Why Pi is Not Three

It’s fun to be whimsical, and one way to achieve that is to take a topic – however silly – and write about it, simply as an exercise to see where it leads and what angles can be illuminated on the journey. Now, various people have expressed the notion that mathematics would be a lot simpler if only pi were exactly three, and there are even stories in which the idea of pi being equal to three forms part of the plot (one modern example being Going Postal by Terry Pratchett). So it occurs to me that it might be fun to imagine a world in which pi really is equal to three, and to mentally explore some of the consequences.

A trivial way to render true the statement “pi equals three” is to invent a language in which the word for “three” is pronounced “pie“, but that is not only cheating, it also lacks subtlety. One way or another, cheating is a necessary part of the exercise, but we can at least be more subtle about it. Pi is the ratio of a circle’s diameter to its circumference, and if we want to imagine a world in which pi equals three we should at least preserve that definition.

A shape for which the perimeter really is exactly three times the “diameter” (defined here as the distance from corner to diametrically opposite corner) is of course the regular hexagon. A good way to show this is to divide the hexagon into six equilateral triangles. If we agree not to mess with the geometry of straight lines, then a universe in which pi equals three must be a universe in which the circumference of a circle is equal to the perimeter of the largest hexagon that can be inscribed within it. How can we make this be?

inscribed-hexagon

Straight away another trivial solution presents itself, which is to redefine “circumference” to mean the perimeter of the largest inscribable hexagon. That makes them equal all right, but is only slightly more subtle than our first attempt, and not very satisfying. It fails to respect the idea that the circumference is the length along a curve, and that the length along half a circle should be half the length around a whole one. Whatever result you get by trying to inscribe a hexagon inside half a circle (not a well-defined problem to begin with), it certainly isn’t that.

Let’s take the image of the hexagon inside the circle, and magnify the bit between one corner and the next, as shown below. If pi equals three, and we’re to insist on a more subtle means of achieving this than we’ve tried so far, then the length of the upper curve must be the equal to the length of the line segment underneath it. Our third and final attempt will involve redefining “length” in such a way that this can be true.

sixth-circumference

Here’s the idea. We’ll redefine the “length” of a path as something along the lines of the energy it takes to draw it, and we’ll put this in the context of an imaginary physics where you get bonus energy for moving in curves. This is now getting much more subtle, but you may already be able to see where it’s going to get unstuck.

We need a conversion factor for translating between “length” as we know it and “length” as redefined in our imaginary world. Given that our bonus energy is acquired by moving in curves, this conversion factor ought to be a function of curvature. Curvature, as defined by mathematicians, is the reciprocal of the radius of the circle that fits most snugly into the curve.

For a circle of radius one (or in other words a curvature of one), the conversion factor has to be 3/pi. This is because in the real world the circle has circumference two pi, whereas in the imaginary world it has circumference six, so 3/pi is the factor we need to make one equal to the other. But for a circle of radius two (curvature one half), the conversion factor is once again 3/pi. We’re forced to the conclusion that the conversion factor is 3/pi for any curve, with the exception of a perfectly straight line (curvature zero), in which case the conversion factor is plain old one.

To convince yourself of this result, study the diagram below and think about both the green circle and the blue one. Incidentally, if the appearance of pi in the conversion factor bothers you, feel free to call it pi subscript mundane. It’s the value pi has in the real world, not in the imaginary one.

circle-sizes

A problem with our third redefinition is that it fails to respect the principle of continuity, that a very small change to a shape should make only a very small change to the length of a path along it. Consider, on one hand, a straight line segment, and on the other hand, a segment of the circumference of a really, really large circle, so large as to be indistinguishable from a straight line (e.g. a line drawn in the sand on the spherical Earth). We are forced to conclude that the curved line is 3/pi times the length of the straight line, no matter how impossibly close to being straight the curved line happens to be. This will not do.

Worse still, in such a universe we could define something I’ll call the pseudiameter, which is the length of an impossibly close to straight but not actually straight curve stretching from one side of a circle to the other. We could then ask about the ratio between the pseudiameter and the circumference of the circle, and … Oh! Look who’s back!

At this point I’m inclined to give up. Whatever you do to redefine pi so that it can be equal to three, one way or another the definition is going to fall to pieces. It isn’t possible to keep a good irrational number down.

Of course, we knew that all along.

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22 Responses to “Why Pi is Not Three”

  1. Greg Laden Says:

    PI could be three if three was the only number.

  2. Flesh-eating Dragon Says:

    In that case, how many numbers would there be? Three, obviously. I can think of three very compelling reasons why that couldn’t possibly work.

    Edit: A statement S is true if there exist three proofs of S. A false mathematical statement has zero proofs. Zero equals three. Therefore all mathematical statements are true, including false ones.

  3. Richard Smith Says:

    PI could be three if the number system was base X(PI/3), where X>3. I’m telling you, that would be one irrational number system.

  4. Flesh-eating Dragon Says:

    I don’t believe that works, because the first digit before the decimal point is always a multiple of one, regardless of the base. With a base of pi, you would make pi an integer, but it would be the fourth positive integer and written 10.

    In the first paragraph I said “various people have expressed the notion that mathematics would be a lot simpler“, and I think you’ve just shattered their dreams into an uncountable number of pieces. Which could be any number of pieces.

  5. Richard Smith Says:

    But that’s why I specified “where X>3.” If X=4, then 1=PI/3, 2=2(PI/3), 3=3(PI/3)=PI, and 10=4(PI/3). Dare I say, easy as..?

    Regarding shattering various people’s dreams, well, I guess I’ve done my job.

  6. Flesh-eating Dragon Says:

    No, it doesn’t work. Changing the base does not change the interpretation of single-digit numbers. The least significant digit in an integer is multiplied by the base to the power of zero, which always equals one, no matter what.

    If X=4 (i.e. the base is 4/3 pi, which is about 4.19), then “1” designates 1, “2” designates 2, “3” designates 3, “4” designates 4, “10” designates 4/3 pi, “11” designates 4/3 pi + 1, and so on.

  7. John Cowan Says:

    [An] excerpt from an Interview with Malaclypse the Younger by THE GREATER METROPOLITAN YORBA LINDA HERALD-NEWS-SUN-TRIBUNE-JOURNAL-DISPATCH-POST AND SAN FRANCISCO DISCORDIAN SOCIETY CABAL BULLETIN AND INTERGALACTIC REPORT & POPE POOP:

    M2: Everything is true.

    GP: Even false things?

    M2: Even false things are true.

    GP: But how can that be?

    M2: I don’t know, man; I didn’t do it.

    Principia Discordia

  8. Flesh-eating Dragon Says:

    A friend from Israel (who reads this blog) once sent me a batch of printed “bearer is a Pope of Discordia” cards, which I dutifully distributed at the next social function I attended.

  9. Joe Says:

    Consider a circle inscribed on the surface of a cone, parallel to its base. There is a certain ratio of cone height to base diameter (about 0.155, I think) where the circle’s circumference in this space is exactly 3 times its diameter.

    Unfortunately: 1) This space has a point with infinite curvature, which seems to defy nature; 2) Only circles centered on the discontinuity show pi = 3.

  10. Flesh-eating Dragon Says:

    Yeah, with curved spaces you can set things up so that the ratio of a circle’s diameter to its circumference is a function of the position and size of the circle, but then the idea of a constant equal to that ratio is no longer applicable.

  11. Procyan Says:

    Pi does equal 3 to one significant figure. And that werks just fine for all my cypherin, danke tres mucho.

  12. Flesh-eating Dragon Says:

    That’s as may be, but you might feel differently if you travelled around the world and found yourself nearly two thousand kilometres from home and out of fuel … :-)

  13. Rasmus R Says:

    all this and nobody mentioned that state made it LAW at one point that Pi=3?

    http://en.wikipedia.org/wiki/Indiana_Pi_Bill

    http://dumblaws.com

  14. Flesh-eating Dragon Says:

    As is pointed out in the links you provide, the relevant bill never became law, and didn’t set pi equal to three in any case. It was, however, extremely silly.

  15. DDeden Says:

    Flesh eating dragon eats 3 pieces of meat pie, measures belly, finds pi!

  16. Flesh-eating Dragon Says:

    At the time of writing I have seven meat pies in the freezer. Two curry, two hungarian goulash, two gourmet chicken, and one chunky beef & mushroom…

  17. Paul-in-Yukon Says:

    I’ve imagined a universe where pi is equal to three.
    How I did it is I pictured a circle that had the circumference of exactly three, then I took a line that was the length of exactly one, attached the line to opposite sides of the circle, and that made a “floppy” line.
    I then placed the circle on an inflatable sphere, and began inflating it until the line was taut. The line was “bent” eg. it wasn’t straight up and down, but was straight side to side. This universe was “bent” by as much as the line was bent. Now that really can’t happen, because once the straight line was bent, the circular line would be bent by the same amount, changing the size of it, but for imagining it, that was the closest I came to picturing it.
    No, I’m not any sort of a mathematician, but like thinking of odd things occasionally. ;)

    Thanks for giving me a forum to post my thoughts.

    P.

  18. Flesh-eating Dragon Says:

    I would quibble with your description in a few places, for example you make it sound as though the line gets less floppy as the sphere is inflated, when actually it would be the other way around. (After all, an infinitely large sphere is flat, whereas a sphere small enough for the circle to be its equator would require the line to be stretched.) But I think I see what you mean. The radius of the sphere, I think, should be twice the radius of the circle.

    I don’t follow the bit about the circular line being bent by the same amount, because if I understand you correctly it wouldn’t be. The real problem is that if you get a bigger circle you have to get a bigger sphere, but that would mean replacing the universe! To put it another way, you have to ignore the fact that circles come in different sizes, which sort of defeats the point of a universal constant.

  19. honey bunn Says:

    wow, you don’t have to go all geeky to prove pi can’t be three, i’m not the smartest person in the world, but pi can’t be three because otherwise a circle would look like you cut a slice out of it.

  20. Flesh-eating Dragon Says:

    Given the time of year, I suspect a touch of deadpan humour in honey bunn’s comment above. That’s cool. Anyway, you might not have to be a geek to convince yourself that pi can’t be three, but you probably do have to be one to enjoy doing so (and to see the point)…

  21. rn ippoldt Says:

    If pi were 3 a sphere would change from 3 dimensions to a fractional; perhaps it would become a 2.9 dimensional object. Any one particular point on the surface on in its interior might exist or might not. Most of the time it would exist, but occasionally it would not.

  22. Flesh-eating Dragon Says:

    I’m not convinced. First, I doubt you can argue unambiguously that this 2.9 dimensional shape (as opposed to some other 2.9 dimensional shape) is a 2.9 dimensional sphere. Second, shapes with fractional dimensions tend to have infinite surfaces, which would make pi infinite, no?


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