yeah. i asked my geometry teacher for help, and he got me past the one little point at which i was stuck. I feel like i has so many smarticles right now…
i haven’t actually plugged it into a calculator, as i lost the one calculator that does tangents, and am too lazy to use the stupid computer one atm, but here it is, formula for pi: draft 1:
S over 2[tan(90-180/s)]+ the square root of {2[tan(90-180/s)]squared/4}
where s is any number, higher for a more accurate calculation of pi. i don’t actually need everything after the plus sign, i could just multiply 2[tan(90-180/s)] by 2, but i think using the diagonal and the apothem is a little better than the apothem twice, for getting the diameter of a regular polygon. if someone wants to plug in any number (preferrably greater than 100) into a calculator and tell me if it works, it would be greatly appreciated. i’m so proud of myself! ^.^
Pi is a ratio. It can not be determined exactly with math. Consider the pyramids. There is a unique function of their height to their base. Now back then measurements were made in cubits, I think that it was the length from the elbow to the fingertips of the emperor at the time. So, when laying out the pyramids, they were X amount of cubits high and the base was laid out in rolling cubits. That is by using a cylinder one cubit in diameter. So you see, it is a ratio, and not an expression that can be exactly defined down to digit dot digit.
Pi is not a ratio in the sense that most people use it: it is not of the form a/b where a and b are integers (that’s what a rational number is). It can be expressed as a ratio of real numbers (well, every real number can…), of course, but it would be a meaningless term if you use it that way. EVERYTHING is a ratio, every integer is simply n/1, every rational number is a ratio by definition and real numbers just the same. It is therefore not right that ratios can not be expressed exactly.
Also, it IS possible to “determine” Pi EXACTLY within abstract maths (using a limes expression, or an infinite sum). But of course not in REALITY. Pi has an infinite decimal expansion, which means that you can not output ALL digits of Pi, however, you can get as close as you want (that means, for every integer n, you can calculate the n-th digit - at least theoretically).
So, sorry but I found your statement a bit strange…(also, I don’t see the point of your example, but that’s another story ).
Sorry to disappoint, but it doesn’t appear to be true, at least not as you’ve explained it. As x goes to infinity, your formula is undefined.
A way to see what this means is to look at the tangent function: tan(90 - 180/s) (presumably in degrees?). If you take s to be very very large, then 180/s is effectively zero. Then it reduces to tan(90 - 0) = tan(90), which is undefined, much like 1/0.
No, pi is irrational; it cannot be expressed as the ratio of two integers. Not all real numbers are rational, of which pi is a common example, as is the square root of 2.
BlenderNewb101, your efforts are very fascinating. Keep it up!
The comment about the pyramids was just for interest. Some clever fellows figured out the ratio of their height to their base, as far as they could tell it equaled pi. Then they started to wonder how was it that ancient builders could have understood pi. Then they figured it out. It was not an understanding of pi, it was in how they laid out the plans.
One of the many definitions of pi is that it is the ratio of a circle’s circumference to its diameter. However, there do not exist any two integers m and n such that m/n = pi, and so it is irrational by definition. Just to be clear.
huh… if infinity was substituted for s, you’d sorta be right. damn.
which is odd, because i’ve gone over it three times now, and i don’t see any fundimental issues with my formula. maybe i can try to explain what i’m doing better than i could before now:
because there’s no way (that i know of) to find both the circumfrence and diameter of a perfect circle without using pi, i started out by using the next best thing: an uber-high sided polygon. for simplicity, i chose 360 sides.
I assumed that each side of this regular polygon was equal to one (again, for simplicity) and scince the perimiter (circumfrence) was obviously then 360, my next step was to find the diameter to as close an approximation as i could get, the apothem+ half of the longest diagonal. using half of one of the sides, the apothem, and the diagonal, i created a right triangle. http://i268.photobucket.com/albums/jj34/yodelingtortellini/tri2.jpg
using trigenometry, I determined that tan89.5=x/.5 (is that right?) and from there, I could find the hypotinuse. i added the hypotinuse and apothem together to get the diameter, and divided 360 by it.
so that was my plan, and i can’t find any issues with it. i then removed the assumption that the figure had 360 sides, and replaced 360 with s, resulting in the formula i first posted.
You’d want to use tan(0.5°) = 0.5/x to find the apothem. Then, the hypotenuse is sqrt(x^2 + 0.5^2).
I must say, this is quite clever! And yes, you will get an approximation of pi from this method (one which is actually quite good). The next step is to generalize it so you can put in arbitrarily large values for the number of sides, then look at it as the number of sides approaches infinity, which will yield pi exactly.
Way out of my league here, but maybe circular geometry?
Why Pi?
The need for pi arises because square or rectangular area and circular area are incommensurable. This means they cannot both be analyzed from the same perspective, or point of view. Instead, it is necessary to adopt a frame of reference that is appropriate to circular area to adequately describe and analyze circular area.
It is time to eliminate the flawed elements of Euclidean-Cartesian geometry. The belief that a line is composed of infinitely many points with no area is simply incoherent, and should be abandoned. Likewise, the idea that it is possible to accurately represent circular or even curved area in terms of the straight-line frame of reference supplied by the Cartesian coordinate system should also be discarded.
seems a bit circular (pun intended!) to find pi through a sinusoidal function - aren’t sinusoids based on pi to begin with? A geometric solution would be better
Jack000- Really? is trigonometry based off of pi? i certainly hope not, that would ruin my entire plan
Laurifer- ohh that’s why thing’s weren’t working properly! in that case, x=.5/tan.5(yes?) and the new formula would be:
S over 0.5/[tan(180/s)]+ the square root of {{0.5/[tan(180/s)]}squared/4}
which gets rid of the tan 90 problem!
gracias, Laurifer. your input is being very helpful.
again, if anyone has a calculator they know how to use better than i can use the one on the computer, manually checking this formula would be extremely helpful.
Maybe it was easy to misunderstand my post - but I meant exactly that. A “Ratio” the way pixelmass used it is not a “Ratio” as you and me understand it. Of course Pi is a ratio (it is obviously Pi / 1) but that is obviously not a “rational number”, just as you say, and a rational number is what I mean when I talk about a “Ratio”.
That’s why I said that the way he used the word is quite confusing and leads to misinterpretations.
BlenderNewb, don’t worry, it’s actually the other way round - Pi is defined by trigonometric functions (as I explained the last time ) and plays a major role if you work with radians (which is standard in mathematics).
@ Myke, I did not invent the definition of pi. It IS however defined as a ratio. Odd isn’t it? Here is an explanation,
the comments are interesting as well. There are probably better/different was of explaining it, but for now this is the one that I found easily.
I think I might have considered defining pi as a proportion.
Uh- oh. this isn’t good. most of that article goes way over my 9th grade head, but they do mention pi multiple times. however, it’s always while using radians, which i know involve pi, so until i’ve finnished the rest of the theorem, i’m going to shove my head in the sand and continue doing what i’m doing, then find some other way to get the apothem.
Here is a link for solving for apothem. Please note the use of pi in the equations. http://www.mathopenref.com/polygonapothem.html
Now consider solving for pi: http://mathworld.wolfram.com/PiFormulas.html
Eventually you will run into that darn summation symbol with infinity at the top. The same holds true for a lot of trig functions as well, either by using pi in the fomulas or the symbol itself. Therefor logic would dictate (to me at least) that they are all mathematically in error as no exact value can be determined for them. Granted that at a billion, and maybe by now, more decimal places, things should be accurate enough unless we discover warp drive and start trying to navigate the universe for incredible distances using flawed math or geometry. Another fun thing to think about is the size of a point when trying to apply two points to determine the length of a line. Now if a point has no size, what would be the minimum distance two points could be separated from each other and still have a line between them? It is sort of like having point A being at an arbitrary distance from point B and then starting at point A dissect the line to point B in half and from that new point do it again and then from that new point, do it again… . You will never get to point B!
Here is a fun little link Talking about golden ratios, circles, pyramids, rational and irrational numbers, geometry and stuff, just food for thought I guess if you want to wade through it all. http://www.hyperflight.com/golden_numbers-proportion.htm
Just my two cents worth, Have fun! And don’t give up trying to find the answer, you never know, you just might! It could be revolutionary (heh, sorry about the pun)or at least eye opening when the left side of your brain meets the right side and start to function as one.
Yeah, well…as I have said, Pi is a ratio, but keep in mind that the ratio you use to define Pi that way needs REAL numbers to express Pi. Whereas rational numbers only use integers, and that’s what I meant all the time. But let’s keep it at that, I’m sorry for confusing all of you and not properly explaining my point
).
) and plays a major role if you work with radians (which is standard in mathematics).
