Here's something that has been bugging me for decades. Well I keep forgetting about it thankfully, but I've never really been able to answer it. Why is Pi actually the value that we have for it and not some other number? If the ratio of a circle's circumference to its diameter was any different, then a regular hexagon wouldn't be composed of 6 equilateral triangles, but is there a fundamental reason why this had to be the case? I know that it can be generated with various series, so if we use that as an answer then the question becomes, why those series? I've mentioned this to a few people over the years and everyone just looks at me blankly or gives some kind of hand-wavey argument. It seems that there's only me that sees it as a valid question. To phrase this a different way, why does flat geometry have 6 equilateral triangles fitting the circle? This happens to be true in our everyday experience of nature, but were it not the case and we were to live in curved space would we consider that curved space to be 'flat' and other spaces curved relative to it? If so then why do we end up with a definition of a flat space with the very precise symmetry of the 6 equilateral triangles fitting the circle?
There is no obvious answer to the question "why is flat space flat" But for a "flat" space, there is nothing "magical" about the 6 equilateral triangles fitting together. The angles in any triangle add up to 180 degrees (that's one way of defining what "flat space" means). If you make a polygon by sticking triangles together, the angles inside a polygon with n sides add up to 180(n-2) degrees. The angles at the vertices of an regular polygon are all equal, so each angle is (180 - 360/n) degrees. So, you can find the values of n where regular polygons can fit together to "fill" the space around a point. The only possibilities are 6 triangles, 4 squares, and 3 hexagons. For n > 6, two polygons are not enough, and three are too many.
Sure, but my point is that there is something very special about the 6 equilaterals making a regular hexagon, and this only occurs in what we call a 'flat' space. If it's only flat following a definition that is chosen to coincide with our experience, then there is still something left to explain.
For many thousands of years, all we knew about was the apparently flat space we lived in. The earth is large enough so that it's curvature is not immediately obvious. In a curved space you can have a triangle with three 90° angles.
There is nothing special about 6 equilaterals making a hexagon it is simply one result of their properties. Why are you looking for more than is there?
Sure, but the 'is because it is' argument relies on us accepting that our definition of flatness is the only possible mathematical definition of flatness rather than an arbitrary definition chosen to match our experience of the physical world. If Pi were slightly smaller then no number of equilaterals centred on a point would form a regular shape, smaller still and we would find that 5 of them would form a regular pentagon and so on. If this were our experience of the physical world would we define this to be 'flat' or would we still find the case of 6 equilaterals forming a regular hexagon to have a more special property worthy of the definition?
Well.... no. This has nothing to do with a definition of flatness and has nothing to do with the physical world. In Euclidean geometry 6 equilateral triangles equal a hexagon. This can be proved. That is the end of the story. If the proof does not supply you with enough of the 'why' that's too bad. The title of the OP was why is pi, pi? and the answer is: because it is. is a nonsensical piece of speculation because its size follows logically from the properties of a circle. Now you can define other geometries if you like and play with them to your heart's content but anything you discover in them will be defined in terms of the axioms and therefore the properties of the geometry. You will still not have a 'why' answer.
As you're aware that's just another 'is because it is' argument. Can we really do no better than that? Perhaps if you take the axioms that lead to the Euclidian plane and derive from them, the value of pi then question becomes more formal and easier to understand.
That's exactly how it is estimated. Pi only applies in flat, Euclidean space. As someone already pointed out, if you measure a circle and its diameter on the surface of a sphere, you won't get a ratio of Pi. So, if you accept Euclid's axioms, then you can derive Pi by looking at regular polygons inside and outside a circle. The more sides the polygon has, the nearer the length of all these sides will be to the length of the circumference. So, you can estimate Pi as accurately as you want by this method. In fact, you can use this method to show that there is a common ratio between the diameter and circumference of any circle. I always think that's an overlooked point: why is there a common ratio for all circles? Proving this using Euclidean geometry might be a good thing to try to do.
What other arguments do you expect in mathematics? The truth of a statement follows logically from axioms.
Take the triangular number 2701 ( figurate number ) and locate in pi digits, starting after the decimal. You'll find that it starts at the 165th place after the decimal 2) 165th position 7) 166th position 0) 166th position 1) 167th position Sum the position numbers ( 165 +....167 ) = 666 = sum of the first 144 numbers in pi While 666 is the 36th triangular (T36), the consecutive triangular 703 (T37) is the 4th part of the tessellation that produces the 73rd triangular 2701 (T73) Neat little bit of pi trivia there :P I grew curious about pi after learning how Archimedes and others used geometric techniques to close in on pi, and wondered,.. if there is a relationship between shapes which can be used to approximate pi,...are there relationships between the numbers in pi and the numbers which produce shapes ( figurate numbers ) ? I have found some quite interesting things about pi which have actually led me to Hipparchus' work pertaining to astronomical calculations, which he supposedly checked against earlier tables of calculations provided by Chaldean astronomers. It may be a rather bold statement, or even premature on my part, but I am going to say that the current things taught regarding how accurately pi was known to ancient civilizations, is tripe. If you can derive approximations of pi by dropping sticks on the ground, you can certainly find other methods that are not intuitive. My main area of focus has been on the repunits { 111,222...888,999 } and their locations in pi, which incidentally have a very curious distribution,...but not surprisingly, the first repunit ( 111 ) is found at the 153rd place in pi, while 153 is the square of New Moons in one calendar year, ~ 12.369... New Moons / Year. By playing around with these repunits { 111,222...888,999 }, pi, their numbers in pi, and some other very curious relationships involving sums of series, figurates and palindromic primes ( emirp ) and using simple combinatorics, it led me to Hipparchus' work on astronomical calculations, totally unexpected, but then again, Hipparchus was thought by some to also have been rather adept at combinatorics himself, hence Schröder–Hipparchus numbers. I've got about 50 pages of work on this, still drafting it bit by bit. Anyway, random crazy guy here, thought I'd drop that on the table.
We can ask the same question about all astronomical constants. I.E Why is the speed of light specifically 299 792 458 m/s? Your question about pi is more simple to understand, because if pi was any different, a circle wouldn't be possible. Pi CAN'T be any different than 3.34... (this can also be applied to pretty much everything else. However, pi is more simple to understand) I'm pretty sure others have answered your question pretty well. cb
PI isn't a constant of nature and circles do exist with a ratio of circumference to diameter other than 3.14... in other curvatures of space. As an example in the physical world, if you could rotate fast enough around the centre of a circle and you attempted to measure Pi this way, it would take a different value because each arc of the circle would appear to contract. You could assign it the same status as the constants of nature, but that would preclude an ontological flatness, which is what the question was designed to determine. Is there a mathematical reason why Pi takes the value it does or is it just a consequence of Euclidian axioms? It seems that it's the latter. So then the question becomes, are the Euclidian axioms just chosen to match observation? The answer to this seems to be yes. From a purely mathematical perspective the remainder of the question is what is the complete set of possible replacements for Euclid's 5th axiom and what if anthing is special about Euclid's choice for it. I haven't been able to establish an argument for ontological flatness, even though I still suspect there must be one.
A lot of the "mysticism" surrounding the value of pi was worsened after Carl Sagan's book Contact. I guess a few people here have read the book? At the end of the book, Ellie is told by the aliens that there are secret messages hidden in transcendental constants like pi. Since she has no physical evidence of her "trip", she resorts to calculating pi to arbitrary precision, and ends up finding an anomaly in base-11 - a sudden long string of ones and zeros. When put on a grid, the string formed a circle of ones against a field of zeros. That gave her a warm and fuzzy feeling that the Universe had been intelligently designed, since pi was "built into the fabric of the Universe" (language very similar to that was used in the book). As much as I respect Sagan, this was an extremely nonsensical part of his book, and can easily mislead people who are already a little cranky to begin with. That scene was omitted in the movie. Thank goodness for that - usually, it's the film version that ends up mangling the better science from books.
For me, the question originated when considering packing structures in chemistry, as a child. My personal prejudice is that any relationship found in the digits of Pi can only be an expression of expansion series and my interest has only ever been a geometric one. I certainly don't consider that mystical or cranky, though I do confess to seeing a certain power afforded to nature by the symmetry of the regular hexagon that isn't available in other spaces. It's encouraging to find non-Euclidean geometry studied in mathematics and playing a significant role in nature.
Everything tells you Pi must be 3.1459... because so many things are based on it. Just think about the confusion it would cause if Pi were 4.32492945!
Technically, you'd need a scientific claim in order to apply it, but I think it's ok so long as you have some form of claim to apply it to. What do you have in mind? You may not have answers to certain questions and you may not even like the questions being asked, but to suggest that someone is a crackpot for asking them is unnecessarily belligerent. I'm used to having to work for answers and if your username is a fair reflection, then I expect that you are too. If you lived in a world where you did measure Pi to be 4.32492945, would you find your own world more confusing than a world where it was 3.1459...? This might sound like a rhetorical question, but it's also a serious question that I genuinely don't know the answer to. It's also, to some extent, the essence of the questions that I'm asking in this thread. Which gives me an idea - how do we define spatial complexity? Is it relative to Euclidean space or does it provide an absolute way to compare spaces? Is there one way to define the complexity of a space or multiple? Does Euclidean space have the lowest complexity or are there other spaces with the same degree of complexity? These would seem like a well formed mathematical questions, right?
My problem with the questions being asked in this thread is that they are not mathematical questions. If they have not already crossed the line they are getting dangerously close to number mysticism. Isn't there a post in this thread linking the decimal expansion of π to the number of new moons in a year? The irrational numbers have a peculiar fascination for some people, these people believing these numbers have special qualities not shared by other numbers. I deliberately used the word 'qualities' and not 'properties'. A 'property' is something I regard as being provable. I am also used to working for answers and I do it in a mathematical way. No hand waving, no 'woo-woo', no metaphysical speculation just good, honest, old-fashioned pen and paper mathematics. This severely restricts my accomplishments but I am under no illusions about my place on the mathematical ladder. There is more glory in a child mastering quadratic equations than all the non-mathematical flights of fancy exhibited in this thread.