The Pi-on

By Sean Carroll | November 11, 2010 6:42 am

I am in love with this comment and want to have its babies:

pi appears as a constant in many formula of physics. General relativity says that it isn’t constant. Is it the origin of the pi particle, aka pion?

A curmudgeonly literalist might, when faced with a question such as this, harrumph a simple “No.” A more loquacious sort might explain that general relativity does not say that π is not a contstant. Pi is not a parameter of physics like the fine-structure constant, which could conceivably be different or even variable from place to place. It’s a universal answer to a fixed question, to wit: what is the ratio of the circumference of a circle to its diameter, as measured in Euclidean geometry? The answer is of course 3.141592653589793…, or any number of representations in terms of infinite series.

But the point of the question is that GR says we don’t live in Euclidean space; we move through a curved spacetime manifold. That’s okay. In a curved space, we could imagine defining the “diameter” of a circle as the maximum geodesic distance connecting two of its points, and taking the ratio of the circumference with that diameter, and indeed it would typically not give us 3.14159… But that doesn’t mean π is changing from place to place; it just means that the ratio of circumference to diameter (defined this way) in a curved space doesn’t equal π. If the circumference/diameter ratio is less than π, you are in a positively curved space, such as a sphere; if it is greater than π, you are in a negatively curved space, such as a saddle. Geometry can also be much more complicated than that, with different ratios depending on how the circle is oriented in space, which is why curvature is properly measured by tensors rather than by a simple number.

Taken from Mathematics Illuminated, which says that pi really does depend on the geometry of space, which is crazy.

Taken from Mathematics Illuminated, which says that pi really does depend on the geometry of space, which is crazy.

(Parenthetically, one of the dumbest mathematical arguments ever given was put forward by the world’s smartest person, Marilyn Vos Savant. The columnist wrote an entire book criticizing Andrew Wiles’s proof of Fermat’s Last Theorem. Her argument: Wyles made use of non-Euclidean geometry, but what if geometry is really Euclidean? Touche!)

However … despite the fact that π doesn’t really change from place to place in general relativity, the geometry does change from place to place, and there is a particle associated with those dynamics — the graviton. Although the formulation of the original question isn’t accurate, the spirit is very much in the right place. And I, for one, will henceforth be perpetually sad that the physics community missed a chance by attaching the word pion to the lightest quark-antiquark bound state, rather than to the particle associated with deviations from Euclidean geometry. That would have been awesome.

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  • Bjoern

    Well, why was the lightest quark-antiquark bound state actually named “pion”? What’s the (hi)story behind that naming? Do you know anything about that?

  • Sean

    New particles are often named by Greek letters, and all of the good ones are taken.

    Nowadays we know that the pion is a pseudo-Goldstone boson of spontaneously broken chiral symmetry, and theoretical models with similar particles often also label them using π. But I believe the actual particle was discovered first.

  • nobody

    Hi Sean,
    Do you remember this April fools-day paper ?

    I think it proves beyond any doubt that Pi is time and space dependent. :)


  • Ned Wright

    > If the circumference/diameter ratio is less than one, you are in a positively curved space, such as a > sphere; if it is greater than one, you are in a negatively curved space, such as a saddle.

    Read “pi” for “one” above.

  • Sean

    Urg. At least somone reads the posts! Fixed.

  • Navneeth

    And while you’re fixing things, let me add that Wyles is actually Wiles.

  • Sean

    What is it with people actually reading the posts? Something in the water?

  • ian

    We’re working in new units where pi = 1.

  • Blake Stacey

    The more hard-core you are as a theorist, the more constants you set equal to unity. The natural conclusion is that we will understand quantum gravity when we accept that $latex hbar = G = c = k_B = pi = 1$.

  • boreds

    How can you forget the ion?

    “What about the imaginary unit i? Against all odds, does it varie and produce ions?”

    Maybe an ict formalism of GR can make sense of both ions and pions.

  • bob

    Of course you were being ironic when you referred to Marilyn vos Savant as “the world’s smartest person”. She may have the highest tested IQ on record – whatever that may mean – but she has written and said a number of things that aren’t smart at all, such as the book you cite.

  • Sili

    Oooooh! Oooooh! Can I play too? Logarithms will be some much more fun if we set e=1 (sorry, can’t do Latex – nor for that matter Maple – I’m only just learning to use Word’s equation editor).

    While we’re at it, could someone please pi(e) Vos Savant? And then point her to Katz.

  • Rory Kent

    This whole article is ludicrous, everybody, even Prof. Frink, knows pi is exactly three.

    “And I, for one, will henceforth be perpetually sad that the physics community missed a chance by attaching the word pion to the lightest quark-antiquark bound state, rather than to the particle associated with deviations from Euclidean geometry. That would have been awesome.”

    I am now dangerously depressed. How amazingly awesome would that have been!

  • Arjen

    CV may have lost a reader… but it was worth it.

  • Sean

    I hope we haven’t lost a reader — I honestly did like the comment, even if the answer is “no.”

    And I love the “ion” idea. That will be the field whose value represents the deviation from the real numbers in an ultimate complexification of all of physics. (E.g.

  • marc

    Come on! Those of us who Fourier transform often use units where 2 pi = 1. This is known as the “small circle approximation”. So if pi is spatially dependent, then either 2, 1 or = must also be.

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  • Alan Kellogg

    Given that mass causes space-time to curve, and that objects follow that curvature, are gravitons really necessary?

  • Matti Pitkanen

    Pi is troublesome number when one tries to generalize geometry to p-adic context. If one wants pi
    one must allow infinite-D (in algebraic sense) of p-adic numbers meaning that all powers of pi multiplied by p-adic numbers are allowed. As such this is not catastrophe but if one tolerates only algebraic extensions then only the phases exp(i2pi/n) make sense. Only phases but not
    angles. Something deep physically (distance measurement by interferometry)?

    In light-hearted mood one might ask whether gravitation could save from this trouble and allow to speak about circumference of circle also in p-adic context. By replacing plane with a cone (this requires cosmic string;-)), 2pi defined as ratio of length of circle to its radius becomes k*2pi and could therefore be also rational.

  • Al Feersum

    Some time ago, I was struggling with the non-Euclidian interpretation of pi, and wondered why pi was constant, even in curved space.

    So I asked a very clever person who I thought might know the answer. Prof. Ian Stewart, FRS, Emeritus Professor of Mathematics, Warwick University replied:

    pi is always the same. The definition in terms of a circle refers specifically to euclidean geometry. In fact pi is usually defined by analytic methods, for example as half the period of the sine function (which is in turn defined as a power series).

    In non-Euclidean geometry, the formula for the circumference of a circle differs from the euclidean one. In general the circumference is bigger than it would be for the same radius in Euclidean geometry, if we work in hyperbolic space (negative curvature). It is smaller if we work in elliptic geometry (positive curvature). And how much bigger or smaller depends on the radius.

    But this doesn’t change the mathematical definition of pi.

    Thanks Ian. I hope you don’t mind me reproducing your reply.

    By the way, his Cabinet of Mathematical Curiosities is worth taking a look at…

  • uhmmm

    A circle is defined in terms of pi and not the reverse. You can do calculations with perfect circles in various geometries, but in nature you can only observe approximations.

    You cannot construct or observe a perfect sphere (or spherical shell, or a planar slice through these) in nature where one or more of the following exists:

    a) non-negligible spacetime curvature;
    b) non-negligibly dynamical spacetime;
    c) non-negligibly dynamical matter embedded in spacetime;
    d) non-continuous matter;
    e) non-continuous spacetime.

    Furthermore the c^2 term in spacetime intervals and the Heisenberg uncertainty principle impose limits on what one can say about the shape of an object carefully set up to be an instantaneous spherical shell of *any* radius. This observational uncertainty dwarfs the uncertainty in our calculations of the number pi.

  • Cody

    The “constant-ness” of pi was first brought to my attention reading a physics critique of the movie and book Contact. In the book, the ETs tell Ellie to go back and look into the digits of pi, and after months of calculation she finds digits that have statistical regularity that they shouldn’t have. (The movie omitted that whole plot line.) In the review they pointed out that even “god” (were there such a thing) couldn’t change the value of pi, as it is abstractly defined and independent of the physical universe.

    But I have another question: I had a professor mention in passing that coordinate transformations in General relativity are not path independent. Could someone elaborate on that somewhat?

  • Brian Too

    OK, most of this discussion is going over my head.

    However isn’t the problem resolved if you accept that pi is a concept? It’s value as we experience it may be 3.14…, but in other, ah, curvatures?, it could have different specific values?

  • Emily

    All I want to comment is: Sean, I love that you don’t stomp on people for asking naive questions, but suss out their spirit and respond to them gently — and teach people a lot in the process.

  • Nullius in Verba

    In mathematics, choice of definitions is not an absolute, but based on convenience and mathematical aesthetics.

    The original definition of Pi was indeed the ratio of the circumference to the diameter, in an era when all geometry was assumed to be Euclidean. This ratio was subsequently calculated as a mathematical constant, and turned up in all sorts of other places that had nothing obvious to do with circles or even geometry.

    So it became convenient to switch the fundamental definition from circles to one of these mathematical alternatives. It gave the same answer, so what harm was done?

    But then mathematicians discovered that Euclidean geometry was not the only alternative, and that you could have curved spaces, and indeed spaces with different metrics (definitions of the distance between points). So all of a sudden, the definitions could conflict. But by now, the earlier value had wormed its way into so many different areas of mathematics that it would have caused chaos to try to extract it and call it something else, so mathematicians just made the switch of definition formal. Pi was now equal to the number, and it just happened to be the ratio of circumference to diameter in the special case of Euclidean geometry.

    But there’s nothing “wrong” about going back to the original definition, especially for the sake of engaging people’s interests in mathematics, so long as you’re clear that it’s not the same as the definition of Pi as a number.

  • John

    Haven’t we proved that the Universe is flat? I seem to recall a number of studies supporting that conclusion.


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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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