Three-point-one cheers for pi !

Pi, that most celebrated of mathematical constants, leads a curiously double life.  On the one hand, we have numerical formulas for pi, like Leibniz’s formula π = 4 × (1/1 − 1/3 + 1/5 − 1/7 + …); imagining a world in which this expression converges to a value other than 3.14… is as hard as imagining a world in which 2+2 doesn’t equal 4. On the other hand, we have a geometric definition of pi as the ratio of the circumference of a circle to that circle’s diameter, and this definition of pi lets us imagine that pi is a physical constant like the speed of light — that it could have a different value in an alternative universe that’s built using a different kind of geometry. Could there be worlds in which geometrical pi equals 3.24…, say, and in which the more open-minded scientists and mathematicians speculate about other worlds in which pi has some crazy value like 3.14…?

EARTHLY CIRCLES

Before we travel to imaginary universes, let’s play a little while in our own universe to get a sense of what’s at stake.  The geometric definition of pi presupposes that if you draw one circle and I draw another, with a different center and a different radius, and each of us measures the ratio of the circumference to the diameter of the circle we’ve drawn, we’ll get the same number.  That certainly matches up with our experiences drawing circles on pieces of paper.  But what about really big circles drawn on the Earth’s surface?  (“Geometry” literally means “measuring the Earth”.)  Imagine drawing progressively larger circles on the Earth’s surface, centered at the North Pole.  (To make it easier to draw our circles, we’ll remove the planet’s inconvenient oceans, and to make the arithmetic come out easier, we’ll shave the planet down a bit so that it’s a perfect sphere with an equator exactly 24,000 miles long.) For each r, we’ll look at the set of points on Earth’s surface that are at distance r from the North Pole as the crow flies, and call it an Earth-circle of radius r.  For instance, an Earth-circle of radius 1,600 miles centered at the North Pole is the Arctic Circle, an Earth-circle of radius 6,000 miles centered at the North Pole is the Equator, while an Earth-circle of radius 12,000 miles centered at the North Pole consists of just a single point (the South Pole).  If we define the π(r) to be the circumference of an Earth-circle of radius r, divided by 2r, then π(r) is about 3.14 when r is small (because the vicinity of the North Pole on Earth’s surface is flat), π(6,000) is exactly (24,000)/(2×6000) = 2, and π(12,000) is exactly (0)/(2×12,000) = 0.  So as r grows, π shrinks!

You’d be right to object that we’ve “changed the value of pi” by changing the meaning of pi, and in particular, by changing the way we measure the radius of a circle.  And if this were still the 19th century, that objection would be the end of the conversation.  But thanks to Einstein and the experimentalists who’ve followed in his wake, we now are pretty sure that the universe we live in is curved at intergalactic scales, and there’s no natural way to view it as a 3-dimensional curved hypersurface sitting inside an uncurved 4-dimensional space; our 3-dimensional space is best viewed as part of 4-dimensional space-time, and space-time is definitely not flat (in fact, in general relativity, curvature of space-time is what gravity is all about).  Euclidean geometry simply fails to be a good description of the universe when we try to describe large patches of it.  Indeed, modern mathematics gives us an overabundance of non-Euclidean spaces.  Some, like the surface of a sphere, give us a π(r) that decreases as r gets large; others give us a π(r) that increases as r gets large (think of a kale leaf).  In positively curved spaces, π(r) decreases as r grows; in negatively curved spaces, π(r) increases as r grows.

Kale: a negatively-curved vegetable.

CURVED SPACES

We can imagine a negatively curved universe in which the resident life-forms are big enough to have direct kinesthetic experience of their world’s curvature.  The geometers of that universe would never ask “Do all circles have the same pi?” (that is, the same circumference-to-diameter ratio) because they obviously don’t.  That world would have circles, but the ratio between the circumferences of a circle and its diameter wouldn’t exhibit constancy.

But here’s the interesting part of this fable: in that universe, scientists interested in probing the microscopic world would discover the curious fact that, as the radius of a circle gets smaller and smaller, the ratio between the circumference and the diameter approaches a curious value slightly larger than 3.  Mathematical physicists in that universe would realize that, on a fine scale, the universe doesn’t look like kale at all.  As an idealization of the world of the infinitely small, kale-geometers would invent Eulidean geometry, and re-invent the number pi we know and love — no longer as a property of everyday circles, but as an asymptotic property of circles that are very, very small.

We can also imagine beings in a positively curved universe (though those universes tend to be finite in scope and hence not so hospitable to the development of intelligent life; organisms big enough to perceive the curvature of the universe would be nearly as big as the universe itself).  To the extent that those beings have the ability to probe the structure of their universe, they too would realize that at a microscale, points and lines and circles come very close to satisfying the axioms of Euclidean geometry, and like their counterparts in negatively curved space, they would re-invent our pi and find the same numerical formulas for it.

L^p SPACES

Maybe you were hoping I’d show you a universe that resembles our own in that pi (defined geometrically) has the same value for all circles, but differs from our universe in that the value is different from 3.14…?  If so, you’ll like normed spaces, and more specifically, the ones called L^p spaces.  (L^p is pronounced “Ell-pee”, by the way, not “Ell to the pee”.) We’ll stick to 2-dimensional L^p spaces.  These spaces look a lot like the ordinary Euclidean plane, and we put a coordinate system on them the same way, but we measure distance differently: the distance between (x₁,y₁) and (x₂,y₂) is no longer the square root of (x₁−x₂)²+(y₁−y₂)² but rather, by fiat, the pth root of |x₁−x₂|^p+|y₁−y₂|^p.  The case p=2 (“Ell-two distance”) is just ordinary Euclidean distance, so our standard kind of geometry is now seen as part of a spectrum of geometries, ranging from L¹ to L^∞ .  At one end we have L¹ distance, also known as “taxicab distance” or “Manhattan distance”. At the other end, even though infinity is not a number, mathematicians define the L^∞ distance between (x₁,y₁) and (x₂,y₂) as the limit of the L^p distance between those two points as p goes to infinity; this turns out to be just max(|x₁−x₂|,|y₁−y₂|).  Kelsey Houston-Edwards has made a great video “When Pi is not 3.14” about L^p spaces and the “values of pi” that they give rise to.  In each L^p space, the ratio of the circumference of a circle to its diameter is a fixed number (independent of the radius); that number equals our familiar 3.14… when p=2 but takes on other values when p≠2 .  For instance, when p=1, “pi” is 4; and when p=∞, “pi” is also 4. A nice gif created by Kelly Delp shows how the L^p “circle” changes as p changes:

You might think that mathematicians living in universes governed by L^p geometry with p ≠ 2 would never be interested in “our” pi.  But you’d be wrong.  That’s because there turns out to be something special about p=2: it’s the p-value that makes “pi” as small as possible.  If you imagine turning the p-knob from 1 to infinity, you find that “pi” decreases from 4 to 3.14… as p goes from 1 up to 2, and then as p continues on past 2, “pi” increases from 3.14… back up toward the limiting value 4.  See the figure below, taken from the stackexchange page “Measuring π with alternate distance metrics (p-norm)”. (The figure has one extra dot that shows “pi” for a value of p less than 1, which won’t concern us here.) If mathematicians in some particular L^p space came up with the idea of L^p spaces in general, they’d be inclined to ask “What are all the possible values of the circumference-to-diameter ratio in such spaces?”, and they’d find that the range of values is [3.14…,4], where that 3.14… is our pi.

How “pi” changes with p.

The moral is, it’s not hard to imagine universes built on a different geometry, but in many of those universes, conscious and curious beings would still rediscover the familiar mathematical constant 4 × (1/1 − 1/3 + 1/5 − 1/7 + …).  They might ascribe a different significance to it, and they’d certainly call it by a different name, but its value would be exactly the same as ours, down to the last decimal digit, except of course that pi doesn’t have a last digit (whereby hangs a tale I’ll tell some other time).

Scholarly punctiliousness compels me to say that even though some people (mostly amateur mathematicians and mathematical popularizers) talk about “other values of pi” (and I’ve engaged in that practice here), in mathematics, pi is 4 × (1/1 − 1/3 + 1/5 − 1/7 + …) by definition. It’s fun to throw off the shackles of convention and declare a mathematical Feast of Fools, but at the end of the day the lords of misrule must surrender their scepters and 3.14… must be re-enthroned as Pi Eternal.

In a related vein, I’ll mention that while there’s a transgressive pleasure to be had from bending or breaking mathematical rules (changing the value of “pi”, or making multiplication non-commutative, or whatever), mathematical breakthroughs seldom arise from the desire to transgress per se; transgression is typically a side-effect of trying to apply existing ideas to new domains, and then seeing what needs to be altered to make things fit together.

Here’s a challenging puzzle for my more assiduous readers: Consider the plane {(x,y,z): x+y+z=0}, with the distance function |x₁−x₂| + |y₁−y₂| + |z₁−z₂|. What is a circle for this geometry? What is “pi” for this geometry? (See Endnote #1.)

NOT JUST CIRCLES

If pi were just about measuring diameters and circumferences, it might be merely an interesting tidbit of mathematical arcana, like Euler’s constant.  What makes pi the superstar of the real number system is the way it pops up all over the place in mathematics, in areas far afield from geometry.

You like number theory? If you look at all the ways you can choose a pair of integers i and j between 1 and N, the proportion of pairs such that i and j have no common factor larger than 1 gets ever-closer to 6 / π² as you make the cut-off N ever-larger.

You like probability? If you toss a fair coin 2N times, the probability of getting equal numbers of heads and tails get closer and closer to as N gets larger and larger.

You like statistics? A central concept in statistics statistics is the normal distribution, given by a formula that contains a pi lurking under a square root sign. Many quantities of real-world interest are governed, at least approximately, by the normal distribution. Eugene Wigner in his famous essay “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” tells a related anecdote that deserves to be true whether or not it really is:

There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate.  The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on.  His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg.  “How can you know that?” was his query.  “And what is this symbol here?” “Oh,” said the statistician, “this is pi.” “What is that?” “The ratio of the circumference of the circle to its diameter.” “Well, now you are pushing your joke too far,” said the classmate, “surely the population has nothing to do with the circumference of the circle.”

I won’t try to explain what the statistics of populations has to do with circles, but I can show you why the pi in the formula for the circumference of a circle is the same constant as the pi in the formula for the area of a disk, and in the formula for the volume of a ball, and for the surface area of a sphere. (In accordance with mathematical convention, I define a circle as the set of points in the plane AT some fixed distance from a center point, a disk as the set of points in the plane UP TO some fixed distance from a center point, a sphere as the set of points AT some fixed distance from a center point, and a ball as the set of points UP TO some fixed distance from a center point.)

Before I explain why all four of these π‘s are the same, or at least give most of the explanation, let me point out that we could imagine things being otherwise.  It’s not surprising that there’s some formula for the area of a disk of radius r of the form A(r) = α r²; after all, we expect the area of a two-dimensional object of linear size r to grow like the square of r.  It’s also not surprising that there’s some formula for the volume of a ball of radius r of the form V(r) = β r³.  But disks aren’t balls, so why should the constants α and β be related in any way?

ONE PI TO RULE THEM ALL

To see why the alpha in A(r) = α r² is none other than the pi that occurs in the formula C(r) = 2πr for the circumference of a circle of radius r, see the nice pictures at Steven Strogatz’s Opinionator essay “Take It To The Limit”.  (But don’t look at the pictures on an empty stomach; they’re likely to make you think of reheating some pizza in a toaster oven!) Henri Picciotto has created an interactive animated version of this way of thinking about the area of a circle.

A slightly different picture involves cutting a disk of radius r into N equal wedges and approximating the area of the disk by adding up the areas of the N triangles obtained by turning those wedges into triangles (without arranging them as artfully as Strogatz does).  Each triangle has altitude close to r and base close to C(r)/N, and hence has area close to (1/2)(r)(C(r)/N); add up N of them, and you get (1/2)(r)(C(r)).  So A(r) = (1/2)(r)(C(r)). In this way we can derive A(r) = πr² as a consequence of C(r) = 2πr. Easy peasy!

(You were expecting me to say “easy as pi”, weren’t you?)

Here’s another way to see the relationship between A(r) and C(r) (which those of you who know calculus may recognize as being related to the fact that C(r) is the derivative of A(r)). Imagine a pizza in which the cheese-and-sauce-covered part of the pizza has radius r, and the bare baked dough goes out a distance d further, so that the pizza as a whole has radius r+d. What’s the area of the outer crust? Here are two approximate answers:

(1) If the pizza has been cut into N equal slices, then each slice has an crusty part that’s approximately a d by C(r)/N rectangle and so has area about d times C(r)/N; since there are N of them, the total area of the outer crust is about d times C(r).

(2) The area of the outer crust equals the area of the whole pizza (A(r+d)) minus the area of the cheesy-saucy part (A(r)), which gives π(r+d)² − πr² = π ((r+d)² − r²) = π (r² + 2rd + d² − r²) = π(2rd + d²) = π d (2r + d). If we assume d is a lot smaller than r, then 2r + d is pretty close to 2r, so the area π d (2r + d) is close to π d (2r), which is about d times 2 π r.

Neither of these two approximations is perfect, but as d gets smaller and smaller relative to r, it can be shown that the relative error of each approximation gets smaller and smaller, so taking the limit as d goes to zero, we get C(r) = 2πr. That is, we can derived C(r) = 2πr as a consequence of A(r) = πr²: it’s the “same pi” in both formulas.

Similar pictures let us see a relationship between the formulas for the surface area of a sphere S(r) and for the volume of a ball V(r); see Endnotes #2 and #3.

But how do we relate the “two-dimensional pi” that governs C(r) and A(r) to the “three-dimensional pi” that governs S(r) and V(r)?  A bridge is provided by the volumes of cylinders and cones.  I won’t give the details here, but the basic idea goes back to Archimedes; I found one explanation on the Web written by Walter Whiteley and Stephen La Rocque, and you can see how math-dad Mike Lawler treated the topic with his kids. I suspect my readers will point out others as well.

C(r), A(r), S(r), and V(r) are far from the end of the story of pi’s relevance to geometry.  Spheres and balls are special cases of surfaces and solids of revolution, often the subject of problems in second semester calculus; if you’re asked to solve such a problem, and your answer doesn’t have a pi in it, you’ve probably done it wrong.

Pi also occurs in higher-dimensional geometry; see Endnote #4.

PI DAY

One reason I’m bringing this up now is that, in four weeks, a whole lot of people are going to be showing off their prowess in reciting the decimal digits of pi, in celebration of Pi Day.  Pi is about 3.14, and 3/14 is March 14; get it?

Another reason I’m bringing up the topic of the digits of pi is that late last year, software engineer Peter Trüb of Switzerland computed over 22 trillion digits of pi, setting a new world record. If some Pi Day enthusiast were to try to recite these digits at the rate of one digit per second, more than half a million years would elapse before the recitation was completed.  Not all mathematicians are impressed by Trüb’s feat; one mathematician I know called the effort “a waste of chips and neurons”.  This particular mathematician has no grudge against Trüb and is in fact a big fan of the sorts of formulas that are used for computing digits of pi.  But the digits themselves don’t impress him.  After all, there’s no obvious pattern to those digits, as Trüb himself painstakingly checked for the first 22 trillion of them; if those digits are fodder for interesting mathematics, it is a mathematics of the future that we cannot even catch a glimpse of now.

My own command of pi’s decimal expansion is limited to the first fifteen digits, as encoded by the handy mnemonic “How I need a drink, alcoholic of course, after the heavy chapters involving quantum mechanics” (count the letters in each word). An error of one part in one quadrillion would lead me to misjudge the length of a circle the size of Earth’s orbit by a fraction of an inch; I can live with that.

The most widely used decimal approximations of pi are 3.14 and 3.14159.  Some other time I’ll write about other approximations, including 355/113, 22/7, 3, and even 1.  A few Europeans have suggested that a more appropriate day to celebrate pi would be 22 July, aka 22/7 (after all, 22/7 is a better approximation to pi than 3.14). But I have a different proposal: Given that 3.1 is precise enough for most purposes, how about we recognize March 1 as “Close Enough Pi Day”?

Thanks to Kelly Delp, Bill Gosper, Sandi Gubin, Kelsey Houston-Edwards, Mike Lawler, Henri Picciotto, Eugene Salamin, Rich Schwartz, and Glen Whitney.

Next month: Band saw blades, bedbug zappers, rubber bands and me.

ENDNOTES

#1: A digression about cones: A not-necessarily-circular cone is specified by a region R in a plane (the base) and a point P sitting above the plane (the apex); you form a solid cone by taking every point that lies on a line segment that joins P to a point in R. I won’t convince you that you can determine the volume of the cone just by knowing the area A of the base and the height h of the apex, but I can convince you that if there is a formula of the form c times h times A, the constant c has to be 1/3. The way to see it is to divide a 1-by-1-by-1 cube into three triangular “cones”. Let P be one corner of the cube, and let R be a square face of the cube that doesn’t have P as one of its corners (there are three such faces). By letting the cube rest on face R, you can see that height of P over R is h = 1, and of course the area of R is 1×1 = 1. So if V = c×h×A, we get V = c×1×1 = c. But there’s another way to compute the volume of this cone, namely by noticing that the cube is made of three of them, one for each of the three square faces of the cube surrounding the corner of the cube opposite P! So V + V + V = 1×1×1 = 1, implying V = 1/3. Combining this with V = c, we get c = 1/3.

#2: Now let’s look at a couple of ways to relate V(r) (the volume of a ball of radius r) and S(r) (the surface area of a sphere of radius r).

Imagine dividing up the surface of that waterless perfectly round Earth into small countries, and imagine that each country claims everything that lies underneath it all the way to the planet’s center.  This divides the planet up into three-dimensional wedges, and if the countries are small, each wedge is approximately a pyramid with height close to r, the radius of the planet.  Each wedge therefore has volume approximately equal to 1/3 times r times its base-area (see Endnote #1). So the sum of the volumes of the wedges is about 1/3 times r times the sum of the base-areas of the wedges.  But the base-areas of the pyramids add up approximately to the surface area S(r) of the sphere, so the total volume of the wedges should be about 1/3 times r times S(r).  When we take this approximation to the limit (as the countries get smaller and smaller), we get V(r) = 1/3 times r times S(r).  (Check it out: (4/3)πr³ = (1/3) r 4πr².)

And here’s another way to see the relationship between V(r) and S(r) (related to the fact that S(r) is the derivative of V(r)). Suppose each country claims airspace up to a distance of d. Then the total volume of the world’s airspace would be about S(r) times d (since each country’s airspace would be an approximate prism of base S(r) and height d). On the other hand, the world’s total airspace would equal V(r+d) − V(r) =  (4/3) π ((r+d)³ − r³) = (4/3) π (r³ + 3r²d + 3rd² + d³ − r³) = (4/3) π (3r²d + 3rd² + d³). Since r is much bigger than d, the quantity 3r²d (the first term in that parenthesized sum) dwarfs both 3rd² and d³ (the other two terms), so we get V(r+d) − V(r) ≈ (4/3) π (3r²d) = 4πr²d. So now we have two approximations to the total volume of airspace: S(r) × d and 4πr² × d. The smaller we make d, the smaller the relative error of both approximations becomes. So we derive S(r) = 4πr² as a consequence of V(r) = (4/3) πr³: it’s the “same pi” in both formulas.

#3: In this geometry, the points at distance r from (0,0,0) form a hexagon with corners at (s,−s,0), (s,0,−s), (0,s,−s), (−s,s,0), (−s,0,s), and (0,−s,s), where s = r/2; the perimeter of this hexagon (using the |x₁−x₂| + |y₁−y₂| + |z₁−z₂| distance function) is 6r, so the circumference-to-diameter ratio of this metric space is 3. The phrase “circumference-to-diameter ratio” is unwieldy; perhaps we should call it the piety of a metric space?

If any of you work out the piety of the plane {(x,y,z): x+y+z=0} with the distance function (|x₁−x₂|^p + |y₁−y₂|^p + |z₁−z₂|^p)^1/p, for p between 1 and infinity, please let me know and I’ll report your findings here!

#4: Rich Schwartz reminds me that pi occur in formulas when we measure the hypervolume of higher-dimensional analogues of spheres and balls:

“In computing the volumes of (unit) high dimensional spheres, my favorite mantra is The surface area of the boundary of the ball  is 2 pi times the codimension 2 diameter of the ball. For instance, the surface area of S³ is 2π times the area of the 2-disk, π, which comes to 2π².  The volume of B⁴ is then 2π²/4 = π²/2.  And so on.  Using the mantra, and the familiar integration in polar coordinates, you can inductively work out the surface area of any unit sphere and the volume of the ball it bounds.”

(Here S³ is the hypersphere {(w,x,y,z): w²+x²+y²+z²=1} and B⁴ is the 4-dimensional ball {(w,x,y,z): w²+x²+y²+z² ≤ 1} that S³ bounds.)

The upshot is, if there are higher-dimensional universes “out there” that look like multi-dimensional Euclidean geometry up close, then beings in those universes would still encounter 3.14…. Pi is truly a trans-universal constant!

#5: Twenty or thirty years ago I chanced upon a video purportedly created by an organization called the Institute for Pi Research, championing the exploration of “alternative” values of pi. It was a curious video, seemingly intended partly as a deadpan hoax and partly as a parody of creationism. Has anyone else come across it? I can’t say that it was as funny as it could have been, but I can say that I’ve never seen anything quite like it. Postscript: A few years after I wrote this essay (and the preceding Endnote), I felt inspired to create my own parody in the spirit of the Instititute for Pi Research. It’s called “The Truth About Pi”.

REFERENCES

Petr Beckmann, “A History of Pi”.

Martin Gardner, “The Transcendental Number Pi”, chapter 8 from “New Revised Edition of Mathematical Diversions”; based on the article “Incidental information about the extraordinary number pi” originally published in the July 1960 issue of Scientific American.

Kelsey Houston-Edwards, “When Pi is not 3.14”.

Evelyn Lamb, “Pi Approximation Day Celebrated July 22, How Much Pi Do You Need?”, July 22, 2012.

Mike Lawler, “Sharing Kelsey Houston-Edwards’s Pi-video with kids”, January 8, 2017.

Steven Strogatz, “Take It To the Limit”, from The New York Times Online, April 4, 2010.

Steven Strogatz, “Why Pi Matters”, from The New Yorker, March 13, 2015.