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?

Critchlow was certainly not trying 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.

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…

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

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.