Ah! But it’s so terribly simple and elegant! I’d love to explain how Pi was and still can be discovered by anyone who has knowledge of basic algebra. Finding Pi is
not some mysterious and esoteric bunch of mathematics that only mathematicians can understand. Anyone who knows algebra and a bit of geometry can grasp it if they follow the logic carefully.
Pi was first formally found by one of the three greatest mathematicians of all time: Archimedes. Archimedes lived in the time of the Ancient Greeks, and was able to discover an incredible amount of mathematics all on his own. There are multiple ways to find Pi, but the following is my favorite.
It’s a lot easier to understand if you have pictures, but since this has to be explained through text alone, try to bear with me. You may want to grab a sheet of paper and draw diagrams of what I describe as you read through it. That should make it a lot easier to understand. As I write this, I’ll try to find images on google to help illustrate what I’m talking about. Here it is:
Suppose you have a uniform circle. You want to know
exactly how many times the diameter of the circle will wrap around the circumference. (Pi) What you’re trying to do is find out what the circumference is
in terms of the diameter (twice the radius). That means, you want to create an equation that reads: C = XD where C is the circumference, D is the diameter, and X is the ratio between them (Pi).
Well, in order to begin this immense task, let’s start out with something we already know how to do. Say you have a pentagon with a dot at its center. If you draw the smallest circle possible that fits exactly around the pentagon, you will have created a circle whose center is the same as the “center” of the pentagon. In mathematics, this is called
circumscribing the pentagon with a circle. Anyhow, here’s a picture of what it looks like:
If I asked you to find the perimeter (total length of all the sides) of the pentagon, could you do it if you knew its radius? (By radius, I mean the length from the pentagon’s center to one of its corners.) Well, after a bit of thinking, you’d say: “Sure.” If you have a regular polygon, and you know its “radius” (length from the center to a corner) you can find its perimeter using geometry. (In geometry, “regular” means that the lengths of all of the polygon’s sides are the same.)
What’s this have to do with finding Pi you ask? Well, imagine in your head (or draw out on a sheet of paper) a regular polygon with 10 sides. Now imagine that same regular polygon (same radius) but make it have 25 sides! If you’re drawing this, the first thing you’ll notice is that it’s hard to make it not look like a circle. Now imagine the same regular polygon with 10,000,000,000 sides! (Remember, since the radius is the same, the size of the polygon remains the same, it can’t get bigger.) This polygon has so many sides, if you saw it, you’d think it was a circle!
The point is that, if we can find a formula that tells us the perimeter of a regular polygon in terms of it’s radius (which we can using geometry) and then we make it have a whole bunch of sides (like a million) it starts to look and act like a circle. Taking this further, we create a formula for the perimeter of a regular polygon with N sides in terms of its radius and N. If we then make say that N is approaching
infinite then we’ve successfully created a formula for a
Circle in terms of it’s radius, and N (Which we say is arbitrarily close to infinite). This is the only abstract part of the problem, so don’t be worried if you’re confused. The basic idea is that a regular polygon with infinite sides is really the same thing as a circle.
Now, using that idea, let’s create a formula for a regular polygon with N sides and a radius R.
We’ll start with a pentagon for simplicity. If you cut the pentagon into triangles, you’ll get five triangles. (One for each side) Here’s what one of the triangles will look like:
You know that the total number of degrees around the center of the pentagon is 360 so the angle at the tip of the triangle is going to be equal to 360 degrees divided by the number of sides the regular polygon has. (In this case five, but let’s just call it N so that we can have a more general expression. That way when we’re done, we’ll be able to find the perimeter of a regular polygon with any number of sides by simply plugging in the number of sides everywhere we see N.)
So,
360/N = A
A = the angle of the tip of each triangle that is a part of the regular pentagon. (The gray triangle in the picture is an example.)
We also know that the perimeter of the regular polygon is equal to the length of the base of one of the triangles times the number of triangles in the regular polygon. Since the number of triangles in the polygon is the same as the number of sides, the Perimeter is equal to the number of sides times the length of the base of the triangle.
P = SN
P = Perimeter
S = Side Length of one of the triangles that makes up the polygon. (See the gray triangle for an example)
Now, using geometry we can cut one of the triangles that makes up the polygon in half to create a new right triangle. It looks like this:
We know that the hypotenuse (the side from point B to point A) of this new right triangle is the the radius of the regular polygon, and we know that the angle at the tip of this new right triangle is exactly half of the angle of the larger triangle. Since we know that the angle of the larger triangle is A (A = 360/N see above) we can say that the anger of this new right triangle is (1/2)A.
Since A = 360/N
(1/2)A = 360/2N.
We now know that in this picture, the hypotenuse is R and the angle at the top is 360/2N. Using basic trigonometry, we can find the base of this new right triangle! Sin(Angle) = Opposite side length over Hypotenuse side length. Remember that the base of the
whole triangle is S. So the base of this
new right triangle is (1/2)S. So, Sin(1/2A) is equal to one half of S times the hypotenuse (R).
Sin(1/2A) = (1/2)SR
We can solve this equation for S to find that:
Rsin(1/2A) = (1/2)S
2Rsin(1/2A) = S
Since (1/2)A = 360/2N.
S = 2Rsin(360/2N)
Remember that the perimeter is equal to the length of S times the number of sides, so:
P = SN
S = 2Rsin(360/2N)
P = 2Rsin(360/2N)N
There! Now we have a formula that tells us the perimeter of a regular polygon in terms of its radius and the number of sides it has!!! Notice that the only variables in the equation are R (The radius) and N (The number of sides the polygon has). This means that you can pick a regular polygon with any number of sides, and find it’s perimeter if you know its radius.
Since we already discovered that a circle is essentially a regular polygon with infinite sides, if we set N to infinite. (Or let N
approach infinite), we’ll have a formula for the perimeter of a circle in terms of it’s radius! Since the perimeter of a circle is its circumference, we’ve discovered a formula for the circumference of a circle in terms of its radius!
When N ~ Infinite:
C = 2Rsin(360/2N)N
In the beginning we decided that we wanted to find a number X so that the circumference of a circle is equal to it’s diameter times X.
C = XD
Since the diameter equals two times the radius.
D = 2R
C = (2R)X
We already found that when N ~ Infinite:
C = 2Rsin(360/2N)N
So, since C = (2R)X and C also =2Rsin(360/2N)N
(2R)X = 2Rsin(360/2N)N
If we solve for X we find that
X = sin(360/2N)N
Pi = sin(360/2N)N
Now we’re done. We’ve found a number X so that X times the diameter of a circle equals the circumference! But we must remember that this is only true when N approaches infinite. What good is it then you ask? Well you don’t need to actually
have and plug in infinite. You can simply plug in very large numbers! Say N = 10,000. If you plug in 10,000 into the formula we’ve created, you’ll get an answer very close to Pi.
Watch, I’ll do it!
N = 10,000
Pi = sin(360/2N)N
Pi = sin(360/20,000)10,000
Pi = 3.141592601912665692979346479289
You see! It’s very close to Pi! Some of you might say, “Hey! The 20th decimal is wrong!” Or something like that. That’s because in order for it be exactly equal to Pi, N must = infinite, which you can’t do.
But! The larger a number you plug in as N, the
closer to Pi your answer will be! If you plug in 1,000,000,000,000 as N, you’ll get an answer VERY FREAKING CLOSE. This is also why Pi can be thought of as an irrational number that goes on forever! Since you can always plug in a bigger and bigger number for N, you can always have Pi correct to more and more decimal places! Thus, Pi continues forever!
The only thing I must warn you about, is be careful that when you plug in some number for N, you plug in
the same number in both places N appears in the formula. That is, don’t plug in 10,000 for N once, and then accidentally plug in 100,000 for N later. Don’t do this:
Pi = sin(360/200,000)10,000
If one were to do that, he would have plugged in 100,000 once, and 10,000 the second time. You must be consistent with what you select N to be since N shows up twice in the formula.
All I have left to say is try it!
[ March 28, 2004, 12:12 AM: Message edited by: Lord X ]
