# Examples of calculus trivia?

August 10, 2006 9:56 PM Subscribe

I need more examples of "interesting" calculus (or other mathy) trivia like this one factoid I mention in here...

Recently I learned that if you take the exterior of a sphere and divide it into n slices with cuts parallel to the y-z plane (or whatever other plane you like, of course) the surface area of each slice is the same. (i.e., cut a melon into slices -- the area of rind on each slice is the same.) I had a fun few minutes proving it to my satisfaction. Do you know other interesting little tidbits I could prove in my spare time, or drop as some awesome conversation at a party? Not necessarily looking for calculus stuff, here.

Recently I learned that if you take the exterior of a sphere and divide it into n slices with cuts parallel to the y-z plane (or whatever other plane you like, of course) the surface area of each slice is the same. (i.e., cut a melon into slices -- the area of rind on each slice is the same.) I had a fun few minutes proving it to my satisfaction. Do you know other interesting little tidbits I could prove in my spare time, or drop as some awesome conversation at a party? Not necessarily looking for calculus stuff, here.

I've always loved fixed-point theorems because on one hand they seem obvious and yet you can easily prove some non-intuitive stuff.

Example: take two sheets of graph paper. Crumple one up as much as you like and place it on top of the flat sheet. No matter how you crumpled it, there will be at least one point in the crumpled sheet that will lie above the flat sheet.

Example: Prove that at any time, there are at least two polar opposite points on the Earth that have the same temperature (actually there are infinitely many of them)

posted by vacapinta at 10:14 PM on August 10, 2006

Example: take two sheets of graph paper. Crumple one up as much as you like and place it on top of the flat sheet. No matter how you crumpled it, there will be at least one point in the crumpled sheet that will lie above the flat sheet.

Example: Prove that at any time, there are at least two polar opposite points on the Earth that have the same temperature (actually there are infinitely many of them)

posted by vacapinta at 10:14 PM on August 10, 2006

Sorry, I'm trying to rack my brain since I really enjoyed proving some of the more interesting things when I was studying physics...

Also, prove the Shell Theorem to yourself, which is an application of the Divergence theorem. Namely, if you were to construct a hollow sphere, there would be no Gravity inside.

posted by vacapinta at 10:41 PM on August 10, 2006

Also, prove the Shell Theorem to yourself, which is an application of the Divergence theorem. Namely, if you were to construct a hollow sphere, there would be no Gravity inside.

posted by vacapinta at 10:41 PM on August 10, 2006

Not sure if it's exactly what you were looking for, but I recently ran across Euclid's Proof of the infinitude of primes and I thought it was a pretty nifty way of demonstrating this fact.

posted by kaefer at 11:08 PM on August 10, 2006

posted by kaefer at 11:08 PM on August 10, 2006

I like Gabriel's Trumpet and it is 100% calculus. Take 1/x rotated around the x axis with x >= 1. It has a finite volume, but an infinite surface area.

posted by fleacircus at 12:08 AM on August 11, 2006

posted by fleacircus at 12:08 AM on August 11, 2006

Best answer: I think this integral which shows that Pi is a little less than 22/7 is particularly neat, as the integral involved is so simple.

If you want to prove it yourself from scratch, don't follow the link. Just try to integrate x

posted by edd at 12:37 AM on August 11, 2006

If you want to prove it yourself from scratch, don't follow the link. Just try to integrate x

^{4}(1-x)^{4}(1+x^{2})^{-1}from 0 to 1.posted by edd at 12:37 AM on August 11, 2006

Best answer: Have you exhausted the Mudd Math Fun Facts? I linked to the calculus section but they have lots of stuff.

posted by Wolfdog at 4:17 AM on August 11, 2006

posted by Wolfdog at 4:17 AM on August 11, 2006

e^(pi*i)+1=0. A classic and a personal favorite. A single formula that contains every number you'll need up to and including calculus: e - calculus, pi - trig, i - imaginaries, 1 - the multiplicative identity, 0 - the additive identity, both for algebra.

The proof is pretty awesome, too.

posted by ChasFile at 7:22 AM on August 11, 2006

The proof is pretty awesome, too.

posted by ChasFile at 7:22 AM on August 11, 2006

Vacapinta— What do you mean "above" the flat sheet? If you place the crumpled piece on top of the flat piece, won't all of it be above?

posted by klangklangston at 10:38 AM on August 11, 2006

posted by klangklangston at 10:38 AM on August 11, 2006

Vacapinta means "that will lie exactly above the corresponding point on the flat sheet."

Crumpling and projecting down on to the flat sheet essentially defines a continuous map of the sheet to itself (both the crumpling and the projection are continuous), and that's where the fixed point theorem kicks in. A fixed point for this map is a point that ends up exactly over where it started after the crumpling is done.

posted by Wolfdog at 10:43 AM on August 11, 2006

Crumpling and projecting down on to the flat sheet essentially defines a continuous map of the sheet to itself (both the crumpling and the projection are continuous), and that's where the fixed point theorem kicks in. A fixed point for this map is a point that ends up exactly over where it started after the crumpling is done.

posted by Wolfdog at 10:43 AM on August 11, 2006

AHHH! OK. Thanks.

posted by klangklangston at 11:09 AM on August 11, 2006

posted by klangklangston at 11:09 AM on August 11, 2006

You can cover a checkerboard with 32 dominoes. Remove two squares at opposite ends of the long diagonal. You cannot cover the remaining squares with 31 dominoes.

posted by Wet Spot at 11:09 AM on August 11, 2006

posted by Wet Spot at 11:09 AM on August 11, 2006

At 11:00, you have a vase with ten pennies inside.

At 11:30, remove 1 penny, and put 10 more pennies inside.

At 11:45, remove 1 penny, and put 10 more pennies inside.

At 11:52 and a half, remove 1 penny, and put 10 more pennies inside.

Et cetera.

How many pennies are in the vase at noon? Zero.

posted by Wet Spot at 11:19 AM on August 11, 2006

At 11:30, remove 1 penny, and put 10 more pennies inside.

At 11:45, remove 1 penny, and put 10 more pennies inside.

At 11:52 and a half, remove 1 penny, and put 10 more pennies inside.

Et cetera.

How many pennies are in the vase at noon? Zero.

posted by Wet Spot at 11:19 AM on August 11, 2006

Best answer: Take any triangle. Trisect all of the interior angles. The trisector lines intersect at three distinct points which form an equilateral triangle.

posted by Wet Spot at 11:20 AM on August 11, 2006

posted by Wet Spot at 11:20 AM on August 11, 2006

Wet Spot's "pennies in a vase" is a variant on the Monty Hell problem and/or the Ross-Littlewood paradox. The "paradoxical" nature of it is due to the fact that the number of pennies "at time t=1" can be expressed as a series which is conditionally convergent, and different interpretations of the order of terms in the series may lead to rearragements of the sum that converge to zero, diverge to infinity, or anything in between. Different interpretations of the physical mechanics of the problem naturally lead to different rearrangements of the sum.

To get the answer "zero" I bet he wants to argue that every penny eventually gets taken out - but whether that's true or not depends on, for instance, how you choose which penny to take out and each step!

posted by Wolfdog at 11:46 AM on August 11, 2006

To get the answer "zero" I bet he wants to argue that every penny eventually gets taken out - but whether that's true or not depends on, for instance, how you choose which penny to take out and each step!

posted by Wolfdog at 11:46 AM on August 11, 2006

Response by poster: I'm marking as 'best answers' all the ones I haven't heard of. They're all good though.

I'm reserving judgment on Wet Spot's pennies in a jar, example, though: if all he's doing is rearranging terms of a conditionally convergent sequence, then he can't really say that there's "zero" pennies left and call that the "right" answer. You can rearrange the terms of a conditionally convergent sequence and get lots of different numbers! Like, dude, I'm getting

on reading the link wolfdog provided: yeah, OK, if that's what he's up to, wet spot should explain exactly which penny we're removing. otherwise I call foul.

posted by evinrude at 11:58 AM on August 11, 2006

I'm reserving judgment on Wet Spot's pennies in a jar, example, though: if all he's doing is rearranging terms of a conditionally convergent sequence, then he can't really say that there's "zero" pennies left and call that the "right" answer. You can rearrange the terms of a conditionally convergent sequence and get lots of different numbers! Like, dude, I'm getting

*one hundred*pennies in the jar -- how are you getting zero? (Oh wait, I just recounted; I actually have*pi*pennies. Sorry.)on reading the link wolfdog provided: yeah, OK, if that's what he's up to, wet spot should explain exactly which penny we're removing. otherwise I call foul.

posted by evinrude at 11:58 AM on August 11, 2006

Best answer: 'Squaring the square,' that is, building up a square from smaller, all unequal squares with integer sides and leaving no spaces, is a difficult task which was accomplished last century, in 1939, but proving whether or not it is possible to 'cube the cube' is much easier and might amuse you (the link addresses this latter problem, so try it before you click or read down too far).

Another problem I like, and which has a family resemblance to your original sphere problem, is the isoperimetric problem in the plane: which geometric shape has the least perimeter for a given area? Everyone knows the answer is a circle, but I used to put my self to sleep anxious nights by trying to construct a proof based as entirely as possible on considerations of symmetry.

Also, given the answer to the isoperimetric problem in the plane, can you extend it to space?

posted by jamjam at 12:45 PM on August 11, 2006

Another problem I like, and which has a family resemblance to your original sphere problem, is the isoperimetric problem in the plane: which geometric shape has the least perimeter for a given area? Everyone knows the answer is a circle, but I used to put my self to sleep anxious nights by trying to construct a proof based as entirely as possible on considerations of symmetry.

Also, given the answer to the isoperimetric problem in the plane, can you extend it to space?

posted by jamjam at 12:45 PM on August 11, 2006

That penny paradox seems a little impossible. It takes some amount of time to take a penny out, and to put ten pennies in. At some point, you'd be too slow to keep up with the terms of the equation. Up to that point, you'd be putting more pennies in than you'd be taking out, so you'd have more than ten in the jar.

posted by owhydididoit at 10:05 PM on August 11, 2006

posted by owhydididoit at 10:05 PM on August 11, 2006

Ohwhy: See also Xeno.

posted by klangklangston at 8:34 PM on August 12, 2006

posted by klangklangston at 8:34 PM on August 12, 2006

Best answer: Here's one that I think is not too well known: suppose you have a continuous (real) random variable; you sample it repeatedly to form a random sequence, maybe something like this:

66.6405, 67.4241, 77.8846, 33.9181, ...

and you let X be the position where the first

posted by Wolfdog at 5:55 AM on August 13, 2006

66.6405, 67.4241, 77.8846, 33.9181, ...

and you let X be the position where the first

*decrease*occurs. (You'd record X=4 for the example). Then the expected value for X is exactly*e*, and that is independent of the distribution you're sampling, as long as it's a continuous distribution.posted by Wolfdog at 5:55 AM on August 13, 2006

This thread is closed to new comments.

posted by CrunchyFrog at 10:02 PM on August 10, 2006