A truly good math-story gets spread rather than scrutinized. And a good story it was : more than a millenium before Plato, the Neolithic Scottish Math Society classified the five regular solids : tetrahedron, cube, octahedron, dodecahedron and icosahedron. And, we had solid evidence to support this claim : the NSMS mass-produced stone replicas of their finds and about 400 of them were excavated, most of them in Aberdeenshire.

Six years ago, Michael Atiyah and Paul Sutcliffe arXived their paper Polyhedra in physics, chemistry and geometry, in which they wrote :

Although they are termed Platonic solids there is

convincing evidence that they were known to the Neolithic people of Scotland at least a

thousand years before Plato, as demonstrated by the stone models pictured in ﬁg. 1 which

date from this period and are kept in the Ashmolean Museum in Oxford.

Fig. 1 is the picture below, which has been copied in numerous blog-posts (including my own scottish solids-post) and virtually every talk on regular polyhedra.

From left to right, stone-ball models of the cube, tetrahedron, dodecahedron, icosahedron and octahedron, in which ‘knobs’ correspond to ‘faces’ of the regular polyhedron, as best seen in the central dodecahedral ball.

But then … where’s the icosahedron? The fourth ball sure looks like one but only because someone added ribbons, connecting the centers of the different knobs. If this ribbon-figure is an icosahedron, the ball itself should be another dodecahedron and the ribbons illustrate the fact that icosa- and dodeca-hedron are dual polyhedra. Similarly for the last ball, if the ribbon-figure is an octahedron, the ball itself should be another cube, having exactly 6 knobs.

Who did adorn these artifacts with ribbons, thereby multiplying the number of ‘found’ regular solids by two (the tetrahedron is self-dual)?

The picture appears on page 98 of the book Sacred Geometry (first published in 1979) by Robert Lawlor. He attributes the NSMS-idea to the book Time Stands Still: New Light on Megalithic Science (also published in 1979) by Keith Critchlow. Lawlor writes

The five regular polyhedra or

Platonic solids were known and worked with

well before Plato’s time. Keith Critchlow in

his book Time Stands Still presents convincing

evidence that they were known to the Neolithic peoples of Britain at least 1000 years

before Plato. This is founded on the existence

of a number of sphericalfstones kept in the

Ashmolean Museum at Oxford. Of a size one

can carry in the hand, these stones were carved

into the precise geometric spherical versions of

the cube, tetrahedron, octahedron, icosahedron

and dodecahedron, as well as some additional

compound and semi-regular solids, such as the

cube-octahedron and the icosidodecahedron.

Critchlow says, ‘What we have are objects

clearly indicative of a degree of mathematical

ability so far denied to Neolithic man by any

archaeologist or mathematical historian’. He

speculates on the possible relationship of these

objects to the building of the great astronomical stone circles of the same epoch in Britain:

‘The study of the heavens is, after all, a

spherical activity, needing an understanding of

spherical coordinates. If the Neolithic inhabitants of Scotland had constructed Maes Howe

before the pyramids were built by the ancient

Egyptians, why could they not be studying the

laws of three-dimensional coordinates? Is it not

more than a coincidence that Plato as well as

Ptolemy, Kepler and Al-Kindi attributed

cosmic significance to these figures?’

As Lawlor and Critchlow lean towards mysticism, their claims should not be taken for granted. So, let’s have a look at these famous stones kept in the Ashmolean Museum. The Ashmolean has a page dedicated to their Stone Balls, including the following picture (the Critchlow/Lawlor picture below, for comparison)

The Ashmolean stone balls are from left to right the artifacts with catalogue numbers :

- Stone ball with 7 knobs from Marnoch, Banff (AN1927.2728)
- Stone ball with 6 knobs and isosceles triangles between, from Fyvie, Aberdeenshire (AN1927.2731)
- Stone ball with 6 knobs and isosceles triangles between, from near Aberdeen (AN1927.2730)
- Stone ball with 4 knobs from Auchterless, Aberdeenshire (AN1927.2729)
- Stone ball with 14 knobs from Aberdeen (AN1927.2727)

Ashmolean’s AN 1927.2729 may very well be the tetrahedron and AN 1927.2727 may be used to forge the ‘icosahedron’ (though it has 14 rather than 12 knobs), but the other stones sure look different. In particular, none of the Ashmolean stones has exactly 12 knobs in order to be a dodecahedron.

Perhaps the Ashmolean has a larger collection of Scottish balls and today’s selection is different from the one in 1979? Well, if you have the patience to check all 9 pages of the Scottish Ball Catalogue by Dorothy Marshall (the reference-text when it comes to these balls) you will see that the Ashmolean has exactly those 5 balls and no others!

The sad lesson to be learned is : whether the Critchlow/Lawlor balls are falsifications or fabrications, they most certainly are NOT the Ashmolean stone balls as they claim!

Clearly this does not mean that no neolithic scott could have discovered some regular polyhedra by accident. They made an enormous amount of these stone balls, with knobs ranging from 3 up to no less than 135! All I claim is that this ball-carving thing was more an artistic endeavor, rather than a mathematical one.

There are a number of musea having a much larger collection of these stone balls. The Hunterian Museum has a collection of 29 and some nice online pages on them, including 3D animation. But then again, none of their balls can be a dodecahedron or icosahedron (according to the stone-ball-catalogue).

In fact, more than half of the 400+ preserved artifacts have 6 knobs. The catalogue tells that there are only 8 possible candidates for a Scottish dodecahedron (below their catalogue numbers, indicating for the knowledgeable which museum owns them and where they were found)

- NMA AS 103 : Aberdeenshire
- AS 109 : Aberdeenshire
- AS 116 : Aberdeenshire (prob)
- AUM 159/9 : Lambhill Farm, Fyvie, Aberdeenshire
- Dundee : Dyce, Aberdeenshire
- GAGM 55.96 : Aberdeenshire
- Montrose = Cast NMA AS 26 : Freelands, Glasterlaw, Angus
- Peterhead : Aberdeenshire

The case for a Scottish icosahedron looks even worse. Only two balls have exactly 20 knobs

- NMA AS 110 : Aberdeenshire
- GAGM 92 106.1. : Countesswells, Aberdeenshire

Here NMA stands for the National Museum of Antiquities of Scotland in Edinburg (today, it is called ‘National Museums Scotland’) and

GAGM for the Glasgow Art Gallery and Museum. If you happen to be in either of these cities shortly, please have a look and let me know if one of them really is an icosahedron!

UPDATE (April 1st)

Victoria White, Curator of Archaeology at the

Kelvingrove Art Gallery and Museum, confirms that the Countesswells carved stone ball (1892.106.l) has indeed 20 knobs. She gave this additional information :

The artefact came to Glasgow Museums in the late nineteenth century as part of the John Rae collection. John Rae was an avid collector of prehistoric antiquities from the Aberdeenshire area of Scotland. Unfortunately, the ball was not accompanied with any additional information regarding its archaeological context when it was donated to Glasgow Museums. The carved stone ball is currently on display in the ‘Raiders of the Lost Art’ exhibition.

Dr. Alison Sheridan, Head of Early Prehistory, Archaeology Department, National Museums Scotland makes the valid point that new balls have been discovered after the publication of the catalogue, but adds :

Although several balls have turned up since Dorothy Marshall wrote her synthesis, none has 20 knobs, so you can rely on Dorothy’s list.

She has strong reservations against a mathematical interpretation of the balls :

Please also note that the mathematical interpretation of these Late Neolithic objects fails to take into account their archaeological background, and fails to explain why so many do not have the requisite number of knobs! It’s a classic case of people sticking on an interpretation in a state of ignorance. A great shame when so much is known about Late Neolithic archaeology.

Never let the truth stand in the way of a good story!

Nice research.

Michel

Very interesting! Good work.

Q: Why are you digging in Scotland?

A: I’m a double major in Archaeology and Mathematics, and I’m hoping to discover Neolithic knowledge of the 13 Archimedean solids.

Nice work! You must have done a lot of digging. I contacted

Dr. Sheridan some years back and was not impressed with

what she said.

The question remains: What was the use of these artifacts?

The best answer so far (from an antiquarian) is: “We do not

know.”

A suggestion is that carrying one conferred speaking status

at meetings.

It would be fun to get a documentary made on the stones

and the mathematics they lead to — up to the monster and

its subgroups and their binary polyhedral group connections.

What impresses me most is the quality of workmanship. What tools were available to do this? How long would it have taken to make a piece?

Excellent detective work!

To me, the leftmost of the balls in Lawlor’s picture doesn’t look like

anyof the balls in the Ashmolean picture. It seems more polished and shiny, and also the wrong shape. What do you think?I’m giving a short talk on this at the AMS meeting here in Riverside…

Oh, maybe this is what you meant by “…but the other stones sure look different.”

Hi John,

even though Im on vacation in the French mountains on a slow internet connection I couldnt escape from the shockwave your post+talk is making… Just got an email saying “now you’re famous in the USA, unfortunately not for your geometry, but for your balls!” Thanks!

Speaking of emails, the UK-curators of archeology departments must have had enough of mathematicians’ questions about the scottish balls. Merely thanking one of them for the info obtained, I got this reply :

“During the week beginning 30 March, NMS Archaeology’s abilities to deal quickly with email messages will be compromised by the systematic deep-cleaning of the Departmental floor surface, followed by spraying with insecticide, as a way of addressing a moth infestation. Normal service will be resumed ASAP.”

I took this as a hint to stop harassing them…

I went to the library and checked out Keith Critchlow’s book Time Stands Still, and discovered some interesting things.

Critchlow was certainly

nottrying to convince people that the Ashmolean Museum contained stone balls shaped like all 5 Platonic Solids – just the opposite. His book does contain the infamous figure reproduced by Atiyah and Sutcliffe, but he says nothing about the source of this photo except that it was taken by one Graham Challifour. It also has lots of other interesting pictures of stone balls, many with ribbons on them “to demonstrate the symmetry”.So, it may be Lawlor who first claimed that the Ashmolean contains stone balls shaped like Platonic Solids. Perhaps more a matter of sloppy scholarship than a “hoax”?

Though one must wonder how Challifour created that picture.

I went to Kelvingrove Museum (Glasgow) today and found three 6-knobbed neolithic stone balls, but not the 20-knobbed one from Countesswells that you mention above. Apparently Victoria White has now left the Glasgow Museums service, but I’ve sent an email of enquiry (or harassment) anyway, asking if I can get a look.

It’s fun to see this story evolve!

The original post is a great piece of detective work, and I greatly enjoyed it, but after a while I got worried….

There is indeed clear evidence of error by several people, but where is the evidence of “hoax”? This word implies deceit, even if not necessarily malicious, and that is a rather unpleasant accusation, and there ought to be evidence for this. Piltdown man was a hoax- is this? Even more so, ” falsifications or fabrications” needs to be backed up with evidence.

A general point first- in the Critchlow book which is the source of the problem it is claimed that “The picture shows a complete set of Scottish Neolithic “Platonic Solids””. This, as noted in the main post above, is incorrect, in that the set is incomplete. However, knobs seem to be considered to be vertices in the book, not faces, so the missing 20- projection object would be considered to be a dodecahedron, not an icosahedron, at least if the conventions were held to consistently. The middle item marked up (inconsistently) as a dodecahedron has the same number of projections, 12, as on the item marked up as an icosahedron, so should really have been shown as a second icosahedron, but that would have spoilt the effect!

As John Baez has found (see above post ), the Critchlow book which is the source of the problem makes NO claim about the museum in which the photograph was taken. You have shown, by looking at the web site for the Ashmolean Museum, that their objects are clearly NOT those photographed. But Critchlow never claimed that they were! Instead, in the text he gives a very detailed description of the Ashmolean item which has 14 projections, and analyses the symmetry carefully- he says it is not an icosahedron; from his description, it appears to be D6v, exactly what Atiyah and Sutcliffe predict for the 14-electron Thomson problem! I think he’s right. I can convince myself, looking at the Ashmolean site original (in colour) that in their picture the C6 axis is almost horizontal, entering the object from the projection which is furthest to the right. Also he says that one of the Ashmolean objects has 7 projections, so it just is not possible that he intended anyone to think his pictures were from the Ashmolean. Unless something wrong with the balls which were actually photographed can be found, the evidence for “hoax” is non-existent.

Lawlor may have assumed that the mention of Ashmolean in the text meant that that was the source of the picture, and if so, I agree that’s sloppy, but that’s no hoax either.

Furthermore, Critchlow is perfectly well aware of the concept of a dual- he spends some time explaining the cube-octahedron duality. He even describes the Ashmolean 6-projection objects as cuboctahedra, on account of the triangles, mentioned on the Ashmolean website, which occupy the 3-fold positions. These can be seen in the central picture above. (Not so sure about ‘isosceles’, though…)

I have no idea whether or not Critchlow and/or Challifour know about this correspondence, but if they do, they might be rather hurt by the accusations of falsifying something. They seem to me to have done nothing wrong, and have probably got the maths right except for the completeness of the ‘Platonic’ set; quite a lot of real mathematicians, not to mention chemists and physicists, seem not to have noticed this incompleteness over the last few years, so I can’t find much fault with them for that. Maybe they are due an apology?

Two minor corrections to the above on points which could cause (more) confusion. The reference to “3-fold positions” is a sloppy way of saying “the positions of the 3-fold axes” Also “D5v” should read “D5d”.

The polyhedral stone spheres. is a secret language of the neolithic people to

inform the people about how the universe was composed by god in the begining of

creation from primordial matter.

I had the inspiration to prepare a scientific paper from these sculptures and is expected to be published in an international scientific journal.

Dtd 20th April, 2010

Great detective work. Just one thing though:

I feel that this is a false dichotomy. There is considerable overlap between, one the one hand, the aesthetics of sculpture and pattern, and on the other, the aesthetics of mathematics. Even if these balls are not platonic solids, the problem of finding attractive layouts of knobs on balls with pleasing symmetry is a geometrical one.

This is Dr Sheridan. I would be interested to hear why Mr McKay was not impressed with what I had written in reply to his correspondence.

Dr Sheridan here. I am sorry that you misinterpreted my automated Out of Office message as a request to stop ‘harassing’ me! That was not the intention – as I think you should have realised, given that it was flagged as an ‘Out of Office’ message!

As a general point, and writing as an archaeologist, I do wish that those who spend so much energy debating whether carved stone balls were representations Platonic solids or not would bother to acquaint themselves with the archaeological background of these objects. They need to be understood within the Late Neolithic of Scotland.