Mathematical Musings of the Neolithic Kind

By Bernie Bell

It began a few years ago when I was ill and couldn’t move from the sofa.  A little carved ball was found, at the Ness, all covered in muck.  Jeanne, bless her!, turned up at our house, as soon as she’d finished at the Ness that day, with a camera full of pictures of the ball.  That’s all I’d seen of it, until I went to the Kirkwall museum.

carved stone balls  Kirkwall MuseumI stood and looked, and looked, and looked, then went round the museum, then went back and looked, and looked and looked, then went to the shop, then went back, and………..and also, to say ‘bye-bye’ to the ball, and its neighbour, the perfect, clear ball, from the year before.  I tried holding my hand over the display case, to try to get the feel of it.  It cries out to be held, and it’s hard that it’s in the case, but, it has to be, I realize that.

So, next thing, Mike and I went to the Ness during the  dig in 2014 , and……………………

Ness of Brodgar 2017 Archaeological Dig: Neolithic Discoveries in Orkney 3,300 BC

We had a good tour with Roy, then, I was talking with a woman in the shop, and asking had she seen the exhibition in the Museum etc.  Annabel over-heard, and told me that Babette’s replica balls, were in the finds hut, and I could go and see them, if I liked.  So, we went over to the finds hut, and there was Nick, who got the little box down and let me/us hold and play with Babette’s replicas of both an ‘ordinary’ six-knobber, and the Extra-Ordinary-A-Symmetrical- Six-Knobber (E.O.A.S.S.K), which was found the year before.

When I was looking at it in the museum, I was trying to see as much as I could of it, and work out how I could fit the Platonic Solids idea around  it, but it can’t be done, when looking at it in the case.  Maybe it can be done, but not by me!  For that matter, even when holding it, I can’t really do that stuff.  I said to Nick, that someone with Maths in their heads, could work out all sorts of things, from that ball.

It was good to hold the copy, but, it didn’t feel like any kind of substitute, for what I saw in the case in the museum.  As to why this should be, I’m not sure.  For one thing, I’m pretty sure, the reproduction is a bit bigger than the original.  I’m not sure, but I think so.  And that might matter.  I think it would matter, for how it fits in the hand, especially a small hand.

For another thing, I think the stone that it’s made from, matters. Nick said it’s a kind of basalt.  Basalt is an extra-ordinary rock – think of how it’s formed.  I was delighted to hold this one, mainly to be able to see it as I turned it about, in my hands.  Turn it this way, you see four knobs, turn it this way, you see three knobs.  I’m not sure, but I think you get 3, four-sided views, and 2, three sided views.  If so……that’s another one for the folk with Maths in their heads.

So, I had a good old go at the replica E.O.A.S.S.K., and a comparison with the replica, other, six-knobber, which only went to show, how very much more-so, the Extra-Ordinary one, is.

I want, I want, I want, to hold the original.  It has a sheen on it, which is from it being ‘dolled up’ (conserved), but, what’s to say that it didn’t have a sheen, back then?  Either from purposeful polishing, or from folk holding it?  Having a sheen, fits it, it works with what it is, and all the other aspects of it.  It’s a very ‘human’ little thing, very friendly.

Complex stuff about a-symmetry and possible Platonic Solids and all that, but also, at the same time, very human.  But, that’s humans, for you….so complex, and so simple, and so much some of both.  As I said earlier, when I went to the museum, I kept going back, again and again, to look at it, and its neighbour, the ‘plain’  little ball, from the year before.

I’d really like a good picture of that Extra-Ordinary-A-Symmetrical- Six-Knobber, but….how can that be achieved?  due to its shape?

I think I’ve hit on what the difference is….or maybe part of what the difference is………..or, what makes the difference, for me……….

The ball in the museum, is really ‘tight’, it’s very compact.  It’s small, but very….compact.  It looks like, if you held it in your hand, you’d feel as though you were holding a little dynamo.  It  looks like it would feel heavier than it looks.  Do you see what I’m on about?

The other one, is good, for inspecting and thinking about the original one, but, I may be wrong, but I have a strong impression of it being a bit bigger, and just not as ….tight. Not as compact.  Not got so much to it, somehow, in such a small object.

Yes, I know, I’m thrashing about a bit, here.  I don’t know, I’ll keep on, running it round, in my head!

Maths – (C.S.E. Grade Four Maths)

3 x 4 = 12.   2 x 3 = 6.   12 x 6 = 72.  7 + 2 = 9.

Threeeeeee, is the magic number!

Using Maths, to illustrate ideas.  Mathematics, are at the root of ….just about everything.  I only wish I’d seen that, when I was at school, and had a chance to pay attention.  My brain just doesn’t work that way, anyway, though.  I never saw what it was all about.  When presented with those problems of  “If  5 men take 3 hours to dig a hole 10 feet deep, how long will it take 7 men to dig a hole 32 feet deep?”  I didn’t see the point, had no interest in these men, digging holes, and looked at the words in total bemusement!

As to the ball, and the other carved balls, being centred on mathematics….Yes, yes and thrice yes. I think so, anyway. I can’t say strongly enough, how much I feel that is so.  Complex, wonderful Maths, at that.

3 to the power of 6, is 729.  7 + 2 = 9.   9 + 9 = 18.  1+ 8 = 9.  All the threes.  Threeeee is the magic number.  Maths, the world, is built, on Maths.  Un-fortunately, I don’t have a grasp of Maths, in my head.  I have the intuition to see what’s there, but can’t work out the calculations to express it.  Those old folk, had both.

Put it all together:-

3 x 4 = 12.   2 x 3 = 6.   12 x 6 = 72.     7 + 2 = 9.

3 to the power of  6 = 729.  7 + 2 = 9.   9 = 9 =18.  1 + 8 = 9.

It goes on, and on, and on, based on the rule of three.


It even works in the garden, always plant in groups of three….it’s everywhere, in everything.


I’m of the opinion, that the carved ball which was un-earthed at the Ness , is a very, very important object.  The concepts encoded in that object, are to do with LIFE, itself.  It may not be the most glamorous find from the Ness, the Sky Stone will be that, but I’m sure that it will come to be seen as the most important find from the Ness dig overall.  Not just in terms of the Ness, but in international terms, in relation to the understanding of just how sophisticated and  Universal, those people were in their knowledge.

And this is where I link the two kinds of carved stones.  For some years, I’ve had the idea that the markings on the inscribed stones, what is referred to as the Neolithic ‘art’, are, in fact, writing.  Marks which are telling us something, if we can only read it.


The more pieces of inscribed stone which are being found, not just at the Ness, but at other sites, too, the more I’m convinced of this.  Put together the inscribed stones, with ‘writing’ which we just can’t read, and the carved objects, which hold, in their structure, basic, true, pure mathematical formulae, which we can read, as they are universal, and we have a key to one of the main things which the Ness was about.  Teaching.  Learning.  The transfer and sharing and spreading of knowledge.  Wrap this in ceremonies, to catch the public imagination, and you have a centre for learning and pilgrimage, in the heart of the Orkney landscape.

What we now call The Ness of Brodgar, was, and is, a centre for teaching and learning.  Why do I say “and is?”  Because, hopefully, by re-discovering these places and the knowledge that they hold, we will begin to re-learn much which we, as a species, have lost, or forgotten, about the part which we play, in The Universe.

What I forgot to say, earlier, is…………

The clear ball,  I actually got the chance to hold that one.  Anyway, point is………….

A clear, rounded, stone ball.  The Void.  Clarity.  Everything, and Nothing.  All that is.

If anything, something ‘bigger’ than the ones with shapes carved, to point you in directions of thought.

A clear stone ball. Clarity.

To Watch:

Mathematics in Neolithic Scotland – Professor Tony Mann

Making a carved stone ball with Chris Gee

Ness of Brodgar

Ness of Brodgar ( F Grahame)


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16 replies »

  1. Picture it…..Ness of Brodgar…..Mid-summer’s Eve, around 3,000 B.C. A group of work-men and artists have toiled all night, putting the finishing touches to the building which is destined to be ‘The OrkneyTwilight Home for Bewildered Shamen.’ Just as dawn breaks, a clear shaft of light, streams down the entrance passageway, illuminating the brightly coloured wall carvings and paintings, which the workmen have prepared, for the delectation of the ‘Old Dears’. One workman turns to the other and says…….”Cor, blimey, would ya’ look at that?!!! Would ya’ Adam & Eve it? Straight down the corridor, lights the place up, something lovely, just as though we’d planned it, or something! Wait ’til I tell me missus about it, she’ll have to come and see this.”

    Esoteric knowledge? Nahhh, pure chance. (I don’t know why the workman, is a cockney – but folk did come from a long way, to visit the Ness, as it was known, nation-wide, as the best facility for Bewildered Shamen, in the country).

    (I’m allowed to take the P. as I wrote it!)

  2. Howie (Firth) sent some observations and said I could add them – here they are………

    “That is a brilliant insight about the ball needing to be held in the hand, and about the deep significance of mathematics for the people of the Neolithic.

    I am sure that 3 is significant, and also 4, and it is fascinating to read what you say about the ambiguity about the three-sided views and the four-sided views.

    There was an old association of the number 3 with the heavens, and the number 4 with the earth (for instance the four corners of the earth).

    The numbers 3 and 4 can then be combined in various ways.

    I think that investigating the properties of numbers might have been a way of building up a kind of cosmology.
    So what you are saying about the stone balls is a really good insight that could open up a lot. I wonder if you had the opportunity to sit for a while holding one of them, if a picture would form in your mind about the way it was used? And then what might happen if you could hold other originals?”

    And this is what I replied……..

    “Some years ago, I think it might have been 2008, Babette (Barthelmess) had an exhibition of her work, in the Kirkwall museum, including her re-creation of the Ball of Towie. Babette said that if Mike and I went in on the day that she was dismantling the exhibition, I could hold her re-creation. I had thought of the carved stone balls as aids to meditation – I thought that, if a person held the carved object, and followed the lines, it could take them…….where they needed to go. But….when Babette gave me her re-creation to hold, I went and sat quietly in a corner, and found myself immersed in the blank disc – the clear side – the Void. That was the aspect I was drawn to, and that was where I stayed!
    The only original carved Neolithic object which I have actually held, is the clear stone ball found at the Ness in….I think it was 2012. The lad who was showing us the finds, said that it was probably going to be a carved stone ball. I’m of the belief, that it wasn’t, that it was meant to stay just as it is, clear, The Void. Just as it is, everything, in one, perfect, stone ball. No carvings, needed. It tells the tale, by being what it is. But, that’s just my idea, of it.
    When I held that ball, I passed it from hand to hand, it was getting warmer and warmer, and very much felt that the next thing, should have been to hand it on, so I passed it back to the lad who was showing us the finds. Hand to hand, then hand to hand.“

  3. And Howie answered………..

    “That is most interesting what you say about the significance of the blank one, that could thereby represent the Void – that would indeed fit beautifully, and the Void would be the deepest one, out of which everything else would come, indeed a kind of primal unity.
    It is so neat to see these as an aid to meditation – it would fit so well as with mathematics today people use pen and paper and set down letter and numbers and symbols, which is not only the outcome of their thinking but an actual stimulus to it. So for a culture with less recourse to writing it down, an ideal way to work could have been to sit and reflect and to hold one of these balls in the hand while doing so could have fitted so well.”

  4. And………….
    If the numbers 3 and 4 are added together, we get 7, which is traditionally significant.
    And they can also be combined in a Pythogorean triangle. So 3-squared plus 4-squared gives 5-squared, and this type of combining brings 3 and 4 together to give 5.
    I think I’ll leave this alone, now!

  5. A bit of an update – 2.07.16. Part of an exchange with friend Hannah, leading to something I hadn’t thought of before…..

    Hannah……”Found an interesting article on the stone balls- apparently a team has made them into virtual objects you can view online.


    Virtual balls:

    The website is a bit cumbersome (or rather, I’m not very good at using it) but its worth a look and a really lovely idea. They mention several theories about what the balls were used for (including weapons- pah!) ”

    Me…”In, I think it was 2008, when Babette had her exhibition in the Kirkwall museum, including her recreation of the Ball of Towie – when Mike and I went to see it, there was a young man ( who had drink taken!) who was very keen on the idea that they were used for throwing, either as weapons, or for hunting. I tried to present the argument, that – would someone go to all that trouble, for something which they might not be able to retrieve? It would be like having highly decorated golf balls. As I say – he had drink taken, and wasn’t really open to persuasion.”

    Hannah….”but no one has mentioned your idea of them as mathematical objects…. ”

    Me….”I must admit, it’s not just my idea! My first impression of them, was that they were aids to meditation, which was strengthened and expanded by Babette’s work on the Ball of Towie, to include the idea of them as ways of encoding and passing on information – that leads to the maths. The maths came in, for me, when I handled Babette’s reproduction of the EOASSK at Brodgar, a few years ago. I think quite a few folk are going for that theory now. Of course, it doesn’t have be either/or – they could be any or all of these things, individually or collectively – that’s just how clever those folk were.”

    Hannah….”Also speaking of balls, I may have mentioned it earlier, but the wood turner in Harray Michael Sinclair is making wooden versions of some of the balls. Worth a look if you’re ever driving past”

    Me……”I’ve seen pictures of them, and also the real thing, when they were on display in Jeanne’s shop in Stromness. Not just the balls – all his works are ….well….objects of beauty and desire, are the words which come to mind. I particularly like the carved ball, with concentric circles. I like wood, I like things made of wood – and he does it all so beautifully. When Eoin Leonard retired, I was sorry to think that there wouldn’t be a wood–turner on Orkney, then Michael Sinclair came to the fore.
    Lovely things – really, lovely things.
    Try getting an archaeologist to say what they think the carved objects were/are for – that’s a challenge!
    Something has just occurred to me, which , for some reason, never occurred to me before. I’ve thought there might have been wooden objects which were also ‘visual aids’, and which have rotted away, sometimes leaving just bits of metal which were attached to them. But…I’ve never thought that there might have been carved wooden balls – same purpose as the stones, lighter and easier to transport – but which will have all rotted away. There’s a thought to conjure with.”

  6. And Mike added………….

    “That is interesting, a thought to conjure with… Yes, why wouldn’t they use wood! Imagine the patina on a wooden ball that got passed from hand to hand, very much taking up something of the users. They’ll have rotted, but perhaps one will turn up from a bog…”

    They’ve just found wooden bowls at the dig at The Cairns in South Ronaldsay – if wooden bowls, why not wooden balls?

  7. And now…..local artist Joan Rodwell is making clay EOASSK reproductions, which un-cannily reproduce the original. They are on sale at the Ness of Brodgar dig site shop – and – I’ve got one! Hurrah!

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