April 29, 2004
Going in Circles
"Dad?"
"Yes?"
"Why is pi what it is?"
"Hmmm... Could you rephrase the question?"
"Pi. This number that appears in the circle area formula, area equals pi times the radius squared. Pi. 3.14..."
"Or approximately 22/7, yes..."
"Why is it what it is, rather than something else?"
"Well, suppose we wanted to figure out what pi was. One way we might start is by taking a circle and covering it with squares. Let's cover the circle with four squares, each with its sides equal to the radius of the circle, and all of the squares meeting in the middle at the center of the circle. What's the area of each square?"
"The area of a square is its side times itself--its side squared."
"And if its side is r?"
"Then the area of each square is r-squared."
"And how many squares are there?"
"Four."
"And so the area of all the squares together is?"
"Four r-squared."
"And do the squares completely cover the circle?"
"Yes they do."
"So now we know that pi must be less than four. Now let's try taking a circle and putting triangles inside it. Let's start with four right triangles that meet in the middle at the center of the circle. What are the short sides of these triangles?"
"Each side is the same r--the radius of the circle."
"And the area of each triangle?"
"Is r-squared over two. They're triangles."
"And the area of all four triangles?"
"Is two times r-squared."
"And do all of the triangles fit completely inside the circle?"
"They do."
"So now we've figured out that pi is more than two. And we already knew that it was less than four."
"Dad?"
"Yes?"
"How long will this take?"
"I'm almost finished. Now I'm going to wave my hands and say that Archimedes did this more than 2200 years ago, and he kept on taking more and more complex shapes whose area he could calculate that covered the circle, and more and more complex shapes whose area he could calculate that fit completely inside the circle, and so by lots of exhaustive work narrowed pi down from 'somewhere between 2 and 4' to '3.1416'..."
"So you're saying that pi is 3.1416 because that's what it is?"
"Yes. It's an experimental fact."
"But could it have come out differently?"
"Yes. There are places in this universe where it does. Draw a circle around an ultra-dense neutron star and you will find that pi is different--or rather that the ratio of the circle's circumference to its diameter is something different from pi. Pi has a unique and stable value only in those parts of the universe where gravity is weak."
"But why is pi equal to 3.1416 in those parts of the universe where gravity is weak?"
"Ummmm...."
"And why is the pi that appears in the circle area formula area equals pi times radius squared the same as the pi that appears in the circle circumference formula circumference equals two times pi times the radius?"
"Ummmm...."*
*Professor Valerie Ramey of UCSD points out that the circumference is the derivative of the area with respect to the radius, and that the radius is the integral of the circumference with respect to the radius. So if you believe in one, you can quickly get to the other.
Posted by DeLong at April 29, 2004 06:05 PM | TrackBack | | Other weblogs commenting on this postHere's an even better question:
Why does e^(pi*i) +1 = 0?
That still blows my mind, and I'm a math major.
Posted by: Jeff on April 29, 2004 06:19 PMIn high school, whenever someone asked a question that was beyond our abilities at that point, the teacher would respond, "You need calculus to figure that out." Then in calculus, we needed differential equations and so on. However, the word "calculus" for some reason, I think maybe it scares kids, satisfies most all their mathematic curiosities until they are ready to learn calculus.
*If you really want to confuse your child, show him/her the third cube root of 27. In doing that you will get to explain Sin, Cos, i, and also the more abstract uses of PI all at the same time. What fun!
>That still blows my mind, and I'm a math major.
According to my wife who is currently a philosophy grad student George Lakoff considers this question at length in his book 'Where God went wrong', nope that was olon Kaluphid.
Lakoff wrote 'where Mathematics Comes from' and his basic thesis is that maths is simply a product of the way our brains are wired and we understand the universe through mathematical models because thats what our brain understands. so mathematics is not a platonic ideal.
My response to this is that after her defense next Wednesday she will have to find a real job.
Posted by: Phill on April 29, 2004 08:01 PM*I meant the second cube root
Posted by: Andrew on April 29, 2004 08:08 PMBrad-
It is refreshing (but not suprising) to see you dealing with non trivial problems.
Sam
Posted by: Sam Taylor on April 29, 2004 08:17 PMMath is a useful tool for understanding the Universe, but it is not the Universe itself. All of our scentific theories are simply testable models which help us comprehend the reality of what is. Sometimes you just have to accept that the Universe is the way it is because it is that way, and move on ...
Posted by: moonbiter on April 29, 2004 08:20 PMThe basic answer is that the value of pi is a property of (flat) space; a result of (analytic) geometry. I do not have room enough in this margin to explain and justify this. :-)
Posted by: Randolph Fritz on April 29, 2004 08:28 PMAfter returning from my abduction by the scatterpoints, I began the process of trying to explain my ordeal. The most painful recollection I had was of the total discontinuity of their ambient space. There was a particular number, it was large -- something like 10^(10^(10^(10^10)))) which was supposed to be the number of points in their ambient space. In any case, they had developed a mapping process whereby objects occupying spaces of dimension >=1 could be transported, teleported would be more accurate, into the scatterpoint domain. And back, fortunately. They had no circles, as we know them ,no differential geometry to apply to their ambient space. Yet surprisingly they knew about the number \pi, --they had to-- in order to develop their teleportation process. Surprisingly, they also had quantum mechanics and had no need for special relativity.
Posted by: CSTAR on April 29, 2004 08:32 PMMmm, pi...
You're a good Dad, Brad.
Posted by: harv on April 29, 2004 09:09 PMYeah, good work Brad. I don't have kids, but I'm pretty sure my answer would have been, "What do you mean by 'why'?"
Posted by: ogged on April 29, 2004 10:40 PMApparently there was once an attempt to legislate a rational value for pi. In Indiana.
See: http://www.straightdope.com/classics/a3_341.html
Posted by: jimBOB on April 29, 2004 11:51 PMa better approximation is the doubles of the first three odd numbers 355/113. The human brain seems to be wired to detect patterns, and can detect patterns in data which do not exist, stockbrokers,etc. I feel that it was to detect animals, such as tigers [bad}young wild pigs [good].
Posted by: big al on April 30, 2004 03:42 AMI'm glad there are kids like yours going forth in the world, Brad.
D
Note that the method of finding pi by sticking more and more little squares inside and outside the circle comes with a lagniappe: as the squares get smaller and smaller, we are working further and further out the decimal string.
Thus we know that the decimal string will never repeat because the little squares further out are nibbling away at bits of the curve of entirely different curvature from those killed off earlier. Hence pi is a trancendental.
About the time they were first laying out Stonehenge (the pits and eclipse calculator, not the megaliths) the British were making what they call "barrows" -- an roughly circular field bounded by a raised earthen wall.
They'd take a couple of stakes with a length of rope -- a hundred feet or so -- and enscribe a semicircle. Then they'd move the center stake a predetermined distance and enscribe a quarter-circle, then do the same thing to finish it off.
The result was an area whose circumference was three times its diameter -- eliminating that silly irrational number by fiat.
A circle is a Moebius strip in a black hole....inside out and twisted while elongated. The numbers are fractured....
And why did the Big Bang happen?
Endless numbers were processed in a blank eternity of no time. All possible combinations of probabilities passed through this blankness. Then all improbable possibilities happened ipso facto and once all possible and improbable mathematical riddles and constructs were processed, the only option left was impossible events.
In other words, Lady Luck threw her invisible dice one time too many and blew up everything.
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Posted by: click here on June 14, 2004 06:54 PM*Professor Valerie Ramey of UCSD points out that the circumference is the derivative of the area with respect to the radius, and that the AREA is the integral of the circumference with respect to the radius.
Sorry to be nitpicky.
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