19th February 2018 

Scottish Neolithic carved stone balls

Scottish Neolithic Carved Stone Balls: an exploration of their origins and possible relationship to the Platonic Solids of the Greek mathematicians.

Some 400 Scottish Carved Stone Balls have been found over a fairly wide area of Scotland with a particular concentration of finds within a 50 miles radius of Aberdeenshire. The stones are nearly all of 70 - 75 mm. (3 inch) diameter. They are made from a variety of different stone material and have knobs carved on their surface. Some have a regular pattern of knobs, others have an irregular pattern and on some the size of knob varies.

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The Towie Stone

The outstanding specimen is the “Towie” stone (right) found on Glas Hill, at Towie in Donside, Aberdeenshire in 1860. It is believed to date from about 2500 BC in the Neolithic (late Stone Age period). It is beautifully illustrated in a drawing published by David Douglas in 1883 in the book “Scotland in Pagan Times” by Joseph Anderson, Keeper of the National Museum of Antiquaries of Scotland.
Professor Colin Renfrew writing a caption to an illustration of the Towie Stone wrote: “A beautiful enigma, a symbolic masterpiece – incised stone ball, c 2500 BC, from Towie, Scotland – but symbolising what?” That question fired me.

Explaining the stone balls

A number of suggestions has been offered to explain the possible purpose of these balls. The suggestions included missiles; weights for fishing nets; or a ceremonial role giving the holder the right to speak in a gathering. I have several hypotheses that I would like to put forward. First, let us examine the simple sphere which in ancient times possibly had spiritual significance.

Several plain spheres - stone balls - have been found in Scotland. It is the simplest of all symbols or objects. Imagine, or better, hold a smooth ball. Every point on the surface of the ball is exactly the same as every other point and all are exactly the same distance from the centre. It takes only one dimension to accurately define the size of a ball. This is usually given as the diameter.

Now, holding the ball, play with it in the hands and fingers (see figure below). Using only the thumb and forefinger of each hand balance the ball and move it until applying reasonable pressure you get the ball settled in equilibrium between the fingers and thumbs. The four points you are touching are where the knobs would be formed on the Neolithic stone. They are separated by four equilateral triangles.
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Ruth Volmer and the sphere

Before going further in this, it may be helpful to quote from a modern Bavarian-born American sculptor whose work was exhibited at Inverleith House in the Royal Botanic Garden, Edinburgh in 2002 and again in 2005. Maybe her approach was similar to the approach of he Neolithic people. Ruth Vollmer (1899 – 1982) wrote :-

”The next step was to explore the sphere geometrically. I was looking inside and I was looking for proportions. After the first series of these spheres, I still felt that I had not got an iota of understanding for this mysterious form.

“I now feel I have made the first step (in what direction I do not have an inkling) of finding forms inside the sphere, hollow and full. Being immersed in this mysterious form, I perceive vaguely an endless variety of cosmic and earthly; biological and crystalline manifestations. I am concerned not to spoil the mystery whilst exploring.

“I am involved with the sphere, exploring it geometrically and finding unexpected forms. I suppose they have existed in mathematics before, but they have not been manifest visibly. The sphere is a bubble – or a drop”

In the exhibition in 2002 there were wire models shown but unfortunately she seems to have had no knowledge of the early Scottish Neolithic work. I’m sure if she’d known of these Scottish carved stone balls she would have been bowled over.

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Experiments with solid geometry

Go back to the ball in your hand. Manipulate it again and this time, using the thumb and two fingers of each hand, get the ball settled in equilibrium between the six points and we find eight equilateral triangles separate the six knobs.

Having established this spacing of equilateral triangles, you can experiment and place twelve knobs equidistant on the surface of the sphere and show the twenty triangular faces of the icosahedron.

In my view, these Neolithic people were experimenting with solid geometry and making wonderful finds. This was empirical exploration, not mathematical. Having measured the distance between adjacent knobs on the twelve knob stone (AS 109 in the Scottish Museum collection) and found only two measurements the same, I am convinced that the knobs were not set out mathematically. It can however be shown that within the twelve knob stone they found the golden proportion. I am not suggesting that they knew this but they were in tune with amazing principles and the beauty of mathematics.In 360 BC, Plato, describing the Greek solids in his book “Timaeus” defined them as having all sides equal and able to fit within a sphere, all points touching the sphere’s inner surface. The Platonic solids illustrate solid geometry in the period of the classical Greek mathematicians, Euclid, Pythagoras and Apollonius and are refined in their presentation.

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Relating stone balls to Platonic solids

The solid models (left column) were carved in wood by the author to gain an understanding of this subject. Four models are shown. The Scottish balls dating from some 2000 years earlier are obviously related to the sphere. Although there are five Greek Platonic solids only four are shown because there has been confusion over whether the dodecahedron is represented in the Scottish stones.

The form of the Scottish balls varies greatly. The majority found have a regular distribution of knobs on the surface. The distribution of the knobs can be linked to the Greek solids as can be seen. The tetrahedron has its equivalent in the four knob stone, the octahedron in the six knob stone, and the icosahedron has its equivalent in the stone with twelve knobs. The knobs relate to the points, and the equilateral triangles between the knobs relate to the faces of the Greek solids. The intriguing thing is that the Scottish stones predate the Greek writings by almost two millennia.

The dodecahedron
The fifth Greek solid is the dodecahedron. No stone representing this solid is known to exist although Keith Critchlow claimed to show that the twelve faces of the dodecahedron was represented in the Scottish stones. (“Sacred Geometry” by Robert Lawlor: Thames and Hudson: 1982). Critchlow illustrated the Scottish stone with pentagonal faces and linked it to the dodecahedron. On all the other Scottish stones the knobs represent the points of the Platonic solids and the spaces between represent the faces. To support his case he reverses this. Critchlow’s interpretation is refuted in Addendum 2009 Lieven le Bruyn to George W Hart found on the Ashmolean Museum site.

Who made the stones?

Who were these people? Almost certainly Celtic, but I would suggest more specifically that they were early Picts. This hypothesis cannot be very strongly defended because of the long time gap, but I think that the gap can be bridged. Pictish art is well recognised. Their medium was frequently stone, like the stone balls and the area of country in which finds were concentrated was very similar. Pictish art is much more strongly two dimensional work, but the weaving line pattern and incorporation of figures and animals is often executed in low relief which does give the work a three dimensional character.

I wonder if these Neolithic people knew that they were working in three dimensions? They did! They must have known even if they didn’t recognise it.

The stone that deserves to shine

Certainly the “Towie Stone” is extraordinary. It’s unquestionably fine art from a very early period. It is Scottish, and the subject, solid geometry, gives it a particular quality of realism. These Scottish stones may prove to be some of the earliest examples of experiments in solid geometry anywhere. Its place in the depths of the new Museum of Scotland where the beginnings and growth of our culture are well displayed is important but its place can be taken by another stone (AS109 in the Scottish Museum collection, perhaps. The “Towie Stone” is exceptional art and should be shining in the daylight.

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