Create a 4-by-4 grid of squares and inscribe a circle in it. The lines of the squares will intersect the circle at 12 points, which appear to be equally spaced at 30-degree intervals around the circumference of the circle. You can then connect selected points to create a triangle, hexagon, or dodecagon.
(The ancient geometers used to take it as a challenge to construct various polygons using only a square and compass. I suppose the modern equivalent would be: Using only a crappy graphics program like MS Paint, construct . . . .)
I can't remember how I happened to discover this -- just one of those happy accidents, I think. I also can't be sure, absent any mathematical proof, that the points are really evenly spaced at 30-degree intervals. (If not, they are certainly close enough that the trick can still be used to create polygons that seem to the human eye to be perfectly regular.) However, I strongly suspect that they are perfectly regularly spaced, because that's just how math is.
The last mathematical proof I embarked on, regarding congruence patterns in triangular numbers (qv) was a matter of straightforward number theory, requiring only everyday algebra, which is why I was never really in any doubt as to my ability to solve it. I am much less confident of my competence in this case, though, being innocent even of basic trigonometry. (Obviously somebody invented trigonometry without having been taught it, and I'm sure I could do the same if push came to shove; the question is whether or not I want to invest so much time and thought in a problem which has elicited only my passing curiosity.)
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Okay, I’ve solved this already. It turns out to be relatively trivial, calling for no mathematical training at all. I’ll probably post the proof later tonight.
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