Sometimes you have to do something for a really long time before you realize what an idiot you are. I have had one of those moments today.
The thing of it is, until you reach that moment when you realize you're an idiot, you keep thinking, nah, I'm not an idiot. I'm just doggedly persistent. So you keep going. Your idiocy is patently obvious to others, but you, in your self delusional cloud, you think to yourself that it's a sweet little problem, that you have to keep worrying it, like some little Jack Russell terrier who knows no better, until your problem yields a solution.
By you, I mean me. You have more sense than I do, and are probably not prone to attempting to solve these problems with the same obsessiveness that I am.
The thing is, and here's where the ratiocination kicks in, I'm pretty well educated. But that is a blessing and a curse. It's mostly a curse. I should know better. I know that you can't invent a perpetual motion machine. Violates the second law of thermodynamics. I know that you can't go backward in time. Violates a whole bunch of statistical mechanics equations and Gibbsean phase space and blah blah. I'm pretty sure there's no God - as Laplace said, "I have no need for that hypothesis." And so this self delusion sets in. Nothing can fool me. I can smell bullshit, and I can smell a good problem, and I know they smell differently.
Der Mensch denkt, und Gott lenkt. Man plans, and God laughs. Or something like that. My German mother was very fond of saying that. No doubt she knew what grandiose castles I was building in my mind's eye.
So today in my random walk through Euclidean geometry I was randomly walking through something called the power of a point theorem, which Euclid proved, oh, 3000 years ago - Book III, Proposition 35 - and that led me to the subject of apothems and regular polygons, and then I started working on a problem that took me most of the day, which I've just now finished, and feel like an idiot because it can't be solved, which I have known since i was about 20, which was like 37 years ago (before man invented the Internet and twittering and video games, so it was impossibly long ago), but did that stop me? No. Because I'm an idiot.
The question I posed to myself was that if you have a circle of radius 1, and inside this circle you build a regular polygon having N sides, what's the side length of the polygon?
For N = 3, you have an equilateral triangle, and the edge length is √3. For N = 4, a square, the edge length is √2. This looks pretty easy, you tell yourself. And so you keep going.
Then you hit N = 5, a pentagon, and you do a few pages of tiny scribbling and you give up in despair, and you say, fuck this, N = 5 is what's known as a bitch, in technical parlance. So you move on to N = 6, which is absurdly easy. E = 1. One! Ha ha ha! Ridiculous! And easy to boot! Press on! You're invincible!
Then worry when you hit N = 7, because 7 is prime, and because primes in geometry usually wind up making proofs and calculations painful. In geometry, when you see a 7, your face starts twitching. Mine did, and so I moved on to N = 8. Mama didn't raise no fool. I'll do the low hanging fruit first, then get back to the hard problems later.
I spent 15 or 20 minutes diddling with N = 8, until I remembered a mathematician's favorite trick: If you can't solve a harder problem directly, you link it to a simpler problem which you can solve. The guy who solved the cubic equation did this. He turned it into a quadratic equation, which he knew how to solve, (kinda sorta... the details are in any math handbook), and voila! So I did N = 8 by turning it into a scaled version of N = 4. And it turns out that it's a nice little bit of recursion.
For any even numbered N, the edge length EN2 = 2 - √{4 - EN/22}. You can show this with about 1/2 page of algebra, if you know the trick. So for N = 8, I already know E4 = √2, so I use the formula to get E8 = √{2 - √2}. I can then do E16 = √{2 - √{2 + √2}}, which is a tad bit gruesome. Nevertheless, if you've done this a few times, you're not very keen on finding E32 or E64 unless you have a good CAS system to do it for you. It's going to involve a train of radicals within radicals.
OK I lied. I added this bit a day after writing this daylog, just for posterity's sake. When N is a power of 2, EN has the very pretty form that is evident when you do a few of the lower powers:
E4 = √2
E8 = √{2 - √2}
E16 = √{2 - √{2 + √2}}
E32 = √{2 - √{2 + √{2 + √{2}}}
E64 = √{2 - √{2 + √{2 + √{2 + √{2}}}}
BUT - you've done it. Furthermore, since you know E3, you can in principle compute any even multiple of 3, to get E6, E12, etc.
It's also easy to know if you're in the right path or not. Whatever the edge length is, you multiply it by N, the number of edges, and the number should less than 2π, which is about 6.28, which is the circumference of a unit radius circle. The perimeter of your N sided regular polygon, N×EN, should be less than 6.28, but it should increase with N and should asymptotically approach 2π. If you tabulate N and EN, then the numbers should always be increasing. If they aren't, then you know you've fucked up somewhere. So there are built-in checks and balances when you do this. The point is, you know you're on the right path when you do a bunch of these cases and they make a pretty pattern, and are internally self-consistent and obey all the rules. When you fuck up, which I do plenty of times, you do some numerical examples so you know you're right or - in my case - wrong. Then you fix the problem and try again. Repeat until perfect. We nerds are nothing if not obsessive.
So I was sailing along pretty well, and the numbers were lining up pretty well, but... well, shit, but there are still all these holes. I got the exact value for E3, but I wasn't able to find any cool, easily used recursive formula for finding multiples of 3. And then what about those prime orders like 5? 7?
Fuck.
Fuckity fuck fuck fuck.
(When I get obsessive like this, I do two things. One is to swear almost constantly, usually under my breath, but sometimes out loud and in dramatic fashion. The other is to drink coffee. The cumulative effect is one of straight ahead cranking. When I make mistakes I swear loudly, get up from the desk, stomp around, make more coffee, get more cranked up, swear some more, whine and beg and plead to the gods of mathematics, pretend to be distracted by... what's this?... the Boston Marathon bombing? Well FUCKITY FUCK FUCK FUCK WHY CAN'T I DO N = 5?)
The pentagon frustrated the hell out of me, so I picked up Euclid, Coxeter, and a few other big geometry books, and no one had exactly what I was looking for. So I spent a few hours and maybe ten pages and a bunch of ink looking at N = 5. I knew how to construct a pentagon with compass and straightedge, but I didn't look very hard at this. Wikipedia had something that was pretty close to what I needed, and Coxeter had something too, but I had to really beat on it a bit to make it yield an answer. I wasn't happy until I had an answer that was an expression of radicals and the four basic math operations. It's ugly:
For N = 5, E5 =
4√{ 5 + 2√5 }
---------------- = 1.1756...
6 + 2√5
The glimmering of sunshine was beginning to make its appearance into my stupid head when I started on N = 7. I kept looking at the Wiki page, for which an approximate value was given. Nothing exact. By which I mean, nothing, nada, zippo, zilch. There did not appear to be an exact answer for N = 7. Let me say this in a positive way: The Wiki writers are usually a pretty sharp bunch. So if they leave a square blank, which in this case it was, for N = 7, that meant that IT WAS IMPOSSIBLE TO WRITE IN CLOSED FORM AN ALGEBRAIC EXPRESSION FOR CERTAIN VALUES OF N. Indeed, I seemed to hit on the only expression for high orders of N that were known, or at least are capable of being calculated by patzers like me. Then I looked at all prime values of N, and all of the squares were blank. No exact solutions. And then...
Somewhere around 1977 I was reading about Evariste Galois, the brilliant young French mathematician, a hothead, who scribbled out some very important ideas on group theory the night before a duel, a duel he knew he was going to lose. He was looking at attempts to solve polynomials of degrees higher than 5, using only the four basic operations plus radicals, and declared that they were insoluble for group theoretic reasons. It was exactly what I was working on, but I wasn't quite so perspicacious as young Galois, and consequently spent a lot of time for naught.
This is where you get the visual of me slapping my forehead with the palm of one hand, and loudly proclaiming myself to be an idiot. Nay, not just any idiot, but the grand idiot of all time. Facepalm. Headdesk. Headdesk. Stomp around. GRRR. Kick the dog, but the dog's already run away, cowering somewhere. Thank god I don't have a dog.
I wailed. There was much gnashing of teeth. I wasted so much time on something that I knew was impossible to solve! Idiot!
Galois would have laughed. Stupid American! He proved that I couldn't solve this about 250 years ago! Ha ha ha! Fool's mate!
All right, Johnny. There's your shitty first draft, and a pretty shitty second draft to boot!