the origin of pi + a perfect circle

hi. i’m new to the board, and fresh from college … so forgive me if i sound ignorant. :slight_smile:

anyway, i have a question that’d been bugging me for ages: what is the origin of natural numbers (or numbers that seem to appear in nature for one reason or another). [the only reason i’m posting this to GD is that ppl seem to be more knowledgeable here for one reason or another] heh.

i do realise that there may be no reason for them to appear ie. if x were to appear instead of pi, i would be asking why does x appear.

but, what i want to find out is, the origin of these numbers… pi, e, golden ratio etc.

i’ve heard of supercomputers calculating pi to the millioneth decimal, but how do they do that? [and pls dont say, becoz they are supercomputers] what i mean is, early men, they draw a circle, and say, oh, look, the circumference of the circle measured = the diameter x this strange number that approximates 3.14.

i know that, but there are problems with calculating pi based on this argument. is it even possible to draw the ‘perfect circle’ [my contension is that it is NOT] but i’ll give it to you. even if it were possible, the measurement itself would pose a problem.

anyone here heard of the ‘fractal paradox’? where a fractal (this figure thiggy) can have a perimeter of infinity but have an area of zero.

yep, thought i’id pop these questions up. hopefully, they don’t pop your brain as well.

p.s. why is a circle 360 degrees? ok, i know there probably will be ppl saying, if it is y degrees, then you’d post hte same question asking why it is y. but anyway, seriously, why 360? anything to do with sun’s revolutions or something??

No idea what a fractal is. but wouldn’t a line have a perimiter of infinity and a area of zero?

as for Pi. If I understand it correct it is intrinisc withen the defined shaped of a circle. (note defined, because WE defined it).

Yo can calculate pi via an infinte expansion of fractions.

read about it from the master

The Koch Snowflake figure has an infinite length and a finite area. Here’s an explanation http://www.math.hmc.edu/funfacts/ffiles/20006.3.shtml

As other’s have said, pi is now computed from a series expansion. Way back when, pi was computed by inscribing a geometric figure inside a circle and circumscribing the same figure outside a circle. By elementary geometric reasoning, the length of the circumference must be between the lengths of the perimeters of the two figures.

For example, place a square around a circle. Its perimeter is 4D, where D is the length of the diameter of the circle. A square inside a circle has the length 2D*sqrt(2) which approximately equals 2.828D so the the circumference of the circle has to be somewhere between 4D and 2.828D. If the number of sides of the bounding figures is increased the lengths of their perimeters gets closer and closer together with the length of the circumference always trapped between them and you can compute pi to any degree of accuracy if you have enough time. Ancient mathematicians used figures with lots of sides to get a really good number.

If you have the time, trot down to your local library and check a book by one of the masters of the subject, namely.
A History of Pi by Petr Beckmann It is a very informative book, though one should note a few things. A) Mr Beckmann is very opinionated, and it does very strongly come out in his writing, and B) it was written in 1971, so a few advances in computing have made some of his statements about Pi obsolete.

Oh, I should point out that the book is written so that a layman without a background in math can follow it fairly easily. It will take a little effort, but I blame that more on the lack of mathmatical education that is common these days.

For a lot of questions, your best bet is to check, first, whether the Master has addressed the issue by searching the Archives:

In this case, Why are there 360 degrees in a circle?

By the way, the circle answer is related to the question Who decided the day should be divided into 24 hours?.

but is not really related to the question Why are there seven days in a week?.

I’m not sure if I’m answering your question, but since a circle is not composed of straight lines its area cannot be exactly known. If the radius is 1, the area is 3.14XXXX.

Say we draw a square in Paintbrush on the computer. If we zoomed in on a square with sides of length 1, the area enclosed would be 1 regardless of how closely we zoomed in and looked at the lines making the edges.

A circle is not composed of lines, but an arc, which causes an irrational amount of area to also be enclosed. If we zoomed in on a circle drawn in Paintbrush, we’d see 3 pixels eventually move up and over to simulate the arc, but never quite mimic it exactly. You must redraw these pixels as an arc to properly draw the circle, thus incorporating an even smaller area. This continues to infiniti, and pi is the closest approximation of this circle’s area.

I agree with you, drawing a perfect circle is impossible, but we could probably draw them down to the atomic level - which is pretty close to perfect for all intents and purposes.

As far as 360 degrees, this probably relates to the same numbering system we adopted for our clocks. 60 is a great number because it can be divided by more numbers than any number below it (1, 2, 3, 4, 5, 6, 10, etc) which probably made math easier back in the days before calculators, since whole numbers are always easier to deal with.

But if they had divided a circle into only 60 degrees you’d have to designate each quadrant as 15 degrees (only divisible by 5 and 3) which doesn’t work well in math. By instead dividing each quadrant into 90 degrees you allow a lot of whole-number based math, and allow the angles of triangles drawn within quadrants to be whole numbers (30, 45, 60, 90), which again, makes doing sin/cosine/tangent easier than if you were using fractions.

While the area of a circle with a rational radius cannot be exactly expressed with a finite number of decimal digits, it can be exactly known: it’s [symbol]p[/symbol]r[sup]2[/sup].

And yes, it’s very often impossible to reproduce a mathematical figure perfectly. They’re ideals, simplified from the real world so that we can deal with them.

Also, the nth hexadecimal digit of [symbol]p[/symbol] can be computed without knowing any of the previous ones. They may use that algorithm these days.

Lastly, all numbers were invented because people wanted a concise way to talk about patterns that occur in the world.

wow thanks!

i’m boshwoggled by the info.

heh

ultrafilter, I’ve seen this formula for the nth hexadecimal digit of pi. But I’ve never figured out how it actually works. I even asked a GQ about it once long ago and never came up with an answer. Do you know of a place I can look to find out how to apply it?

I understand that to get it to be a decimal one will need to know the (n-1) previous terms, but my problem was that when I did the math, I seemed to get a fractional result. I must admit it has been some time since I’ve even looked at it, but I was quite boggled at how one actually applies it.

Thanks for any help in the matter.

All I could find in a quick search is this article. That may give you something to search on.

I’ve got a supplemental question related to this if anyone knows the answer…

In the answer by Cecil on computing pi (link above) he says that certain infinite series converge on pi, or a fraction of it.
Eg 1 - 1/3 + 1/5 - 1/7 + … converges on pi/4

I seem to remember (correct me if i’m wrong) that there are actually quite a few series that converge on pi or a fraction/multiple of it. That seems like a pretty big coincidence…
Is there any simple reason (ie can be understood by a layperson, not just number theorists) why this is?

Or is it just because there are an infinite amount of infinite series, so for any random number there will be any amount of infinite series that converge on it? Does infinity divided by infinity equal any number between one and infinity? I’m starting to wander a little now, but would be grateful if anyone knows the answer…

It’s a coincidence. There are an infinite number of series that converge to any number. We just notice the ones that go to some rational multiple of [symbol]pi[/symbol] because we’ve decided that [symbol]pi[/symbol] is interesting.

Infinity divided by infinity, btw, is not defined, because infinity is not a number.

Others have detailed the sequences that converge to pi or fractions of pi. On reason pi is so darn useful is because it helps measure the unit circle (circle of radius 1). The unit circle is kind of like the prototype cycle–and nature loves cycles. In fact, one of the reasons I bailed from my physics major was because I got tired of every course coming back to oscillations every time.

As for e, it’s a useful number because it’s the solution of the equation e[sup]x[/sup] = d/dx(e[sup]x[/sup])–it is its own derivative. In other words, if you graph y=e[sup]x[/sup], you get a function where the value y at any point x is the same as the slope of the function–the rate of change is proportional to how big it is. This is how many things in nature work, like the growth of bacteria in a plentiful food source (each individual bacterium reproduces at the same or similar rate, hence the rate of reproduction of the colony depends on how big the colony is). Also, radioactive decay follows this model, as each atom of a radioactive isotope has the same chance of decaying, the number of atoms that decay in a given time period depend on how many atoms there are at the time.

i was defined to solve the equation: x[sup]2[/sup]+1=0. Hence i is defined such that i[sup]2[/sup]=-1. This was important in algebra (for instance) because it solves that equation, and it means that a polynomial (equations of the type: x[sup]3[/sup]+x[sup]2[/sup]+x+1=0) of degree (highest power) of ‘n’ has ‘n’ values which satisfy the equation.

Note that powers of i have interesting properties.
i[sup]1[/sup] = i
i[sup]2[/sup] = -1
i[sup]3[/sup] = -i
i[sup]4[/sup] = 1
i[sup]5[/sup] = i

Indeed, multiplying imaginary numbers results in cyclic sequences like the above. Remember how pi is useful in describing cycles? Well it turns out that with a little effort, it’s possible to show that:

e[sup]i*x[/sup]=cos(x) + i sin(x) (Euler formula)

and that

e[sup]i*pi[/sup] + 1 = 0 (which relates all the important constants in one equation!)

The classical name for these things is “Mathematical Monsters.” :slight_smile: Examples are the Sierpinski Gasket or Sierpinski Sieve, the Sierpinski Carpet, and the Menger Sponge. (The Koch Snowflake has already been mentioned.) The Menger Sponge for instance has infinite surface area but zero volume.

The neatest (IMO) example of this type is called the Cantor Set. It works like this: take the closed interval 0 to 1 (written [0,1], which means it includes all the numbers between 0 and 1, as well as 0 and 1 themselves. The open interval which doesn’t include 0 and 1 is written (0,1)). Now remove the middle third of the set, the open interval (1/3, 2/3) (which doesn’t include 1/3 and 2/3, but does include the numbers between). Now we have two parts left, [0, 1/3] and [2/3, 1]. Take each of them and remove the middle third, ad infinitum. In the end, how many points are left? Answer: the same number as you started with! Or, that is, for each point in the interval [0,1], you can find a corresponding point in the Cantor Set. Pretty funky.

Anyway, my favorite “monster” is derived from the graph of y=1/x. (You can graph it at a site like http://people.hofstra.edu/staff/steven_r_costenoble/Graf/Graf.html). The natural logarithm of a number (say ‘z’) is defined as the integral from 1 to z of the function 1/x. The natural logarithm (spelled ‘log’ or ‘ln’) is the inverse of the e[sup]x[/sup] function. Hence ‘ln(e[sup]x[/sup])=x’. Anyway, spin the graph around the x axis, and you get a kind of funnel-like structure. If you evaluate the funnel, you find that the funnel has finite volume but infinte surface area. Which means you can fill it with paint, but you can’t paint it!

Math is neat! :slight_smile:

Great question-no debate.

Moving this to GQ.