Today is “Pi Day”! This post is part of a series of posts on Pi, infinity, and other things mathematical. Click here for the series.
What is Pi (π)? We celebrate “Pi Day” on March 14 (3/14). That is because Pi is (roughly) 3.14. Indeed, I scheduled this post to appear at 1:59 AM, because Pi is (a little less roughly) 3.14159! How nerdy can a guy get?
So Pi is a mathematical idea that has its own day. That is good, because astronomy involves a lot of mathematical ideas. But still, what is Pi, and how do we know what it is?
This question started bugging me after I bought my previous laptop. For some reason the calculator “app” on this laptop was striking in a way that older calculator apps were not. Perhaps this is because the new app was large and bold. But at any rate, note how the button for π (that is, Pi) on the calculator app, when pressed, returned quite a long string of digits—3.1415926535897932384626433832795. That is thirty-two digits of Pi! Wow! (The app on my current laptop does the same.)
This bold string of digits raises the question: How do we know Pi to that many digits? Despite being a physicist, astronomer, and historian of science; despite using Pi all the time; despite teaching many students to use it regularly; I had only a vague idea how the value 3.1415926535897932384626433832795 might be obtained. So, I decided to figure this out, both because I should know, and because I thought a write-up on this would make a good hallway poster for the Science Building at Jefferson Community & Technical College in Louisville, Kentucky, where I am a professor. Math is a big stumbling block for JCTC students. At the college we are always looking for ways to advise students toward truly engaging the subject of mathematics, rather than viewing it as an onerous and pointless hoop (placed before them by the Authorities of Education) that they must simply jump through on the way to their degrees. But Pi is something everyone has heard about; it has its own Day, after all. This post then is based on that little JCTC poster project about what Pi is and how to determine its value for yourself.
Now you, O Reader, may view math no more positively than does the typical JCTC student. Maybe you do; you are reading an astronomy blog, after all. But plenty of people who like astronomy still care little for math, and think it to be beyond them. Indeed math is often portrayed as beyond the understanding of even the best-educated. But hang in there with this post, despite the counter-cultural implications implicit in reading stuff about math. Pi is cool. Mathematical reasoning is cool. And moving beyond just liking astronomy, and toward understanding how it works and how people over time have been able to figure out so much about the universe, is very cool.
So, if you wanted to figure out the value of Pi for yourself—if you wanted to make Homemade Pi, so to speak—you might just measure a round thing and calculate the value. Suppose you go find a nice sphere: a basketball. The ball’s Circumference C you could measure by wrapping a tape measure around the ball; its Diameter D you could measure by putting the ball between a couple of blocks. Since the Diameter D is twice the Radius R, the formula C=2πR that relates Circumference to Radius is also C=πD. You can calculate Pi by dividing the ball’s Circumference C by its Diameter D: π=C/D.
Left—a ball; middle—measuring the circumference of the ball; right—measuring the diameter of the ball.Of course, that basketball might not be truly round. So suppose you could borrow one of the 1 kg silicon spheres made by the Australian Centre for Precision Optics. If you measured one of those you would be using something that qualifies as “the roundest object in the world.”
An ACPO sphere: the roundest object in the world.You use your tape measure and other paraphernalia to measure the Circumference and Diameter of the ACPO sphere. The smallest division on a tape measure is usually 1/16 inch (or 1/16”). Let’s imagine that you can determine Circumference and Diameter to half of one of those divisions, or 1/32”. With the tape measure you might find the Diameter of the sphere to be D=3 & 22/32”, which is 3.6875” in decimal form. You measure the Circumference to be C=11 & 18/32”, or 11.5625”. And now you proceed to calculate Pi from your measurements:
π=C/D=11.5625/3.6875=3.135593…
Obviously this only gets you a few digits of Pi!
So, you upgrade the equipment, getting some precision measurement tools from a machine shop. These let you measure to the nearest thousandth of an inch. Using these, you might find D=3.685”, C=11.577”. Again you proceed to calculate Pi from your measurements:
π=C/D=11.577/3.685=3.1416553…
Still not great!
Finally, suppose you could actually measure the ACPO sphere by counting individual atoms. According to ACPO, the spacing of individual atoms in the sphere is 0.9 nanometers, or a few hundred-millionths of an inch. This is the highest precision possible in measuring this roundest object in the world, because the object is made up of individual atoms; you will not find a half or a tenth or a hundredth of an atom in the sphere. At the single-atom level of precision you might get D=3.68503937”, C=11.57689261”, and
π=C/D=11.57689261/3.68503937=3.141592652780…
So, even if you could count individual atoms in the roundest object in the world, you could not determine Pi to even half the thirty-two digits given by the calculator app. And any achievable measurement of a normal round object, even at machine-shop-level precision, will only give Pi to a handful of digits.
Yet Pi has been determined to far more than thirty-two digits. If you want to make that Homemade Pi, you will have to use pure reason, not measurement. Reason can take you further than experimentation and measurement can go—as much further as a trillion is beyond thirty-two.
Only recently have I learned how to reason out Pi, as the knowledge seemed potentially useful in helping students at my college to view mathematics more positively. And you, O Reader, may not view mathematics so positively yourself. Further down in this post you will find some serious high-school math—hexagons, triangles, and the Pythagorean Theorem, and a lot of repetition. The result obtained from bearing with the math and using reason to determine the value of Pi is pretty darned cool, and I encourage you to read this post to the end. But, for those who would rather not make Homemade Pi—who are perfectly happy if someone else does the baking—I will now cut to the bottom line, and save The Homemade Pi Recipe for the end (for those who want their Homemade Pi to be truly homemade, entirely from scratch)!
The bottom line is that we determine Pi to be 3.1415926535897932384626433832795 not through measurement but through reasoning, either as demonstrated in The Recipe below, or through one of the other methods mathematicians have devised for reasoning out the value of Pi. According to Scientific American and the University of Chicago Research Computing Center, Christoph Grienberger calculated Pi out to forty digits—3.141592653589793238462643383279502884197—by the method described in The Recipe, and these are the most digits to ever be worked out by hand. He published this value in his Elementa Trigonometrica of 1630. And, pertinent to this Vatican Observatory blog, Grienberger was a Jesuit astronomer at the Collegio Romano (the V.O. is home to many Jesuit astronomers), and he is also the astronomer who invented the equatorial telescope mount.
Left—Title Page of Grienberger’s 1630 Elementa Trigonometrica; center—the value of Pi; right—‘Laus Deo’ (‘Praise to God’) at the end of Grienberger’s tables, which was doubtlessly an expression from the heart after grinding out Pi to so many digits, plus doing all the other calculations in the Elementa.
Left—Grienberger’s Equatorial mount (used for solar observations) as illustrated by Christoph Scheiner in his Rosa Ursina Sive Sol of 1630; right—Scheiner’s crediting of Grienberger.Indeed human beings can reason Pi out to far more digits than even Fr. Grienberger did, especially if they do not have to do all the calculations by hand. The Pi Day web page gives one million digits of Pi. Moreover, it states that Pi has been determined to “over one trillion digits.”
A million digits of Pi is a lot. If printed out in a normal font, a million digits would fill a substantial book. A trillion digits is incredible. A trillion digits printed would fill a million books—as many as are in a university library. Yet that same mathematical reasoning that gives us a trillion digits of Pi also tells us that there is no end to the digits of Pi. Pi is what is known as an irrational number—its digits extend on infinitely. A trillion is as nothing compared to infinity.
Thus in the case of Pi we have reasoned our way to the infinite. We know it is there. We can learn more and more about it. For example, within the infinite digits of Pi must be contained any string of digits you care to think up. There is even a web page that will let you search for strings of digits within the first two billion digits of Pi. According to the web page, Grienberger’s birthday, 7-2-1561, occurs at the 182,220th decimal digit of Pi, while the speed of light in miles per second, 186,000, occurs at the 2,352,935th decimal digit of Pi (if you prefer that in kilometers per second, 300,000, look to the 215,286th decimal digit).
But, as much as we learn about Pi, we will never know even the smallest fraction of the infinity that is there. And Pi is just one number—the Circumference of a circle divided by the Diameter of that same circle. And you cannot go out and prove this, or the trillion digits, or the speed of light occurring at the 2,352,935th decimal, by measurement. Measurement gets you only a handful of digits of Pi.
Now consider astronomy. What can we prove by measurement about, for example, the Andromeda Galaxy? Can we roll out a tape measure and use it to check our estimations of that galaxy’s diameter? Can we use a balance to check our estimations of its mass? Do we have a calendar and baby pictures by which we can make sure we got its age right? In every case, we cannot. We know what we know about Andromeda through reasoning. Yes, our reasoning is based on measurements made from Earth, but we will never be able to measure the mass of the Andromeda Galaxy the way biologists measure the mass of lab rats fed on one diet versus another.
So take a look at The Recipe below. Take a gamble on trying to understand how you might determine the value of Pi for yourself, through reason, even if you don’t like math. This simple number says much about how we come to know what we know.
The Homemade Pi Recipe
(courtesy of Archimedes, tested by Fr. Grienberger)
First, start with the circle shown here. Its Radius R (purple) is 1. Its Diameter D (green) is 2.
Next, inscribe a hexagon in the circle, like this:
Now, divide that hexagon into equilateral triangles, like this:
Because these are equilateral triangles, each side of each triangle is equal to the Radius R of the circle, or 1. Therefore the perimeter of the hexagon (blue), which is composed of six triangle sides, is 6.
Now, the hexagon perimeter (blue) and the circle circumference (black) are approximately the same length, or at least in the same ball-park. So, the circle Circumference C is approximately 6. Pi is the ratio between the Circumference C and the Diameter D: π=C/D. Since C is approximately 6 and D is 2, then Pi is approximately
π=C/D=6/2=3.0000…
But that is a poor approximation! So now draw another radius R (purple, below) on the circle, perpendicularly dividing one of the sides of the hexagon (green) into two equal parts.
Next, draw lines (red, below) from the end of that radius to the vertices of the old hexagon.
In this way you can start to construct a twelve-sided figure, a dodecagon. The dodecagon is a closer approximation to the circle.
But what is the length of a side of the dodecagon? To reason this out, think about a triangle ABC that has two equal sides, AC and AB, each of length 1 (because AC and AB are also radii of the circle).
Then think about dividing side CB perpendicularly into two equal parts, eC and eB, using a new line AD (also of length 1, in purple below), and forming new triangles AeC and CeD. These are both Right Triangles. This means you can use the Pythagorean Theorem on them.
CB is a side of the hexagon, so its length is 1. The length of side eC is half that, or 0.5. The Pythagorean Theorem applied to triangle AeC says
AC2 = eA2 + eC2
12 = eA2 + (0.5)2
12 – (0.5)2 = eA2 = 0.75
eA2 = 0.75
eA = 0.86603
Now, the line AD is divided up into Ae and eD.
AD = eD + eA
AD – eA = eD
so
eD = 1 – 0.86603 = 0.13397
Again, back to the Pythagorean Theorem, this time applied to triangle CeD:
eC2 + eD2 = CD2
(0.5)2 + (0.13397)2 = CD2
0.26795 = CD2
0.51764 = CD
You have now reasoned out that the length of one side of the dodecagon is 0.517638. Therefore the perimeter of that 12-sided dodecagon is 12×0.517638=6.21166. Supposing once again this perimeter to be approximately the Circumference C of the circle, you can calculate that Pi is approximately
π=C/D=6.21166/2=3.10583…
This is a better value. But if a dodecagon with its 12 sides is better than a hexagon with 6 sides, it stands to reason that a 24-sided figure will be better still!
You repeat the Right Triangles and Pythagorean Theorem process. Again AC=1. But now eC is half of the length of the dodecagon side (half of .517368): eC=.258819. Following the process gives eA=0.965926 and eD=0.034074. The side length CD in the 24-sided figure is CD=0.26105238. This value multiplied by 24 gives an approximate Circumference of C=6.265257. Then
π=C/D=6.265257/2=3.13262…
This approximation is better still. Repeat the process a third time (for a 48-sided figure), a fourth time (96-sided figure), a fifth (192-sided), and so on. At the thirteenth repetition of the process you are doing calculations for a 49,152-sided figure and your value for Pi is
π=3.141592651450…
Note that by means of this process you have through reason already determined Pi to as many digits as might be possible through measurement, even if you could measure at the atomic level in the roundest object in the world (which you can’t)! And, whereas measuring at the atomic level would be the best you could do with measurement, you can keep going with reason, as shown in the table below, and as Grienberger did.
Through repeating this process, you can, with patience, make Homemade Pi to as many digits as you want. We know what Pi is through reasoning! And mathematicians have come up with many other, more manageable methods of calculating Pi than this, all based on pure reason.
This post is based on two posts that originally appeared here in April of 2016.
