Inscribed Regular Polygons

[sdi]

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.

Comments

[boccaderlupo]

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

[sdi]

WHAT A FASCINATION QUESTION. I'll have to think about this.

[boccaderlupo]

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