There are nearly 400 of these objects found in various sites in Scotland
and now in various museums and private collections. They include various
regular and semi-regular solids, alternatively they can be seen as
arrangements of knobs - from 3 to 10 and then various numbers up to 160!
Some of them are decorated, notably the tetrahedral Towie stone now in the
Edinburgh museum (which stocks an excellent coloured postcard). . I have a
baked clay model of this stone, bought from a shop in Avebury, Wiltshire.
There is a picture of it in J. Purce, The mystic spiral (Thames and Hudson,
1974) plate 46, where J Frazer is quoted as taking the grooves to be meant
for thongs so that the balls could be hurled through the air "uttering
oracles in a whistling voice which a wizard was able to interpret". I have
been unable to trace this quotation, and the author was unable to help me
at the time I inquired.
The Ashmolean museum, Oxford, has a number of balls, kept in a drawer - I
have handled these and they are certainly as remarkable as Keith Critchlow
has pointed out. "These neolithic objects display the regular mathematical
symmetries normally associated with the Platonic solids, yet appear to be
at least a thousand years before the time of either Pythagoras or Plato."
(K Critchlow, Time stands still, London - Gordon Fraser, 1979, p133 -
this book has some splendid photos of various stone balls.)
There have been various attempts to guess at what the balls might have been
made for. The nineteenth century archeologists who excavated them thought
they might be weapons - whether as pike heads, or hurled from slings, or
used in games or perhaps for divination. It seems that the balls were
never found in personal graves, so it has been suggested they were a sort
of ceremonial conch, a prized possession of the tribe. Contemporary
archeologists tend to be more cautious. According to Dorothy Marshall,
"there is so little hard fact to be extracted from the evidence available
about the carved stone balls that postulation as to their evolution and use
if very difficult." ( D Marshall, Carved stone balls, Proc Soc Antiq
Scotland, 108 (1976-7) 40-72 - this is the most up-to-date and
authoritative account. Some previous papers in the same journal are to be
found in 11 (1874-6) 29-62 and 48 (1913-4) 407-20.)
I corresponded briefly in 1989 with a German mathematician, E Brieskorn,
who was interested in the stone balls for a book he was writing at the time
on classical shapes and modular structures in geometry and crystallography.
I do not know if this book has been published, whether in german or
english. I seem to recall he was particularly interested in this
ethnomathematical precursor of problems of arranging points on a sphere.
(Cf L Danzer, Finite point sets on S^2 with minimal distance as large as
possible, Discrete Maths, 60 (1986) 3-66.)
Finally, perhaps I may be excused from quoting myself: "The issue here is
not of course to show that our culture is not entirely due to Mesootamian
man.... What is important is the forceful reminder that mathematics is not
dependent on its written record." (D Tahta, About geometry, For the
learning of mathematics, 1.1 (1980) 4 )
Dick Tahta