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

[sdi]

Very good question. It really is a mix! Some constructions, like Moss's Egg and the dodecagon, were found entirely by hand (though I used a computer afterwards to convince myself that there were no better solutions). Others, like that inscribed regular pentagon, were found entirely by computer. Most of the constructions I've made, though, are a mix: either I used a computer to guide my own search by hand (usually by asking it to find a circle with a given radius, and then studying it to see how I might do so in general), or I found something interesting by hand which I then explored using the computer (e.g. to prove that something that looks right actually is right).

The computer program I wrote to search for constructions can be found here (though it's not very user-friendly, and you'd have to be a computer programmer to get anything out of it). I've also been making pretty heavy use of Wolfram Alpha to verify constructions. (I really need to spend more time with Euclid; I'm not very good at geometric proofs, and my arithmetical ones are unsatisfying.)

My program is handy, but it's pretty limited; it can tell you that something is possible, and it can tell you how to do it, but it never tells you why. I'd say 80% of the time I've spent on this has been spent going back over things to try and make sense of them, and often I fail at it. So, the computer is an idea generator, but you have to go back over all those ideas with a lot of contemplation. Much of the reason I've been posting these has been so others can explore the constructions and maybe get a sense for how to construct, say, √2 without needing a program to search for it.

I've also destroyed most of a great, big sketchbook of newsprint and will need to get more soon :)

[temporaryreality]

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.

[sdi]

I would be very surprised if someone could get very far by just eyeballing these! The pictures get complicated with even a few circles—I think the only way to understand them is to start with the two given points on a blank piece of paper, and draw the circles one at a time (figuring out which circle can get drawn next based on the points available).

But then, I'm a very analytical sort of person. I get overwhelmed very easily—no less in life generally than in geometric problems!—so I'm pretty used to breaking things down into steps and taking them one-at-a-time.

That program, incidentally, was very tricky to put together! The reason is that every time you draw a circle, you add a LOT of new points where new circles can be drawn... so the number of possibilities grows extremely quickly, meaning it takes a long time to search them all. My first attempt took around 400 seconds to solve Napoleon's problem, which just has six circles, and solving problems with any more than that were out of reach. I think I'm on something like version 20 of that program, now, and it's able to solve Napoleon's problem in 40 milliseconds—some five orders of magnitude faster—but even then it's only really up to tackling problems involving up to maybe twelve circles. Sort of the holy grail I've been aiming at working out is Gauss' masterpiece, the heptadecagon, but my program's not up to the job, and so I'm trying to understand what the constructions mean better before I can make the attempt.