[SoCal Pubbie]

“Am I overlooking some obvious Bronze Age problem that would have drawn attention to that particular ratio? “

They needed it to figure out the area of the crop circles the ancient astronauts left when their ships landed to teach primitive man how to levitate giant stone blocks to build large structures.

[sphinx]

Thanks for the speculations. There will probably be more. It's an interesting puzzle. In my modest attempts to find an answer, I've stumped quite a few math teachers as well as several acquaintances with heavy-duty professional math credentials (a couple of physicists, a math Ph.D doing advanced acoustic work for the Navy, a couple of other scientists -- no slouches). No answers so far, so I thought I'd pose it here, at the world's greatest font of obscure knowledge.

I do like SoCalPubbie's speculation, but on reflection, there's an obvious problem. If the aliens taught the ancients to levitate rocks to build large structures, they would risk the power of pi being turned on themselves, with large rocks and even astroids being hurled intergallactically to smash the home plant. This is presumably how the Bugs did it in Starship Troopers -- is there a more plausible explanation? -- and since the aliens had undoubtedly time-travelled, we must assume they'd seen the movie and would know better than to incur that risk. So as attractive as your theory is, I'm still open for additional explanations.

The ancients had their fair share of Einsteins. I don't know how the uber-brilliant think, but perhaps such questions just occur naturally to them as they sit around fires poking at the coals with a stick: "H'm. The ratio of the circumference of a circle to the diameter. That's an interesting question. I wonder if I could figure it out, and without algebra and decimal notation. That would be way cool." If you are smart enough, maybe you really think that way. But I'm still looking for a more mundane theory.