilateral triangles, according to Plato’s description.

© 2010 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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Lloyd, D.R.
Symmetry and Beauty in Plato. *Symmetry* **2010**, *2*, 455-465.
https://doi.org/10.3390/sym2020455

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Lloyd DR.
Symmetry and Beauty in Plato. *Symmetry*. 2010; 2(2):455-465.
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**Chicago/Turabian Style**

Lloyd, David R.
2010. "Symmetry and Beauty in Plato" *Symmetry* 2, no. 2: 455-465.
https://doi.org/10.3390/sym2020455