John Derbyshire has a little maths problem at the tail of his August Diary
in National Review Online. Here it is:
Draw a circle of radius 1 unit (foot, meter, mile, whatever). Inscribe an equilateral triangle in the circle, the three corners all on the circle's circumference. Now inscribe a smaller circle in the triangle, its circumference just touching the sides of the triangle at their mid-points.
Starting with that smaller inner circle, repeat the process; but this time inscribe a square instead of a triangle. Then inscribe a yet smaller circle in the square, its circumference just touching the sides of the square at their mid-points.
Starting with this yet smaller circle, inscribe a regular pentagon in it ... and then a circle inside the pentagon, touching its sides. Keep going in this fashion, with a hexagon, a heptagon, and so on.
The circle you inscribe inside an N-gon will be the (N-1)-th circle you've drawn. (The circle inscribed in the square, for example, is the third one you've drawn.) The radii of all the circles you have drawn up to that point will form a sequence of (N-1) numbers, each smaller than the one before: 1, r2, r3, r4, ... rN-1. What is the limit of this sequence?
First, let's make sure it has a limit. The ratio of the radii of the incircle to the circumcircle (call 'em r and R respectively) of a regular n
-gon is r/R = cos(pi/n
). So the second circle (the one inside the triangle) is cos(pi/3) = one half the radius of the circumscribing circle. The next one has cos(pi/4) = 1/Sqrt the radius of that. The next one is cos(pi/5) = (1 + Sqrt)/4 of that and so on. So the radius of the n
th circle is the product of cos(pi/k) as k
goes from 3 to n
. Does this converge in the limit? Yes. Why?
Recast the product. Take the natural log of each term and switch the product to a sum. If you take e
, the base of the natural logarithms, to this power then you get the same answer as the product. Does the series converge? Use d'Alembert's ratio test. In the limit as k
goes to infinity, is the ratio of the k
+1th term to the k
th term less than unity? Yup. Series converges. In the interval in which we're interested, log of cos(x
) as x
goes to zero is a monotonically increasing, strictly negative function so d'Alembert is applicable and gives the answer that the sum converges.
There's no closed-form solution for the radius of the inner circle in the limit. Its numerical value is approximately 0.1149420448532, but using the product of cosines formula converges horribly slowly. However, using some evil jiggery-pokery and the product of terms -> sum of logs technique adumbrated above, you can get a fast converging series using, mirabile dictu
, the Riemann zeta function.
What the hell was I thinking? The ratio test is useless. Limit as kth
term goes to infinity is unity, and d'Alembert has nothing to say in this case. Fortunately, the Integral Test rides to the rescue. Recast the sum as minus sum of log of sec of pi/x for x = 3 to infinity, check that in the limit log of sec of 1/x goes to zero (it does), and check that in the region we're talking about, the integral is bounded. It is. There's no closed form solution for the integral of the log of an elementary trig function, but numerically it converges. So we're OK.
Sorry for all the bad language. But sometimes it's apposite, y'know?
On a lighter note: here's what I had for supper tonight, with enough leftovers for lunch tomorrow. It's my interpretation of a bunch of cookbook recipes:
Chicken Tabouleh Salad
1 cup Bulgar wheat
2 cups water
1 large or 2 small boneless, skinless chicken breasts
1 cup chopped parsley
1 cup chopped cucumber
2 tsp finely chopped mint leaves
1/2 cup finely chopped spring onions
2 tbsp freshly squeezed lime juice
3-5 cloves garlic, minced
4 tbsp extra virgin olive oil
Season the chicken breasts with salt and pepper and leave them to sit for a couple of hours. Cut a cucumber into quarters lengthways and remove the seeds with a teaspoon. Julienne the cucumber quarters and cut the pieces into 1/4" dices. Set aside. Finely chop 4-6 spring onions (white plus some of the green) and set aside. Finely chop the parsley and mint and set aside.
Heat a skillet with two tbsp of olive oil to medium high heat and add the chicken breasts. Continue cooking over medium high heat for 10-12 minutes, turning occasionally, until the chicken is lightly browned and cooked all the way through. Remove the chicken to a bowl or saucer and leave to cool, preferably in a refrigerator.
When the chicken is quite cold, dice it into 1/3 inch dices. Bring the two cups of water to a boil. Place the Bulgar wheat in a heat-proof dish and add the boiling water. Place it to one side and cover it with a clean tea-towel.
For the dressing: mince the garlic as finely as possible, and mix it with the olive oil, lime juice and a pinch of salt in a bowl. Leave it to infuse. Lemon juice can be substituted for lime, but use three tbsp.
After twenty minutes or so most of the liquid will have been absorbed and the wheat cooked. Drain any excess and make sure the wheat has dried properly. Place it in a large bowl and allow it to cool thoroughly. A half-hour spell in the fridge or freezer here can help (but don't freeze it!). When it is cool, add the parsley, mint, spring onions, diced cucumber and diced chicken and mix well. Finally, toss the salad in the garlic and oil dressing and allow to sit for three minutes more. Serve with warm pitta slices, natural yoghurt and dipping sauces (hot ones are good here: investigate thick tomatoey salsas which I may or may not reveal here).
This quantity serves 6 people according to cookbooks, or three if you are a regular human being.
: or in some Bizarro universe where Michael Moore eats dishes as low fat as this, 0.07 of the disgusting fat fucks.
Ancient Greece had it right when it came to bureaucrats
I suppose using Blogger is faster than mimeographing your own newsletter and handing it out to hoi polloi. Only fucking barely, mind you. So get your own website, you cry.
Problem: to get a credit card that US companies will accept over the Internet via the bank into which my salary is paid requires a level of bullshit that frankly makes me want to kidnap senior bank executives and play with them with a blowtorch for fourteen or fifteen years. Apparently, before my DEBIT card (ideally, picture the word DEBIT written in three light-year-high neon letters floating through the cosmos) can't be used outside Costa Rica unless I a) have a $1000 minimum balance and b) have been a legal resident for five (count 'em) five years. It's a fucking debit card you stupid shitstains! I CANNOT spend more than is in my bank account. You CANNOT lose money. Problem solved.
Oh, I forgot, you dickless relics of a bygone age: this is not the UK or US, and so your silly shoulder-shrugging denial of a cerebral cortex does not, in general, earn you a sharpened screwdriver in your left eye. But it fucking should.
I swear, the best thing as competition approaches is the liquid sensation these useless morons get in their bowels every time payday is due. They call capitalism 'creative destruction', and one of the most beautiful and creative aspects is its destruction of these twats.
There's a strike of sorts going on here right now. As touchy-feely and liberal this place is, rather than pepper-sprayng the indolent shits and firing them en masse, the government is flippy-flapping harder than Kerry in a Trivial Pursuit match. I love this place most of the time, but Jesus God we need a Maggie.
A memorable speech
I missed the John McCain speech at the RNC. I saw bits of it in highlight. I still think he's poison, but I admit his utility in attracting a segment of the electorate that otherwise would not vote Republican.
Then there was Rudy. Rudy Giuliani. Oh. My. God. I think that was one of the best pieces of political theatre that I have seen in the past thirty years. It's not often I get teary-eyed when it comes to a political speech (a corny ending in a movie, on the other hand, can have me be weepy in a heartbeat). I think the last time I was moved to tears by a speech was the re-runs of Ronald Reagan's plea to Gorbachev to "tear down this wall". I'm not saying that Rudy's speech was of similar import, but it was amazingly effective. He mixed vernacular with some pretty high-flown rhetoric. He addressed the crowd in a folksy manner, while introducing some of the most idealistic statements you will have heard recently. Things like 'freedom', and 'self-determination'.
He also whetted a shiv and stuck it in John Kerry. This wasn't grandstanding, ad hominem stuff. Oh, no. Giuliani hurt John Kerry much worse than that. Rudy quoted him. This was devastating. He took a few pairs of Kerry's public utterances and laid them side by side. Giuliani took pains to praise John Kerry's wartime service. That gave him the opportunity to attack his flip-flopping. Excellent. Of course I know he's preaching to the choir, but a lot of his speech was an attack on Kerry that while mild-mannered on the face of it, really drove to the heart of some of the issues that the Dems would rather stay under wraps. I can see this having traction with the swing voters.
I'm not naive enough to think that this will be a tipping point. But from my second-hand observations, RNC 2004 is going a lot better at this stage than its Democrat counterpart.
P.S. A combination of my sucky broadband, my idiotic browser and Blogger ate the first version of this, which was much closer to live blogging. In particular, I observed how Judy Woodruff thought Giuliani's speech was too long. For her, I imagine, it must have been forty minutes of nails scraping down blackboards.