Who Discovered the Icosahedron?

November 4, 2009

Posted by John Baez

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This weekend we’re having a meeting of the American Mathematical Society here at Riverside. Julie Bergner and I are running a special session on Homotopy Theory and Higher Algebraic Structures, and there will also be two special sessions on knot theory, one run by Alissa Crans and Sam Nelson. It should be fun! And it’s starting already: Khovanov will be giving a colloquium talk today.

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But I’m giving a talk in another session — the session on History and Philosophy of Mathematics, run by Shawnee McMurran and James J. Tattersall. Shawnee was a grad student here at UCR back when I first arrived.

My talk is not very profound or professional, but I hope it’s at least fun:

John Baez, Who Discovered the Icosahedron?

It’s designed to look best in full screen mode, at least on my small laptop.

As usual, comments and corrections are eagerly awaited! I hope to keep delving into these issues as the years go by. I’m already trying to recruit my Scottish friends to investigate the mysterious stone balls at the National Museum of Scotland in Edinburgh and the Glasgow Art Gallery and Museum. And I’m going to find out more about Scholium 1 in Book XIII of Euclid’s Elements.

Posted at November 4, 2009 5:02 PM UTC

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Prehistoric Scottish Carved Stone Balls

Here are the carved balls I found in the National Museum of Scotland when I was last there: carved balls.

Posted by: Dan Piponi on November 4, 2009 6:04 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

That’s the Kelvingrove Art Gallery and Museum, in Glasgow. The official website is here.

Posted by: Tom Leinster on November 4, 2009 6:16 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

Lieven le Bruyn links to a page of 21 ancient Scottish stone balls kept at the Hunterian Museum (part of the University of Glasgow). Amazingly, the museum has provided online 360 degree animations of every one of them. But as Lieven points out, none is an icosahedron.

Posted by: Tom Leinster on November 4, 2009 6:31 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

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I made an interesting discovery today! I found Keith Critchlow’s book Time Stands Still — the oldest book I know that contains these pictures of Scottish balls dressed up in ribbons to look like the 5 Platonic solids:

Later authors, including Atiyah, claim these balls came from the Ashmolean Museum in Oxford. But reading this book, I found something interesting.

Posted by: John Baez on November 5, 2009 3:50 AM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

My prejudice in this is that of a chemist. Chemists talk a lot about octahedra, icosahedra etc, but chemists don’t synthesise Platonic Solids as drawn in textbooks, they make molecules, which from a math-physics viewpoint are collections of blobs of electron density, rather like the blobs, or knobs, on the Scottish Balls (I hope your server’s obscenity filter is off).

An octahedral molecule has an electron density map remarkably like the far right hand object in the Graham Challifour/Atiyah and Sutcliffe picture reproduced above, and chemists would have no difficulty in recognising this as octahedral. The point of all this is that the convention which Critchlow is using in his book is exactly the same as the one chemists use: the balls represent VERTICES, not faces, despite the comments in the post by Lieven LeBruyn referred to in John’s lecture on this topic. The ribbons in the photograph are emphasising the symmetries, but are not used totally consistently- see the left hand object, which clearly is a cube with eight not very deeply sculpted ‘balls’, but the ribbons do not connect these.

There is an important error in the interpretation by Critchlow, carried over by Atiyah and Sutcliffe into the wider world. The set of five is NOT a complete set, as LeBruyn points out; there are five because there is a duplication. However, what is being duplicated? Look at the object second from right in the picture. This, in chemist’s language, and in Critchlow’s, is an ICOSAHEDRON. But so is the object next to it, although it has been dressed up to look like a dodecahedron! The mistake probably comes from the grooves on this middle object being much shallower than on the other icosahedron with bigger balls, so to speak.

Possibly another contributing factor here is the perfectly human desire to find what you are looking for. If you know what you are looking for, it can be all too easy to find it, see cold fusion, N-rays, polywater, flying saucers, etc. If you want to find a dodecahedron in an object with icosahedral symmetry, of course you can find something with 20 points, but you cannot convert 12 projections into 20. However Critchlow does know about the concept of the dual of a polyhedron, so he shouldn’t be let off too lightly; probably he should have caught this before putting that photograph into his book. However, so should all the rest of us, chemists included, who have been to lectures on polyhedra and have seen the photograph reproduced! All credit to LeBruyn for showing us all how gullible we can be; as he says, it’s a great story….

Three points emerge:

1) There is no fakery going on- see my contribution to the LeBruyn thread, which is an expansion of a comment by John.

2) There ARE stone balls which are reasonable representations of icosahedra, and the missing 20- knob object is a dodecahedron with knobs at the 20 vertices.

2) Theaetetus almost certainly was the mathematician who discovered the concept of regularity (see Waterhouse), who recognised the octahedron as being an example of this concept (Waterhouse), and discovered the construction of the icosahedron (Sachs, Artmann and others). He almost certainly REdiscovered the icosahedron as a shape, unless some unknown Mediterranean trader got much further north than we think and brought a sample home from Scotland which landed up in Plato’s Academy- possible, but so are flying saucers. However, I think we have to give credit for the original making of icosahedra to the Scottish carvers. There seem to be at least three examples, since Critchlow also shows a picture of objects which are in the Dundee museum, one of which he says is icosahedral, and I don’t see any reason to doubt him any more. Theaetetus, of course, doesn’t lose anything in all this, except a label.

Posted by: Bob Lloyd on January 18, 2010 11:11 AM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

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Thanks for the email and the link to your paper:

D. Robert Lloyd, The chemistry of Platonic triangles: problems in the interpretation of the Timaeus, HYLE–International Journal for Philosophy of Chemistry, Vol. 13, No.2 (2007) 99-118.

It looks very interesting!

The real problem with the famous Critchlow-Challifour picture of stone balls is that Critchlow’s book doesn’t say where those stone balls came from. Not the Ashmolean, apparently, since Critchlow writes:

… the author has, during the day, handled five of these remarkable objects in the Ashmolean museum…. I was rapt in admiration as I turned over these remarkable stone objects when another was handed to me which I took to be an icosahedron… On careful scrutiny, after establishing apparent fivefold symmetry on a number of the axes, a count-up of the projections revealed 14! So it was not an icosahedron.

But he never says where the 5 stone balls in the picture do come from.

By the time we reach Lawlor’s 1982 book, the myth has arisen that they can be found in the Ashmolean:

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 spherical stones 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…

Maybe Lawlor just made this up?

Anyway, it’s pretty clear by now that the stone balls photographed by Challifour are not the stone balls housed at the Ashmolean. My friend Tom Leinster has been hunting for Scottish stone balls shaped like icosahedra or dodecahedra… but so far, no luck.

So the problem is one of bad scholarship, at the very least: if you claim you’ve seen 5 Scottish stone balls shaped like Platonic solids, you have a duty to say where you found them, so other people can take a look. But I’m not really interested in taking people to task for their sins against good scholarship. I really just want to know who discovered the icosahedron! So if there are Scottish stone balls lurking around shaped like Platonic solids, I want to know where they are.

Posted by: John Baez on January 18, 2010 5:59 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

Now here is the obligatory dumb question: Why is the icosahedron a pure mathematical creation in contrast to the other four platonic solids?
Is it because it does not have an “associated crystallographic point group” (meaning perfect crystals with it’s symmetry group do not exist) unlike the dodecahedron?

Posted by: Tim vB on November 5, 2009 12:37 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

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Tim wrote:

Why is the icosahedron a pure mathematical creation in contrast to the other four platonic solids?

I was quoting Benno Artmann on this because it’s an interesting notion, not because I necessarily agree with it.

I think his idea is that the Greeks could have seen things in nature shaped like regular tetrahedra, cubes, regular octahedra and roughly like regular dodecahedra. Or else ordinary folks could have invented some of these shapes without any mathematical thought. On the other hand, the icosahedron might have been discovered as part of the classification of regular polyhedra.

This classification theorem appears as a remark right near the end of Book XIII of Euclid’s Elements. It seems quite possible that Theaetetus was the first to prove this theorem, and that he discovered the icosahedron at about the same time.

If so, the icosahedron might be the first example of an ‘exceptional object’ in mathematics: roughly speaking, an object that’s not part of a systematic family of objects as its type, which is discovered while proving a classification theorem.

A more sophisticated example of an exceptional object is the simple Lie group $E_8$ . And interestingly, the group $E_8$ is deeply connected to the icosahedron!

Is it because it does not have an “associated crystallographic point group” (meaning perfect crystals with it’s symmetry group do not exist) unlike the dodecahedron?

The regular dodecahedron doesn’t have an associated crystallographic point group either! The symmetries of the dodecahedron are the same as those of the icosahedron.

So what’s the difference?

Iron pyrite commonly forms ‘pyritohedra’, which look similar to regular dodecahedra, but are made of little cubic crystal cells, like this:

See:

Steven Dutch, Building isometric crystals with unit cells.

The pyritohedron has 12 pentagonal faces, orthogonal to these vectors:

(2,1,0)   (2,-1,0)   (-2,1,0)   (-2,-1,0)
(1,0,2)   (-1,0,2)   (1,0,-2)   (-1,0,-2)
(0,2,1)   (0,2,-1)   (0,-2,1)   (0,-2,-1)

On the other hand, a regular dodecahedron has 12 pentagonal faces orthogonal to the following vectors, in which the number 2 has been replaced by the golden ratio Φ

= (\sqrt{5} + 1)/2

(Φ,1,0)   (Φ,-1,0)   (-Φ,1,0)   (-Φ,-1,0)
(1,0,Φ)   (-1,0,Φ)   (1,0,-Φ)   (-1,0,-Φ)
(0,Φ,1)   (0,Φ,-1)   (0,-Φ,1)   (0,-Φ,-1)

These vectors are also the vertices of a regular icosahedron!

But if we use the number 2 instead of the number $\Phi$ , we get the vertices of a so-called ‘pseudoicosahedron’:

Apparently iron pyrite can also form a pseudoicosahedron! But I think this happens very rarely. I’ve asked around to see a photo of such a crystal, but nobody has ever shown me one.

In short, it’s probably just some subtle property of iron pyrite that caused the Greeks in Sicily — where pyrite is abundant — to see almost-regular dodecahedra, but not almost-regular icosahedra.

I wrote a lot more about this in week241. It turns out the pyritohedron is just one of a sequence of ‘fool’s dodecahedra’ in which $\Phi$ is approximated better and better by ratios of Fibonacci numbers. And some of the better approximations do arise in nature!

Also, the connection between $E_8$ and the icosahedron is equally well a connection between $E_8$ and the dodecahedron. For more on that see the end of week241, and also the appendix to my talk on the number 5 — the appendix entitled Quaternions, the dodecahedron and $E_8$ .

Posted by: John Baez on November 5, 2009 9:09 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

Nice, I do promise myself to take the time to understand all of this one day.
But, if I understand your explanation, from a pure abstract point of view both icosahedron and dodecahedron deserve to be called exceptional (or none of the platonic solids does).
I learned about them first from playing the advanced dungeons and dragons games that use them as dice. The infinity engine computer games like Baldur’s Gate implemented a subset of those rules - the question is, if computer archaeologists living in the 27th century will figure out why the number 20 was so important to these programs.

Posted by: Tim vB on November 5, 2009 11:20 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

Tim wrote:

But, if I understand your explanation, from a pure abstract point of view both icosahedron and dodecahedron deserve to be called exceptional (or none of the platonic solids does).

That’s right: from a mathematical point of view the regular icosahedron and dodecahedron belong together, since they’re duals of each other and thus have the same symmetry group.

(This duality is why the vectors that represent the 12 vertices of the icosahedron are perpendicular to the 12 faces of the dodecahedron.)

I think they both deserve to be called exceptional, since if you look at Platonic solids in all dimensions, you’ll see that 5-fold symmetry arises only in dimensions 2, 3, and 4.

I learned about them first from playing the advanced dungeons and dragons games that use them as dice.

These have a long ancestry! There exist Egyptian icosahedral dice dating back roughly to 300 BC.

Posted by: John Baez on November 5, 2009 11:36 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

There exist Egyptian icosahedral dice dating back roughly to 300 BC.

Christie’s recently auctioned off a Roman D20 for ~$18000.

Posted by: Mike Stay on November 6, 2009 12:53 AM | Permalink | Reply to this

Viruses discovered it first; Re: Who Discovered the Icosahedron?

Putting on my Mathematical Biology hat for a moment…

Virus Polyhedral (Icosahedral) Symmetry.

“Crick and Watson have shown that the polyhedral capsids can have three possible types of symmetry, viz. tetrahedral, octahedral and icosahedral.”

“It has been shown that an icosahedron is the most efficient shape for the packing and bonding of the subunits of a near spherical virus Therefore viruses are icosahedral rather than tetrahedral of octahedral….”

“According to the rules of crystallography, only a certain number of capsomeres can be present in an icosahedral capsid.”

“The minimum number of capsomeres can theoretically be 12, followed by 32,42,72,92,162,252,362,492,642 and 812. Of these capsomeres, 12 are pentameres occupying the 12 corners, while the rest are hexameres.”

“The actual number of capsomeres found in different viruses are: φX174, 12; turnip yellow mosaic virus and poliovirus, 32; polyoma virus and papilloma virus, 72; reoviruses, 92; herpesviruses, 162; adenoviruses, 252, and tipula iridescent virus,812.”

“φX174. The bacteriophage φXI74 contains 12 capsomeres. It has been suggested that each capsomere is actually a cluster of five units. Therefore the capsid is probably made up of 60 identical units.”

“Turnip yellow mosaic virus (TYMV) has elongated capsomeres. Clusters of 5 or 6 protein molecules project from a core of RNA. Twelve clusters have 5 units each (pentameres) and 20 clusters 6 units each (hexameres)….”

Posted by: Jonathan Vos Post on November 6, 2009 5:50 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

A marginally related question: why does the octahedron have equators (three of them) when none of the other four Platonic solids do? The icosidodecahedron on the other hand, a non-Platonic solid midway between the dodecahedron and the icosahedron, has six equators.

Posted by: Vaughan Pratt on November 11, 2009 6:25 AM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

The cuboctahedron also has a bunch of equators.

I don’t know ‘why’ the octahedron is the only Platonic solid with equators. It’s an interesting question, but I don’t know how to get a handle on it.

(An easier question: the Poincaré dual of the octahedron is the cube. What’s special about the cube, corresponding to how the octahedron has equators?)

Posted by: John Baez on November 11, 2009 8:12 AM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

Is the answer to Vaughan’s question something to do with the fact that the octahedron is the only Platonic solid whose dual has faces with an even number of sides?

If you scroll down from here to the convex regular polyhedra with a spherical geometry, that seems to make sense.

Posted by: David Corfield on November 11, 2009 9:15 AM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

David wrote:

Is the answer to Vaughan’s question something to do with the fact that the octahedron is the only Platonic solid whose dual has faces with an even number of sides?

You’re right!

Let’s say it directly: the octahedron is the only Platonic solids where an even number of edges meet at each vertex. For Platonic solids, this is a necessary and sufficient condition for there to exist equators — since in this case, and only in this case, we can draw an edge path that goes ‘straight through’ each vertex. When we can draw such an edge path, a symmetry argument shows it must divide the polygon in half. So, it’s an equator.

A similar rule applies to tilings of the plane or hyperbolic plane by regular polygons. We can draw a ‘straight line’ in the tiling — or more precisely, a geodesic edge path — if and only if an even number of edges meet at each vertex. So there are lots of geodesics in {3,8}:

but none in {3,7}:

Posted by: John Baez on November 11, 2009 4:59 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

Do you happen to have these figures in the projective representation?

Thanks!
Christine

Posted by: Christine Dantas on November 11, 2009 5:25 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

No. I’m not even sure what you mean! I got these figures from the Wikipedia article on regular polytopes — the one that David cited.

Posted by: John Baez on November 11, 2009 5:31 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

Thanks for the link!

I’m not even sure what you mean!

Do you refer to the projective representation? If this is the case, see, here or Penrose’s first chapters (The road to reality), with beautiful pictures by Escher on both the projective and the conformal representations of the hyperbolic geometry. These representations (or models) have other names, so I guess this may be confusing.

Best,
Christine

Posted by: Christine Dantas on November 11, 2009 5:45 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

No, he’s using the Poincaré disc representation, in which geodesics appear as segments of circles, not the Klein representation in which they appear as straight lines. The Poincaré disc representation is conformal, so maybe that’s what’s meant by “conformal representation” (but the Poincaré half-plane representation is conformal, too, as are an infinite number of other representations … )

Posted by: Tim Silverman on November 11, 2009 6:57 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

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Another way to say this is that the second number of the Schläfli symbol needs to be even. Wikipedia seems to have the triagonal dihedron’s Schläfli symbol wrong where I linked to above – it should be $\{3, 2\}$ .

Posted by: David Corfield on November 11, 2009 9:40 AM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

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Wikipedia seems to have the triagonal dihedron’s Schläfli symbol wrong where I linked to above – it should be $\{3,2\}$ .

You know, one of the nice things about Wikipedia is … well, I fixed it.

Posted by: Toby Bartels on November 11, 2009 4:30 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

Here’s the latest installment in the tale of the missing stone balls.

As you know if you’ve been reading Le Bruyn’s blog, there are just two balls with 20 bumps among the 387 carved stone balls listed in Dorothy Marshall’s paper on the subject:

National Museum of Scotland, in Edinburgh: item AS 110, found in Aberdeenshire.
Glasgow Art Gallery and Museum: item 92 106.1, found in Countesswells, Aberdeenshire.

Fellow café host Tom Leinster has been on a quest, trying to get a look at these stone balls. In November he went to Edinburgh and visited the National Museum of Scotland. I know this because one day I was at the beach and I saw a bottle floating in sea. I swam over and grabbed it… and there was a note inside! It said:

Hi John,

I went to the National Museum of Scotland in Edinburgh on Sunday with Christina Cobbold, who you may remember from your visit here. Two incredibly helpful, friendly members of staff helped us search for the 20-bumped stone ball. Unfortunately I didn’t have the ID number with me at the time… and we didn’t find any ball with 20 bumps. The closest we got was one with about 15, and then there was one like a squashed chunk of pineapple with dozens of bumps in no special pattern. (If you want photos, I can send them, but they’re not very enlightening.)

But some of the objects were gone, replaced by little cards saying “this object has been removed for photography”. I don’t know what the story is there.

And he gave me the email address of someone who might give me a photo of item AS 110. I have been remiss in contacting them, but I will do it now, because of the following incident.

The other day, while I was eating breakfast in my back yard, a pigeon fluttered down and landed on my table. I tried to shoo it away but it cast me a significant glance and held out its leg.

It was a carrier pigeon! Attached to its leg by a little bronze ring was the following note:

Hi John,

Here’s the latest on the icosahedron quest. But before you can have the latest, you have to have the second-to-latest, because there was a development in November. Well, in fact it was more of a non-development, which is why I never got round to telling you about it.

So: Lieven said in his blog that there were apparently two stone balls with twenty bumps in Scotland, one in the Glasgow collection and one in the Edinburgh collection. The Glasgow one was, as you will remember,

GAGM 92 106.1. : Countesswells, Aberdeenshire.

As it wasn’t on display in the Glasgow museums, I arranged to go and see it at the museum repository, which is in a pretty humdrum outer district of Glasgow (“suburb” sounds too cosy). You sit on the train for 10 or 15 minutes, the houses get less and less appealing, and there’s less and less that looks interesting. The main point of attraction is a station rejoicing in the name of Crossmyloof. Then you get off at Nitshill, which again is pretty dull-looking, and you walk through it for a bit, then very implausibly there’s this big, shiny building complex with a huge banner welcoming you to the Glasgow Museums Repository.

So anyway, I went. The curator, Tracey Hawkins, had got the ball out of the collection just before I arrived. And she greeted me with profuse apologies, because there aren’t twenty bumps on it at all… there are six. It’s a cube. And it’s the right object, because the serial number is actually written on. (I was surprised.) I took photos, but… do you really want to see another photo of a stone ball with the bumps arranged like the faces of a cube?

This was the morning of 27 November, and the same afternoon I was organizing the first meeting of the Scottish Category Theory Seminar, so the experience got rapidly swept out of my mind.

But then I got the mail below… watch this space!

Tom

And in the attached message, Tracey Hawkins wrote: “I may have some good news on your quest to find the 20 sided figure - our archaeology inventory assistant has uncovered what appears to be a carved stone ball with 20 ‘bumps’.”

She said that she’d get back to Tom in a couple of weeks to arrange a viewing — after she’d confirmed that the museum’s information on the object is correct.

So, we’ll see what happens!

Posted by: John Baez on January 18, 2010 8:35 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

John Baez wrote:

It was a carrier pigeon!

But carrier pigeons fly from a location of your choice back home, you can’t send them to a location of your choice!…and wouldn’t it have to cross the Atlantic and the north American continent to reach you?

But I came back because I just read an article about the dungeons and dragons games and the dice they use: it would seem that the first editions of the pen-and-paper games used dice that were produced for math school teachers, as illustrative material.

And I learned about “The Royal Game of Ur” that seems to be the oldest complete board game ever found, and it uses thetrahedons as dice. (This post would be far more interesting if it would use icosahendrons, but it does not, sorry!).

And did you ever hear about the Zocchihedron?

Posted by: Tim vB on January 19, 2010 1:27 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

Tim wrote:

But carrier pigeons fly from a location of your choice back home, you can’t send them to a location of your choice!…and wouldn’t it have to cross the Atlantic and the north American continent to reach you?

It was a magic carrier pigeon.

Posted by: John Baez on September 27, 2010 3:31 AM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

Thanks for the great detective work!

Posted by: John Baez on July 22, 2010 3:22 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

Yes indeed - thanks for looking up all those museums in England!

Posted by: Frieda on July 22, 2010 7:45 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

England…?

(From an Englishman living in Scotland, who has learned to be very careful about the distinction.)

Posted by: Tom Leinster on July 23, 2010 12:03 AM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

oh! Great Britain is just such a tongue-breaking notion! I hope the Scots will forgive my laziness.

Posted by: Frieda on July 23, 2010 8:53 AM | Permalink | Reply to this

Kissing problem; Re: Who Discovered the Icosahedron?

I’d love it if some prehistoric sculptor did in fact build a model which shows awareness of the 1694 Newton/Kepler/David Gregory kissing problem argument. Put one sphere on the bottom, then arrange five balls in a pentagon around the central ball, just below its equator, place another five balls more or less in the interstices of the lower five balls (this puts them slightly above the equator of the central ball), and finally top it off with the twelfth sphere on the pinnacle. So there you have it: another arrangement of a dozen spheres around the central ball. You may note that the spheres sit more or less on the vertices of an icosahedron, which is why this configuration is called the icosahedral arrangement.

Wouldn’t it be cool to find a prehistoric sculpture to show that someone anticipated the 1611 proposal by Kepler that close packing (either cubic or hexagonal close packing, both of which have maximum densities of pi/(3sqrt(2)) approx 74.048%) is the densest possible sphere packing, and this assertion is known as the Kepler conjecture. Finding the densest (not necessarily periodic) packing of spheres is known as the Kepler problem.

Posted by: Jonathan Vos Post on July 23, 2010 4:44 PM | Permalink | Reply to this

Re: Kissing problem; Re: Who Discovered the Icosahedron?

Well, the model you describe is there in among the stone balls, at least in outline, so I think you can say they were getting somewhere near to the kissing problem. We have at least three examples of icosahedral packing structures around a central object (my January post). No, the central ball is not the same size,and the outer knobs are at best hemispheres, but imagine the hemispheres completed and the central sphere reduced to the same size; a uniform small reduction in the spacings will then generate the somewhat promiscuous kiss. I can’t say whether or not any of these thoughts occurred to the carvers, but I doubt it; the evidence for their having a mathematical sophistication of the sort claimed by Critchlow has been evaporating rather rapidly since Lieven le Bruyn’s original comment on his website.

Your previous post (pre- Post post?) mentions the very similar icosahedral packing of virus capsomeres around an RNA-filled centre. This is not exactly the same since the symmetry here is lower: as Crick pointed out, the handedness (chirality) of the units removes all the symmetry operations except rotation axes. Nevertheless I think you could say that Nature anticipated the Scottish carvers here. However, if you look at what these capsomeres are doing to each other, ‘kissing’ is way short as a description, and even ‘heavy petting’ is an underestimate.

This may be the place to mention that despite the long history and pre-history of the icosahedron, icosahedral symmetry has pitfalls, and neither Crick nor David Hilbert (!) got it quite right. I will try to get a link to this set up next week.

Posted by: Bob Lloyd on July 24, 2010 9:55 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

Yet more on the 20-knob (20K) question!

My last post acknowledged considerable help from Dr. Alan Saville at the NMS, Edinburgh (that’s Edinburgh, Scotland…). Now he has surpassed himself. I had suggested to him, as as a very remote possibility, that there might be something strange on the far side of the apparently (5K+12K) object NMS X.AS 110, perhaps decorations which divided one of the large balls into four, but I did not expect anything to come of it. After my posting on July 21, I had another email to say that he would be visiting the store where it is kept and would have a look for himself.

He has discovered gold, and the object is much more interesting than I thought. His excellent photograph of the reverse side shows large and small knobs, but also two small triangles on the approximately 3-fold sites between adjacent large knobs, and he says there is a total of three triangles, the last of which is probably on the edge of the photograph.

I hereby publish my apology to the late Dorothy Marshall for maligning her record-keeping! 5 + 12 + 3 does indeed add up to 20 somethings, though maybe she was just a little economical with the truth in writing 20K. However, the main point remains- this is no dodecahedron!

This object is really strange. I can only guess that the carver got bored with yet another 6K ball, many of which do have these ‘interstitial’ triangles. After doing 5 of the 6 knobs, he decided to do something really interesting with the sixth position. His solution means that if the ball is held the correct way with one hand, you can do a fine trick by swapping to the other hand, first showing (1+4)K to the audience, then, with the other hand, showing at least 12K.

Finally, given that most of the balls have high symmetries, here is an early example of broken symmetry….

Posted by: Bob Lloyd on July 23, 2010 9:46 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

Fascinating!

Will we ever get to see a picture? I think the copyright on these Scottish balls should have expired by now, given that they were made in the Neolithic. I know, I know, it doesn’t work that way… finders keepers.

Posted by: John Baez on July 24, 2010 5:26 AM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

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First, before I get to the real news, there is an important point, not about “Who discovered the Icosahedron?” but “Who first noticed that there was one Platonic Solid missing in the (in)famous photograph?” The entries above, including mine, have credited Lieven le Bruyn with this, but it turns out that he was scooped several years earlier by George Hart:

George Hart, Neolithic carved stone polyhedra.

In a recent addendum to this George notes that there is no reason to assume any dishonesty in the original photograph; I agree with him.

He also notes that there is a question about the shapes of the two reported 20-knob balls. As we saw above, the one at the National Museum of Scotland at Edinburgh might be better described as a 17-knob with three additional decorations. This is certainly not a dodecahedron.

I now have a very nice set of pictures from Tracey Hall at the Kelvingrove (Glasgow) museum of their object 1892.106.l. No, it does not have 5-fold symmetry, it is NOT a dodecahedron. The shape is somewhat irregular, but from two different directions it appears to show sets of 7 knobs arranged as six-sided pyramids; one of these is below. (Thank you, Tracey!)

We can now be pretty certain that the Scottish carvers, good as they were at carving, had no awareness of the existence of the five “Platonic Solids”. What Lieven le Bruyn calls the “Neolithic Scottish Math Society” was not responsible for the fourth most beautiful theorem of all time. To be fair, Critchlow’s book does not claim this much, but he does claim a high degree of mathematical sophistication, and believes at least that the carvers saw something special in the Platonic Solids. Since they only found four of them I think we can discount this. Theaetetus is still the most likely discoverer of the proof that there are five and only five (convex) regular polyhedra, as well as the other items mentioned above, but he was scooped on the icosahedron shape!

Posted by: Bob Lloyd on September 27, 2010 3:34 AM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

Congratulations! I think you’ve pretty much settled this ancient mystery! Thanks for putting in all that work. And thanks for giving George Hart his due.

But now I have to ask: what are the three most beautiful theorems, in your opinion?

Posted by: John Baez on September 27, 2010 3:37 AM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

I’m only a chemist, so I couldn’t possibly comment! I’m quoting the opinion of those who replied to a poll in The Mathematical Intelligencer. See:

http://planetmath.org/encyclopedia/Top10MostBeautifulTheorems.html

Posted by: Bob Lloyd on September 30, 2010 7:57 AM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

Given that there are several examples of the Theaetetian Solids in nature as seedpods and radiolaria, some are as big as a millimeter:

I personally will be surprised if the icosahedron has not been noticed by numerous people over several millennia.

A question I am interested in is who started calling them Platonic. I suspect it was not by contemporaries of Plato but much later. Probably sometime between the Neoplatonic philosophers of the Middle Ages and the 16th Century. If anyone knows I would appreciate early references of the term Platonic Solid. Did Kepler use the term Platonic? Does anyone have the exact citation?

Any help would be appreciated.

Eleftherios Pavlides

epavlides@rwu.edu

Posted by: Eleftherios Pavlides on July 9, 2011 3:19 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

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Visiting Lisa’s mom in Montréal, I had coffee with John McKay and he reported some new developments in the saga of the Scottish stone balls. Like others here, he’s trying to track down the dodecahedral and icosahedral balls in this picture from Keith Critchlow’s book Time Stands Still:

He had tea in London with the photographer who took this picture: Graham Challifour. Challifour said he has hundreds of photos of these stone balls. Unfortunately, it seems McKay did not press him on where these balls came from! So, given the discrepancy between Challifour’s photo above and the photo from the Ashmolean, this remains a mystery.

Interestingly, McKay mentioned that the Ashmolean has a “statutory obligation” to show the items in their collection to any interested party.

McKay has also gone to the Kelvingrove Art Gallery and Museum to look at their stone balls. He says there was “nothing interesting” there. Of course Bob Lloyd here has already given us a more detailed report on the Kelvingrove stone balls.

Finally, McKay mentioned an interesting NOVA program on Stonehenge. In this show, an archaeology professor illustrated the possibility of easily moving very heavy stones on wooden platforms by the trick of digging grooves in the ground and putting stone balls in them!

Could this be the reason for all these stone balls?

I don’t know… but this archaeology professor is Andrew Young. He’s at the University of Exeter, and his webpage there says:

Andy obtained his BA degree prior to completing an MA in Experimental Archaeology, both at Exeter. His undergraduate and post-graduate dissertations focussed on Scottish carved stone balls and the techniques which may have been used to manufacture them. He showed that they could be made from a wide range of rock-types without metal tools: confirming they might be an entirely Neolithic phenomenon.

Andy’s research has profound implications for the study of protomathematics as he demonstrated that prehistoric people in the British Isles were numerate and confirms they were creating radially-projected Platonic duals some 1500 years before Plato described the five regular polyhedra in Timaeus.

It would be very interesting to read Andrew Young’s papers on this subject!

Posted by: John Baez on July 9, 2011 6:41 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

I came across this (and similar) sites by accident when looking up Grahame Challifour. He’s an old friend of mine. All this contention makes me curious: why doesn’t someone ask him about it? He’s listed on Linkedin, on Facebook and is readily accessible as a Director of this and that in the Bach flower world of healing. Critchlow (who continues to research and write and talk) is surely similarly accessible. Hasn’t anyone bothered to directly ask them about it all?

Posted by: ken allinson on May 10, 2012 8:11 AM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

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I’m not on Facebook or Linkedin. But, I’ve been in correspondence with Bob Lloyd, one of the commenters above, who became very interested in these issues and sent an email to Graham Challifour in January 2010, asking him about some of these puzzles. At the time, Challifour was just about to go travelling, so he postponed a serious reply until later. I never heard anything more about that end of the story. If you want to contact him, please do!

Posted by: John Baez on May 12, 2012 4:19 AM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

In answer to Eleftherios Pavlides, the first reference to Platonic Solids is surely the Scholion to Euclid XIII- see Baez week 236:

http://math.ucr.edu/home/baez/week236.html

This mentions 130-60BC, a bit before the Middle Ages!

There is a bit more on the (in)famous photograph which started a lot of this discussion, in “How old are the Platonic Solids?”, see:
http://www.tandfonline.com/doi/abs/10.1080/17498430.2012.670845.
This summarises my comments above and adds a few suggestions.

Posted by: Bob Lloyd on May 30, 2012 9:31 AM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

The above link again:

D R Lloyd
How old are the Platonic Solids?,
BSHM Bulletin: Journal of the British Society for the History of Mathematics (2012) 1–10. DOI:10.1080/17498430.2012.670845

Unfortunately behind a paywall, but a preview of the first page.

Posted by: David Roberts on May 30, 2012 11:43 PM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

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I don’t have any trouble getting Lloyd’s paper for free from the publisher here (PDF) or here (HTML or PDF). But perhaps this is a mistake on their part, so take advantage of it soon!

Posted by: John Baez on June 3, 2012 5:08 AM | Permalink | Reply to this

Re: Who Discovered the Icosahedron?

Sorry about the paywall. However, although the publishers want you all to persuade your institutions to take out subscriptions, they have provided me with what they call e-reprints. If anybody is interested enough to send me an email,
(dlloydATtcd.ie)
I will happily send one of these on.

Posted by: Bob Lloyd on June 1, 2012 9:15 PM | Permalink | Reply to this

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