How Old Are the Platonic Solids?
DAVID R. LLOYD
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.
Introduction
The flippant and rather too obvious answer to the question in the title is “as old as Plato.” However, a little investigation shows that the question is more complex and needs some firming up. If it is taken as referring to any of the solids, then there is ancient testimony (Heath 1981, 162) that some of the five predate Plato. This tradition associates the tetrahedron, cube, and dodecahedron with the Pythagoreans.1 The same source (Heath 1981, 162) claims that the remaining two, the octahedron and icosahedron, were actually discoveries made by Plato’s collaborator Theaetetus, and although there have been doubts about this claim, particularly for the octahedron (Heath 1956, 438), there is now substantial support for the idea that this ancient testimony, giving major credit to Theaetetus, is reliable (Waterhouse 1972; Artmann 1999, 249–251, 285, 296–298, 305–308).
Nevertheless, Plato is responsible for giving us the first description we have of the complete set of the five regular solids, in his dialogue Timaeus (about 360 BCE), and probably this is the reason we speak of Platonic solids. Plato was not claiming any originality here; rather he seemed to be assuming that the hearers in his dialogue, and his readers, would already be familiar with these solids. For Plato, their importance was that these were the “most beautiful” (kallistos) structures possible, and that from these he could construct a theory of the nature of matter. It is probable that this kallistos description alludes to what Waterhouse (1972) claims was then a relatively new discovery, that of the concept of regularity, and to the demonstration, which is recorded later by Euclid and which is probably also due to Theaetetus (Waterhouse 1972; Artmann 1999, 249–251), that there can only be five such structures.2 Thus, at least until recently, the consensus has been that although Plato is not responsible for the solids themselves, the discovery of the set of five, and of the fact that there are only five, are contemporary with him.
However, a belief has spread recently that the complete set of these five regular solids has been known since much earlier times, in the form of decorated stone balls from Scotland, all from the Neolithic period. The origin of this belief is a photograph of five of these objects (Figure 1), which first appeared in a book published more than 30 years ago (Critchlow 1979). In this book, it was alleged that the discovery of the set of the five regular solids, and at least some of the associated theory, predates Plato and his collaborators by a millennium3; other claims for ancient mathematical knowledge also appear in this book.4
The claim that the five solids were known at this time was accepted by Artmann, who comments that “All of the five regular solids appear in these decorations” (Artmann 1999, 300–301), though he is very skeptical of the other claims. He notes that a photograph of five of the objects had appeared in Mathematics Teaching (1985, p. 56). More recently, Atiyah and Sutcliffe (2003) published a paper that reproduces the photograph shown by Critchlow, with a caption describing these objects as “models” of the five solids, but they gave no attribution for the source of the illustration. This paper has also appeared in the collected works of Atiyah (2004), and in a review of this collection (Pekkonen 2006), it was claimed that the stone balls “provide perfect models of the Platonic Solids one thousand years before Plato or Pythagoras.” The word “models” suggests some sort of mathematical understanding on the part of the makers of the objects, and, almost certainly unintentionally, echoes Critchlow’s earlier claims. It is probably the appearance of the photograph in the paper by Atiyah and Sutcliffe, and the authority of the authors, which is responsible for the recent growth within the scientific and mathematical communities of the idea that the set of the five Platonic solids was known long before Plato. The photograph (see Figure 1) now makes occasional appearances in lectures on aspects of polyhedra and their symmetries, as an illustration of the idea, which can also be found on various websites, that the solids are much older than had been thought until recently.
FIGURE 1. Five stone balls, decorated with tapes that pick out some of the symmetry elements (copyright G. Challifour).
Clearly the appearance of the set of all five solids at this early time, if true, would be quite remarkable, and should the suggestion of some sort of mathematical understanding be justifiable, there would be substantial implications for the history of mathematics. I examine some aspects of the story in more detail here.
Archaeology and Statistical Frequencies
It is important to emphasize at the outset that the objects shown in Figure 1 are genuine Neolithic artifacts, almost certainly from the large collection belonging to The National Museum of Scotland (NMS),5 which the museum dates to within the period 3200 BCE to 2500 BCE. More than 400 such objects are known (Edmonds 1992), almost all roughly spherical, though there are a few prolate spheroids, and most are carved with approximately symmetric arrangements. With few exceptions, these objects are small enough to be held comfortably in one hand. The principal study of them, with a complete listing of those known at the time, is by Marshall (1977, 40–72). On many but not all of these objects, the carving is quite deep, so that there is a set of protrusions or knobs with a common center, as in the examples in Figure 1. They have been found in a variety of locations, but almost all within the boundaries of present-day Scotland.
Of the five shown in Figure 1, only the second and fifth types (reading from the left) have been discovered in large numbers. It is tempting to classify these two types as “tetrahedral” and “octahedral.” However, such terms are not used by archaeologists, who normally call these “4-knob” and “6-knob” forms. This label is sometimes abbreviated to “4K” and “6K,” and this convention is used here. The symmetries of the objects are discussed below. Some 43 of the 4K objects, and 173 of the 6K, are known. The other numbers given by Marshall, up to 16K, are 3K (6), 5K (2),6 7K (18), 8K (10), 9K (3), 10K (4), 11K (1), 12K (8), 13K (0), 14K (5), 15K (1), and 16K (1). Although there were carvers who were prepared to tackle spheres with larger numbers of knobs,7 perhaps to demonstrate their skill, most seem to have preferred to deal with 14K or fewer. Given that these objects could only be created by slow hand grinding, perhaps with the help of leather straps and abrasive scraps of rock, this preference is not surprising; it must be difficult, if not impossible, to achieve any precision when the knobs become small.
The first object in Figure 1, with 8K, is an example of a rare species. Although 10 examples of 8K are known, by no means all of these have the cube form shown here. Marshall, who had personally examined almost all the known examples, notes that for the 8K objects, “This group of balls has variety in the disposition of the eight knobs” (Marshall 1977, 42). No such comment is made about any other group, and examination of published illustrations of the 8K objects shows a total of only three that have this approximately cubic shape.8 There are substantially more (18) examples known for the 7K version, so it does not seem that the cube form of the 8K objects held a particular significance for the makers or users.
There is a small peak in the numbers at 12K and 14K, suggesting that these were interesting to the carvers. However, purely on statistical grounds, it seems unlikely that the makers of such a wide variety of objects would have made any connection to the particular selection shown in Figure 1, since two of these are commonly found, while the others are all unusual. The choice of this particular set of five for the photograph is probably dependent on knowledge that the carvers did not possess.
The Provenance of the Objects in Figure 1
The illustration is taken from Critchlow (1979, 132, Figure 114), who believes that he can demonstrate a high level of mathematical skills among the Neolithic peoples of the British Isles. The carved stone balls are only part of this narrative. The original caption to the photograph reads “A full set of Scottish Neolithic ‘Platonic solids’—a millennium before Plato’s time.”
Oddly, the book gives no indication of which museum owns the five objects shown, and because the text associated with this illustration gives a detailed description of the five balls that belong to the Ashmolean Museum at Oxford, some have concluded, incorrectly, that Figure 1 shows objects from the Ashmolean collection (Lawlor 1982; Atiyah and Sutclifffe 2003). It is evident from Critchlow’s text that he is describing a completely different set, with different numbers of knobs, from that shown in Figure 1. The Ashmolean set can be seen on the web, together with the original drawings made by a curator at the time of acquisition,9 and it is clearly different from Figure 1.
It is therefore of some interest to try to establish which objects were photographed, if only to see whether they might have been found together, since that could imply that the carvers or owners saw some sort of connection among these objects. Accompanying the original of Figure 1 there is another photograph (Critchlow 1979, 132, Figure 113), apparently taken on the same occasion, of a group of 4K objects, several of which carry markings that identify them as belonging to what was then the Scottish National Museum of Antiquities (NMA in the Marshall list), now The National Museum of Scotland (NMS) in Edinburgh. The fourth object in Figure 1 is also shown in a close-up view (Critchlow 1979, 149, figure 146), with the caption “on show in Edinburgh.” Comparison of this object with one of the illustrations on the NMS website10 makes it very likely that these two photographs are of the same object, recorded as having been discovered in Aberdeenshire. Thus it seems almost certain that the five objects in Figure 1 are from the NMS.11
There are so many examples of 6K objects that it is not possible to identify which particular one appears at the extreme right of the photograph. However, inspection of the images of the 8K objects on the NMS website shows that the leftmost object in Figure 1, the cube, is almost certainly one that was found in Ardkeeling, Strypes, Moray.12
From comparison with the group of 4K objects shown by Critchlow (1979, 132), and the museum’s own photograph, the second object in Figure 1 (4K) is probably one13 that is described as of “unknown origin.” Finally, the fourth object in Figure 1, like the third object, is recorded as having been discovered in Aberdeenshire.14 However, unlike the third object, this one is noted as having belonged to a John Rae (who died in 1893).
The museum descriptions vary considerably for these objects and indicate that they have been accumulated from four or five different sources over the years. Thus the five could not have been found together; they were brought together in the photograph, probably for the first time, in order to support a particular view.
How Many Platonic Solids?
Figure 1 was claimed, and has been widely accepted, as evidence that the carvers were familiar with all five of the Platonic solids. However, there is a problem with the tapes in this figure, which are not used consistently. For the second, fourth, and fifth objects, they are used to connect the knobs, though for the second one (the tetrahedron), connections are also shown between the interstices. However, for the first object, the tapes are connecting four-fold interstitial positions, and for the third object, they connect the three-fold interstices. This inconsistency has tended to hide the fact, first pointed out by Hart (1998) and later by le Bruyn (2009),15 that Figure 1 actually includes only four of the Platonic structures. In his text, Critchlow defines the knobs as vertices of the Platonic solids; this convention agrees with that used by chemists in describing atom positions in molecules. According to this convention, Figure 1 shows, from left to right, a cube, a tetrahedron, two icosahedra, and an octahedron. Although on the third and fourth objects the tapes are arranged differently, each is a 12K object. The main difference between these two is in the form of the knobs, one pronounced and well-rounded, the other flatter, so that the eye picks out pentagons, at least when helped by the tapes added to this object. If the photographs on the museum’s website, without tapes, are compared, then the difference between the two is much less obvious.
Thus there is no evidence here for a structure related to the fifth Platonic solid. If such a structure were to exist, it would be a 20K object with the knobs at the corners of a dodecahedron. There is a reference to “the dodecahedron on one specimen in the Museum of Edinburgh” (Artmann 1999, 305), but since this occurs in a paragraph which mentions the photographs in Critchlow, this almost certainly means the close-up photograph of the third object in Figure 1, with only 12K, which was referred to above (Critchlow 1979, 149, figure 146).
According to the Marshall list, there are only two balls known that have 20K; one of these balls is at the NMS. Alan Saville, senior curator for earliest prehistory at this museum, has provided a photograph which shows that this object is complex, and certainly not a dodecahedron. It could be considered as a modified octahedron, with five large knobs in the usual positions, but with the sixth octahedral position occupied by 12 small knobs, and in addition there are also three small triangles carved at some of the interstices, the three-fold positions of the “octahedron.” These bumps make up a total of 20 “protrusions,” though the word “knobs” is hard to justify.
The other 20K object is at the Kelvingrove Art Gallery and Museum in Glasgow. Photographs taken by Tracey Hawkins, assistant curator, show that this object also is far from being a dodecahedron, though this time there are 20 clearly defined knobs of roughly the same size. The shape is somewhat irregular, but two six-sided pyramids can be picked out, and much of the structure, though not all, is deltahedral in form, with sets of three balls at the corners of equilateral triangles.
No dodecahedral form with 20K has yet been found. There is therefore no evidence that the carvers were familiar with all five Platonic solids. Even for the four that they did create, there is nothing to suggest that they would have thought these in any way different from the multitude of other shapes that they carved. Thus there is no evidence that the concept of the set of five regular solids predates Plato. Nevertheless, the carvers have come up with a variety of interesting shapes, almost all with high symmetries, and I now turn to an examination of a wider grouping of these objects and suggest how the symmetries may have arisen.
The High Symmetries of These Ancient Objects
The five objects in Figure 1 show, at least approximately, the symmetries of four of the Platonic solids. However, most of the other known carved balls up to 14K, and many of the larger ones, also have approximations to high symmetries. Among the balls with relatively small numbers of knobs, up to 14K, almost all have deltahedral structures, formed by the linking of equilateral triangles. (A few of the balls have become worn or damaged, so the original form is not always obvious.)
Even though there is no evidence for the full set of Platonic solids among these objects, it might still be claimed that their symmetries suggest some sort of mathematical competence at this time. Thus it seems worthwhile to enquire if there could be other, nonmathematical reasons why such symmetric structures might have been created, but any such attempt at “explanation” needs to be as simple as possible.
The carvers are likely to have wanted to make their carved knobs as distinct as possible, and since close grinding work is required, using only the simplest of technologies, there would have been a need to keep the knobs as far away from each other as possible, simply to have room to work. Also, they would probably have seen the equilateral triangles that are generated automatically by packing in a pile of roughly spherical pebbles, or of fruits. They would have been familiar with seed heads of plants, where similar packing effects can be seen over part of a sphere, and it does not need much imagination to try to extend these triangular patterns to cover a complete sphere, which would generate a deltahedral structure.
If we assume that a guiding principle for the carvers was to keep the carved knobs as far away from each other as possible, or, equivalently, to produce an even spacing between these knobs, then the structures can be modeled in terms of repulsion between points on a sphere. Within chemistry, there is a well-known qualitative approach called “electron pair repulsion theory.” This theory rationalizes the observed geometry of simple molecules as the result of minimizing interactions between bond (and other) electron pairs by maximizing the distances between them, and in the commonest cases of four and six pairs, tetrahedral and octahedral molecular geometries are often rationalized in these terms. However, the numbers of knobs in these objects span a far greater range than that found for electron pairs in molecules. A much more useful set of data for the present comparison is available from calculations on the rather similar classical problem of the distribution of N electric charges over the surface of a sphere (Erber and Hockney 1997; Atiyah and Sutcliffe 2003). This problem is sometimes referred to as the Thomson problem (Atiyah and Sutcliffe 2003); an alternative name is the surface Coulomb problem (Erber and Hockney 1997).
Encouragingly for chemists, the results of such calculations of the minimum energy configuration of the N charges accurately reproduce the predictions of electron pair repulsion theory, as N varies over the normal range of numbers of electron pairs found in molecules. They show an almost total preference for deltahedra as the minimum energy configurations, and it is noticeable that the majority of the carved stone balls are also deltahedra, though for balls with large numbers of knobs, other patterns appear.
However, the only Platonic solids that are generated by minimizing interactions over the surface of a sphere are the three that are also deltahedra. The cube is not a minimum energy configuration; N = 8 gives the square antiprism, in which opposite faces of a cube have been rotated against each other by 45°. Also, the dodecahedron is not a minimum; the calculation for 20 particles shows a deltahedron with threefold symmetry.
The calculations become quite difficult as N increases because the number of false energy minima increases rapidly, particularly after N = 12. For this reason alone, we should not expect the Neolithic stone carvers to have discovered the optimum solutions for keeping the knobs apart for larger numbers, but a comparison between what they achieved and the calculations is interesting. A complete analysis would require examination of the objects themselves, which are spread across several museums; here only a partial analysis, based on published illustrations and descriptions, is attempted.16
The least number of knobs recorded is three, and there are five examples of this. The Marshall description of the set reads, “Two of the balls are atypical, having rounded projecting knobs making a more or less triangular object which is oval in section. The others have clear cut knobs.” It seems likely from this description that all the balls are more or less of planar trigonal symmetry, as expected from the surface Coulomb calculations. Many 4K balls have been illustrated, and all are clearly approximations to the expected tetrahedral symmetry. However, the execution is of variable quality; the one shown on the Ashmolean Museum website has one of the “three-fold” axes at least 20° away from the expected position. Tetrahedral geometry is difficult to work with since there is no direction from which a carver could check his or her work as it proceeded by “looking for opposites” (Edmonds 1992, 179), as in the next example.
In contrast to the geometrical variations with the tetrahedral 4K, the octahedral 6K balls seem to be more accurately executed, as well as being by far the commonest type. This situation may be due to the fact that a pair of poles and an equator are relatively easy to mark on a sphere by eye. After this step, two perpendicular polar great circles can be added, and there is one beautifully decorated ball, with no deep carving, on which exactly this pattern of three great circles is seen.17 Such a marking with perpendicular great circles allows the construction of octahedral 6K objects by using the intersections as markers for the centers of the knobs to be carved, but they could also provide a route to the 4K objects by using four of the centers of the spherical triangles as markers.
After the 4K and 6K balls, there are more examples of 7K than of any other.
Electron pair repulsion, and the surface Coulomb calculations, both predict a pentagonal bipyramidal structure, with five knobs in a plane, and most carvers seem to have discovered this structure. Seven of those illustrated show this; particularly good examples are at the Ashmolean Museum18 and at the Hunterian Museum.19 The latter museum also has an example of 7K that has an asymmetric structure with very oval shapes for the knobs20; this shaping might be a consequence of the carver having to compensate after missing the symmetric structure.
The 8K objects clearly presented the carvers with some difficulty, since they seem to have been uncertain about the best form to use. Because this object is the only small unit that does not work as a deltahedron, it is hardly surprising that they did not discover the optimum square antiprismatic structure. However, there is an apparently clear path to the cube through the polar circle construction described above, which divides the sphere into eight spherical triangles, as illustrated by the tapes on the 8K object at the left in Figure 1. The fact that so many tetrahedra but so few cubes have been found almost suggests an avoidance of the cube structure; more equal numbers would have been expected if the set of Platonic solids had any significance for the carvers.
There is only one 9K object for which an illustration is available,21 and this object is badly worn. There is a l0K object in the collection of the Ashmolean Museum, which can be seen on their website. A detailed description of this particular object by Critchlow (1979, 147) makes it very clear that the form is essentially a pair of square pyramids on a common four-fold axis in a “staggered” configuration (that is, rotated against each other by 45°). This method creates a deltahedron that has the same symmetry as that calculated (D4d in the Schoenflies convention used by Atiyah and Sutcliffe 2003). There is a similar object in the Hunterian collection.22
From the carver’s point of view, the form they chose for 12K might well have seemed to be an extension of the 10K structure, a staggered pair of pentagonal pyramids, but the result has the much higher symmetry of the icosahedron, the Platonic structure predicted by the calculations. There are two examples of this in Figure 1, and there is a third in the Dundee museum (Critchlow 1979, 145).
Some of these suggested structural principles outlined still apply to the two 14K objects that have been illustrated. Both are pairs of hexagonal pyramids, and the one in the Ashmolean Museum has the pyramids in a staggered configuration, as in the calculated structure (D6d symmetry).23 The other example has the pyramids related by a reflection plane,24 an “eclipsed” geometry. The simple “keep the knobs apart” principle seems to be joined by other influences now, but as noted earlier, even modern calculations of structure become difficult after N = 12. For higher numbers, the predicted deltahedral structures become less common among the carved spheres; one possible reason for this phenomenon may be that eclipsed structures are easier to construct. The knobs have all been generated from spheres by grinding away grooves, and a reflection plane allows two or more knobs to be ground together with a single longer groove.
It seems clear that the high symmetries of the objects with relatively small numbers of knobs arise quite naturally from the simple principle of keeping knobs away from each other, or, equivalently, maintaining even spacing. Three of the Platonic solids are generated automatically in this way, and the large numbers found for the tetrahedron and octahedron suggest that the carvers, or their sponsors, found something attractive about these structures, quite possibly connected to what we would call “aesthetic” considerations. However, the gross disparity in numbers between the octahedra and the cubes is a fairly clear indication that neither the concept of regularity, nor any of the other mathematical aspects of the Platonic solids, were understood at this time.
Conclusions
Although there is high-quality craftsmanship in these Neolithic objects, there seems to be little or no evidence for what we would recognize as mathematical ideas behind them. In particular, there is no evidence for a prehistoric knowledge of the set of five Platonic solids, and it seems inappropriate to describe the objects as “models” of anything. The conventional historical view that the discovery of the concept of the set of five regular solids was contemporary with Plato can still be taught, and it is unchallenged by the existence of these beautiful objects. Reasons for producing them should be sought in disciplines such as aesthetics and anthropology rather than in mathematics.
Postscript
There is no evidence that the Neolithic sculptors ever made a stone dodecahedron. However, the set of five stone Platonic solids has now been completed, after a delay of five millennia, by the contemporary British sculptor Peter Randall-Page. His “Solid Air II” is shown in Figure 2. The base is one of the pentagonal groups of balls, so one of the five-fold axes of this dodecahedron is vertical. The first photograph is taken close to one of the other five-fold axes, and shows a pentagonal face with five others surrounding it.
FIGURE 2. “Solid Air II” by Peter Randall-Page. Used by permission of the artist.
I thank Graham Challifour for permission to reproduce his photograph (Figure 1) and Alan Saville of The National Museum of Scotland, Tracey Hawkins of the Glasgow Museums, and 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 claim 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. Figure 1 is taken from Lawlor’s version.
5. The 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 objects are not considered further.
7. There are examples with much higher numbers of knobs, extending to well over 100, but for almost all of these examples, 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, and 000-180-001-328-C.
9. http://www.ashmolean.org/ash/britarch/highlights/stone-balls.html.
10. NMS 000-180-001-363-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.
14. NMS 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 and Art Gallery 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 Carnegie Inverurie Museum.
18. Note 9; the curator’s drawing shows this structure rather better than the photograph.
19. Two 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 phenomenon in Critchlow (1979, 147) with a description that 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).
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