Q: How would the universe be different if π = 3?

Physicist: We sometimes get questions about physical constants changing, and those questions make sense because there’s no real reason for the constants to be what they are.  But π is mathematically derivable; it kinda needs to be what it is.  You can’t, through the power of reason alone, figure out what the gravitational constant or the speed of light are, but you can figure out what π is.

π is the distance around any circle, C, divided by the distance across that circle, D.  All of the weird places that π shows up track back to this definition.

So this question is doubly profound!  Unlike other constants, if π were different, then scientists (mathematicians especially) would continually have the sneaking suspicion that there’s something deeply, deeply wrong with the universe.

The definition of π seems pretty innocent (the ratio of the circumference of a circle to its diameter), but it shows up over and over in the middle of calculations from all over the place.  For example, even though \left[\int_{-\infty}^\infty e^{-x^2}dx\right]^2 doesn’t, on the surface of it, have anything to do with circles, it’s still equal to π (there’s a loop floating around halfway through the calculation).  A surprising number of calculations and derivations involve “running in a loop”, so π shows up all the time in electromagnetism, complex numbers, quantum mechanics, Fourier analysis, all over.  In fact, in that last two, π plays a pivotal role in the derivation of the uncertainty principle.  In a very hand-wavy way, if π were bigger, then the universe would be more certain.

Aside from leading almost immediately to a whole mess of mathematical contradictions and paradoxes, if π were different it would change the results of a tremendous number of (one could argue: all) calculations, and the fundamental forces and constants of the universe would increase or decrease by varying amounts.  π shows up in way too many places to make a meaningful statement about the impact on the universe, one way or another.


Mathematician: The idea of giving π a new value of could be interpreted in a few different ways. For example:

1. That circular physical objects, as you make them progressively closer to perfect circles, approach a circumference to diameter ratio of something other than 3.14159… If this were the case, it might indicate something about the geometry of spacetime. If space is not flat, that can change geometric relationships. For instance, imagine drawing a circle on the surface of an orange. If we allow distances to be measured only along the orange’s surface (disallowing paths that penetrate the orange or go into the empty space around it), then the ratio of the circle’s circumference to diameter is no longer going to be π. It will, in fact, depend on the size of the orange itself. If our universe is not flat, but a curved surface, that could distort the geometric relationships that we measure on physical objects resembling circles.

2. That we change what we mean when we say π. Of course, π is just a symbol referencing an idea, so if the underlying idea that it references were to change, that would change the value of the symbol. But this is an extremely boring way to answer this question, reducing it merely to the redefinition of a symbol.

3. That we change which mathematical axioms we use. Most people think of math as a single, coherent set of rules. But when you get down to it, there are different possible sets of axioms that you can use to define mathematical concepts. By switching axioms it becomes possible to prove different things. If we were to choose a set of inconsistent axioms (i.e. axioms that lead to contradictions) then it would be possible to use this system to “prove” any mathematical statement true. In that case, you could show that π = your phone number, if you wanted.

If, on the other hand, you choose a set of axioms that are consistent with each other, but different than our standard math, you have to be more precise about what you mean by π. Usually, it makes no difference whether we define π to be the ratio of a circle’s circumference to its diameter, or whether we define it as

2 \arcsin(1)

or

Log(-1)/i

or

4 \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1}.

But when you start messing with axioms, it is going to affect what is provable, and therefore you have to be careful to specify precisely which definition you are using for π, since definitions that normally are equivalent using standard sets of axioms may no longer be. For some sets of axioms, there won’t even exist a mathematical entity that you reasonably could identify as being π.


Answer Gravy: There are a couple of ways to derive the value of π.  Archimedes estimated it by sandwiching the circle between regular polygons that he could find the exact sizes of.

One step in Archimedes’ work-intensive method for estimating π.

However, we can use methods younger than a few millennia to derive cute formulae for π.

\begin{array}{ll}\int \frac{1}{x^2+1}dx\\= \int \frac{1}{tan^2(u)+1}\left(\frac{1}{cos^2(u)}\right)du &\left\{\begin{array}{ll}x=tan(u)\\dx=\frac{1}{cos^2(u)}du\end{array}\right.\\= \int \frac{1}{sin^2(u)+cos^2(u)}du\\= \int 1 du\\= u+ C_1 &\textrm{Where "}\,C_1\,\textrm{" is a constant of integration}\\=arctan(x)+C_1\end{array}

Coming from another angle:

\begin{array}{ll}\int \frac{1}{x^2+1}dx\\= \int \frac{1}{1-(-x^2)}dx\\= \int 1-x^2+x^4-x^6+\cdots dx\\= x-\frac{1}{3}x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\cdots +C_2\end{array}

So, arctan(x) +C_1 = x-\frac{1}{3}x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\cdots +C_2

Since arctan(0) = 0, if you plug in zero you find that C1 = C2, so you can get rid of them and arctan(x) = x-\frac{1}{3}x^3+\frac{1}{5}x^5-\frac{1}{7}x^7+\cdots.  Now, tan\left(\frac{\pi}{4}\right) = 1, so arctan\left(1\right) = \frac{\pi}{4}.  Therefore, \pi = 4\left(\frac{\pi}{4}\right) = 4\, arctan(1) = 4\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots\right).  That tan\left(\frac{\pi}{4}\right) = 1 is based on the definition of the radian, which tracks back to the circumference of the unit circle being 2π, but the fact remains:

\pi = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \cdots and changing the value of π means, among other things, that this summation would need to somehow equal something different.  But it is what it is.

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22 Responses to Q: How would the universe be different if π = 3?

  1. Idran says:

    It’s too bad the question wasn’t pi=4, or you could just say we live in a universe where the overruling distance metric was taxicab rather than Euclidean :D

    Actually, is there a distance metric under which pi=3? If so, you could interpret the question that way.

  2. The Physicist The Physicist says:

    I really wish there was, because that would tie up the question nicely. However, the amount that the ratio of the circumference, C, to the diameter, D, deviates from pi is, in some sense, a measure of the amount of curvature inside of the circle. For a tiny circle on the surface of the Earth, C/D ≈ π. For a “great circle”, like the equator, C/D = 2.
    The only way I can see to make it so that increasing the size of a circle doesn’t change C/D, is to either use a flat space (in which case C/D = π) or have a flat space with a single point of infinite curvature. A more succinct term for this is “a cone”: the surface is flat, but the tip is infinitely curved. Any circle centered at the tip will have a constant C/D different from π, regardless of size. However, any circle that doesn’t contain the tip will have C/D = π, and any circle containing the tip, but not centered on it, won’t be a circle at all (it’s all weird and dented).
    Point is, there’s no space with a constant value of C/D for all circles everywhere, other than flat space.

  3. chaoky says:

    During the pi series derivation in the answer gravy, you incorrectly stated tan(1)=pi/4 whereas you should have put arctan(1)=pi/4. Just a simple typo, but besides that, great article!

  4. Will says:

    Terry Pratchett has his own take on what would happen if you could create a circle where Pi = 3 :P

    http://discworld.wikia.com/wiki/Bloody_Stupid_Johnson#The_Post_Office_Mail_Sorter

  5. The Physicist The Physicist says:

    Thank you!

  6. Idran says:

    Oh I think you slightly misinterpreted my question/comment. You’re right about the value of π based on curvature, but you’re still sticking with the Euclidean distance metric in all those situations. I mean a different distance metric in that a different way of measuring distance between points.

    Take the Taxicab metric then, where the distance between (x1,y1) and (x2,y2) is just (x2-x1)+(y2-y1). If you define a circle (in R^2) as “the set of points equidistant to a given point”, and you define π as “The ratio of a circle’s circumference to its diameter”, then under this distance metric the value of π is 4, and that is constant for any size circle. (It can be easily shown that a circle in this distance metric is a square rotated 45 degrees and centered at your chosen point. Take then a circle in this metric with radius r, and so diameter 2r. Then the length of a single side is 2r (each is up r and over r), so the length of the entire circumference is 8r, and so π=4.)

    I’ve heard from an undergraduate professor that it was proven that, given this definition of circle and this definition of π, under any metric that satisfies the metric axioms, 3<=π<=4, but he couldn't find the article that proved it, and I've never seen it since, so I'm not sure if that's true or not.

  7. Will says:

    It is important to remember that different distance metrics are purely mathematical constructions; actually applying a non-Euclidian distance metric to the universe would be just as complicated as applying a value of 3 to Pi :P

    Using non-euclidian geometry doesn’t actually make the question any easier to answer, it just rephrases the question.

  8. Idran says:

    To be fair, they aren’t necessarily a purely mathematical construction. It’s called the taxicab metric for a reason, after all; if you’re in a large city and someone asks you how to get to such-and-such building nearby where you are, you’ll probably answer “X blocks up and Y blocks over” or something of that nature. And if someone asked you how far away that building is, you’d probably say X+Y blocks rather than sqrt(X^2+Y^2) blocks. There’s some situations where we naturally use an alternate distance metric just because it’s more convenient to describe things that way. :D

    (Granted, I can’t think of any other real-life situation where a person would use an alternate distance metric, but it’s something, at least. :P )

  9. Locutus says:

    When you hover your cursor over the pie, the caption reads pi(e). I would like to point out that pi(e) is my favorite number, and it is approximately 8.5397.

    Love Long and Prosper

  10. The Wonderer says:

    Took me a second Loctus but that funny of you =P. But these constants show up a lot in almost every sort of math. I have known of pi since I was a wee little kid but the real question is why do we always run into them?

  11. Ken says:

    Suppose you use taxicab metric on a triangular grid city (rather than square). Then a circle looks like a hexagon and pi = 3.
    Maybe this is cheating by going to 1D, but on a number line, pi=2, since a “circle” radius r centered at the origin is just the points r, -r, and the circumference is 4r.
    By using a 2D metric such as d((x1,y1),(x2,y2)) = min(|x2-x1|,|y2-y1|), you get pi=infinity.

    You can get all sorts of weird values for pi.

  12. Oli says:

    That’s not a metric Ken; d((0,0),(0,1)) = 0 and (0,0) doesn’t equal (0,1). So still no examples where Pi is outside of [3,4].

  13. Martin says:

    What about pi for a circle rotating at relativistic speed?

  14. Ryan Callahan says:

    If our universe were curved in such a way that pi equalled 3 or 4, wouldn’t we perceive that universe to be flat and this one to be curved?

  15. Ryan Callahan says:

    I suppose that it would still be impossible to square the circle, so the area of a circle with integer radius would be rational, while that of a square with integer sides would be irrational. The entire meaning of multiplication would thus have to change, and we could no longer talk about 8 sets of 8 being 64 and so forth… So I guess this place wouldn’t work, because you wouldn’t be able to have numbered sets of indivisible objects… Unless the concept of quantity itself was fundamentally different.
    What might work is altpi = 2pi, so when you turn completely around you are facing a different direction than if you don’t move or turn completely around again. At least your first face would be. Your second face would be where your first face started… If you rotated around your outstretched hand, it would be rotating with you so you’d see the same side(s), but if you rotated around a mulberry bush you would change which of your two faces was facing which side of the mulberry bush.
    The only problem I see with that is that each indivdual particle would have one face each in rotationiverse 1 & 2, so in fact your face in 1 would have to be made of the same particles as in 2. You could get around this conundrum by making the different faces of the particles have different properties, so it would actually matter which face was facing where, because only one of the faces can see while the other shoots lasers from its eyes.

  16. Idran says:

    It wouldn’t still be impossible to square the circle, because the reason it’s impossible to square the circle is that pi is transcendental, and so it can’t be the solution to any polynomial with rational coefficients. The square root of any transcendental number is also transcendental, and squaring the circle would require a length that is a rational multiple of the square root of pi. However, a compass and straight-edge can only construct lengths that are solutions to polynomials with coefficients over the field of rationals with minimal degree some power of two (since you can take repeated square roots through constructing squares with diagonal equal to a given length). As a side note, the proofs of impossibilities of the other two classic Greek problems are similar; doubling the cube requires being able to construct a length of cubert(2), which has minimal degree 3 over the rationals, while trisecting an arbitrary angle also requires constructing a length with minimal degree 3 over the rationals by the triple-angle identity.

    If pi weren’t transcendental, then that proof would no longer apply, and squaring the circle would be simple.

  17. andre says:

    Hi ,
    Pi(0)=4 seems a reasonable starting point for flat space . 4-x^27 =pi gives x= 0.99476
    Space seems to be much more curved because of the limitation to 27 dimensions . You can derive the rest yourself .

  18. Mario says:

    Pi = 3. The Universe is a cylinder.
    I can think that, only at the “beginning”, pi=3, then the universe was 2 dimensions only. After that, I have a curved development of the space, and the 3 dimensions result. If the Plank Time is true, and I have been able to measure how the universe was expanding from “t=0″, at every Plank time interval, I’d have a “decimal” of pi from the beginning, to the moment that I’m writing this comment…..
    In other words, the simple Circle’s Circumference as a no infinite irrational number value, is perfectly defined in a “point” of the time. Pi, never has end in its decimal values, and it show how is the curvature of the space at every moment.

  19. Brian says:

    Spot on Mario, I agree.
    The curvature of space time is inflating. If we could measure Pi in real time for a real circle to enough decimal places it would be constantly updating.

  20. Mario says:

    Hi Brian, and may be every decimal would be a number that mean how the space-time expansion is “working” at every instant. Why it isn’t a constant?, I don’t know.
    We would need create mathematical formation about this assumptions.
    Regards.

  21. Gregory says:

    pi is 3 1/7 for a reason. In the beginning Pure time spacialized into 3.00 dimensions and 1/7 chronological time. The universe as it ages 6/7 of it’s lifespan adds to this original amount and becomes a 4.00 (New Jerusalem construct in the book of the revelations). Why so many people make it more complex than this?

  22. Jim says:

    At the end of one of Fred Saberhagen’s Berserker stories, one of the main characters receives a ring from a super-advanced race which, when you divided the circumference by the radius, yielded 3 rather than pi.

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