I am very grateful for Nick Tahta's account of the neolithic
sone balls discovered in Scotland, of particular interest being those
that instance the symmetries of all five of the regular spheroid (as
Hilbert put it) solids, including the icosahedron.
Is their extreme age (2500+ years ago) reasonably well assured
(this wasn't quite clear to me from the post).
Is it reasonable to assume that the starting point of whoever
carved them was a sphere, and that they were created out of something
like a intuitive visual understanding of the symmetries of the sphere?
I am in general strongly against speaking of such artefacts as
parts (so to speak) of the history of mathematics; but I think that in
this case there must have been (even if not verbalized) a kind of
puzzling out (rather than an immediate intuition or visualization or the
like) of the variety of symmetries of the sphere, and then perhaps this
puzzling out might be called a kind of primitive (but surely not easy)
mathematics. It seems to me better to speak of mathematics beginning
when the thinking/understanding is at or beyond the point of asking the
question of (some version of) what are the finite symmetries of the
sphere, and are in fact _these_ all of them?
By the way, need there have been a purpose or a use? Maybe
there is a pure and divine aspect of the hominid mind that just does go
in for joy of the abstract hunt?
That the sphere carvings were beautiful in the good sense of
concretely embodying the solution to a brain puzzle. (Re: the silliness
of finding fractal graphics, etc., "beautiful" when one just beholds
them with no clue to the underlying mathematics; that's a kind of
irrelevant and stupid beauty; in contrast with viewing in terms of the
mathematics behind them.
Robert Tragesser
424 Old Clinton Road
West(running)brook, Connecticut 06498 usa
860-399-6305
RTragesser@compuserve.com