How old are the Platonic Solids?

Abstract

Recently a belief has spread that the set of five Platonic Solids has been known since prehistoric times, in the form of carved stone balls from Scotland, dating from the Neolithic period. A photograph of a group of these objects has even been claimed to show mathematical understanding of the regular solids, a millennium or so before Plato. I argue that this is not so. The archaeological and statistical evidence do not support this idea, and it has been shown that there are problems with the photograph. The high symmetry of many of these objects can readily be explained without supposing any particular mathematical understanding on the part of the creators, and there seems to be no reason to doubt that the discovery of the set of five regular solids is contemporary with Plato.

Acknowledgements

I thank Mr Graham Challifour for permission to reproduce his photograph (Figure 1), and Mr Alan Saville of the National Museum of Scotland, Ms Tracey Hawkins of the Glasgow Museums, and Dr Sally-Ann Coupar of the Hunterian Museum of Glasgow, for help with various objects in the collections of their museums.

Notes

¹ Certainly the tetrahedron, in the form of an early gaming die, is much older. For an entertaining account of this and other aspects of the history of the regular solids, see du Sautoy (2008, 40–58).

²  It can also be argued that the description ‘most beautiful’ suggests that Plato had some idea of what we call symmetry (Lloyd 2010).

³ The original claim was ‘by a millennium’. In fact, given the current dating of the objects, this could be extended to ‘more than two millennia’.

⁴ The claims in Critchlow's book were repeated by Lawlor (1982, 97), whose book includes a better reproduction of the photograph, and Figure 1 is taken from Lawlor's version.

⁵ National Museum of Scotland, Edinburgh. Items from this collection are referred to by their NMS numbers in subsequent notes. Illustrations can be found on their website: http://nms.scran.ac.uk/search/?PHPSESSID=nnog07544f585b9idc2sftdc13

⁶ The 5K objects are atypical, in that these two are heavily decorated with additional carvings, and a third is described as ‘oval’ rather than spherical; these will not be considered further.

⁷ There are examples with much higher numbers of knobs, extending to well over 100, but for nearly all of these only one, two or zero are known (three each for 27K and for 50K).

⁸ NMS 000-180-001-392-C; 000-180-001-369-C; 000-180-001-328-C.

⁹ http://www.ashmolean.org/ash/britarch/highlights/stone-balls.html

¹⁰ NMS 000-180-001-392-C.

¹¹ According to a personal communication from G Challifour, all the objects in his photograph (Figure 1) were from the same museum.

¹² NMS 000-180-001-392-C.

¹³ NMS 000-180-001-719-C.

¹⁴NMS 000-180-001-368-C.

¹⁵ The title of this web page includes the word ‘hoax’. In the correspondence on this page I have pointed out that there is no evidence to support any such suggestion; see also Hart (1998 addendum; 2009).

¹⁶ In addition to those in the NMS (see footnote 5), several useful illustrations can be found in Marshall 1977, and a few in Critchlow 1979; five more are available at the Ashmolean site (footnote 9). The Hunterian Museum in Glasgow also has a large collection, but access to the illustrations requires the museum reference number.

¹⁷ Marshall 1977, Figure 9:3; the object is AS 16 at the Inverurie museum.

¹⁸ Footnote 9; the curator's drawing shows this structure rather better than the photograph.

¹⁹Two views of this item, GLAHM B.1951.245d, are available at http://tinyurl.com/yhqhlxj

²⁰ Hunterian Museum GLAHM B.1951.112.

²¹ Hunterian Museum GLAHM B 1914 349.

²² Hunterian Museum GLAHM B.1951.876.

²³ There is another illustration of this in Critchlow (1979, 147) with a description which confirms this symmetry.

²⁴ As photographed, the six-fold axis is almost vertical, and enters through the left-hand top knob, see: Hunterian Museum GLAHM B.1951.245a.