sdi: Oil painting of the Heliconian Muse whispering inspiration to Hesiod. (Default)
sdi ([personal profile] sdi) wrote2022-11-29 01:51 pm
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Inscribed Regular Polygons

A very common traditional construction was, given a circle and a point on it, inscribe a regular polygon within it with a vertex at that point. I've completed my collection of non-exotic inscribed polygons using (I believe) as few circles as possible, and figured I'd share them in case anyone is interested.

(Regular decagons (10 sides) and pentadecagons (15 sides) are also commonly encountered, due to their appearance in Euclid, but I've omitted them for being of less interest with compass-only constructions.)

Triangle: Despite being so simple, the equilateral triangle is very wasteful, requiring four circles. (There are a number of ways to do it with four circles, though! This one happens to be identical to the hexagon's construction, below.)

Square: I've talked about Napoleon's Problem before, but here's a clean, easy-to-prove construction using six circles which I found while working on Moss's Egg.

Pentagon: Michel Bataille also gives a construction for this, but this one (using eight circles) is, I believe, the simplest possible. It is also very surprising! I've verified it using symbolic computation tools, but I haven't managed a geometric proof.

Hexagon: This construction is trivial: circles just love to form hexagons.

Octagon: This construction (using ten circles) isn't terribly elegant, but it has the merit of starting from the square construction, above, and being no worse than any other construction I've found.

Dodecagon: This construction (using nine circles) also starts from the square construction, above, and uses the fact that placing a circle at each of the square's corners forms a dodecagon. There's lots of other ways to make a dodecagon, but I don't believe any are simpler than this.

temporaryreality: (Default)

[personal profile] temporaryreality 2022-11-29 09:38 pm (UTC)(link)
When you're "doodling," are you doing it on the computer to begin with, or on paper? Is there an underlying logic that tells you you should be looking for an octagon (say), or do you just look at your doodle long enough to figure out which figure is made by whichever assortment of points?
temporaryreality: (Default)

[personal profile] temporaryreality 2022-12-01 01:06 am (UTC)(link)
Since I'm not a very analytical person, I've been eyeballing what you're putting up - of course only after the fact of your having drawn in the polygon shape am I able to see that, "yep, those points do construct a shape" ... except for the seven-sided figure. That had me scratching my head... I was thinking, "how did sdi figure out where the points go?? I'm not seeing how that was constructed" and then I read to the bottom and whew! I wasn't wrong to be puzzled.

Here, I'm like an appreciator of any art form who doesn't really know the finer details - looks neat, but the whys and wherefores are way over my head. :)

It could make me feel a bit better about my very unsophisticated way of looking at any of this, knowing you used a program to figure some of it out, but, then, you had to write the program! :D

Well done, I must say.
boccaderlupo: Fra' Lupo (Default)

[personal profile] boccaderlupo 2022-12-01 03:36 pm (UTC)(link)
Would love to map the sides of these polygons and the number of circles onto a Pythagorean number scheme (or perhaps a Cabalistic scheme) and see what shakes out...

I find it intriguing that circles "love" to form a 6-sided figure, for instance, if we regard 1 as Kether and 6 as Tiferet (the "crown" and the "heart," as it were)...

Axé,
Fra' Lupo
boccaderlupo: Fra' Lupo (Default)

[personal profile] boccaderlupo 2022-12-01 06:38 pm (UTC)(link)
It's even more interesting, to me, that you can derive a 12-sided figure (with all its significance (Olympians, Zodiac, months, et al.)) from 9 circles (3 x 3, 3 + 3 + 3, with all its significance), also including the number 4 (will all its esoteric significance (four elements, archangels, et al.))...

May be a form of occult meditation, after all... ;)

Axé
Edited (Had to edit my addition...not a math guy, by any stretch!) 2022-12-02 15:57 (UTC)