Happy Birthday, Albert Einstein!

What a year. Einstein may have famously called his own birthday a natural disaster, but between the discovery of gravitational waves in February and the 100th anniversary of the general theory of relativity this past November, it’s been a big year for the renowned physicist and former Princeton resident. Throughout the day, PUP’s design blog will be celebrating with featured posts on our Einstein books and the stories behind them.

HappyBirthdayEinstein Graphic 3

Here are some of our favorite Einstein blog posts from the past year:

Was Einstein the First to Discover General Relativity? by Daniel Kennefick

Under the Spell of Relativity by Katherine Freese

Einstein: A Missionary of Science by Jürgen Renn

Me, Myself and Einstein by Jimena Canales

The Revelation of Relativity by Hanoch Gutfreund

A Mere Philosopher by Eoghan Barry

The Final Days of Albert Einstein by Debra Liese

 

Praeteritio and the quiet importance of Pi

pidayby James D. Stein

Somewhere along my somewhat convoluted educational journey I encountered Latin rhetorical devices. At least one has become part of common usage–oxymoron, the apparent paradox created by juxtaposed words which seem to contradict each other; a classic example being ‘awfully good’. For some reason, one of the devices that has stuck with me over the years is praeteritio, in which emphasis is placed on a topic by saying that one is omitting it. For instance, you could say that when one forgets about 9/11, the Iraq War, Hurricane Katrina, and the Meltdown, George W. Bush’s presidency was smooth sailing.

I’ve always wanted to invent a word, like John Allen Paulos did with ‘innumeracy’, and πraeteritio is my leading candidate–it’s the fact that we call attention to the overwhelming importance of the number π by deliberately excluding it from the conversation. We do that in one of the most important formulas encountered by intermediate algebra and trigonometry students; s = rθ, the formula for the arc length s subtended by a central angle θ in a circle of radius r.

You don’t see π in this formula because π is so important, so natural, that mathematicians use radians as a measure of angle, and π is naturally incorporated into radian measure. Most angle measurement that we see in the real world is described in terms of degrees. A full circle is 360 degrees, a straight angle 180 degrees, a right angle 90 degrees, and so on. But the circumference of a circle of radius 1 is 2π, and so it occurred to Roger Cotes (who is he? I’d never heard of him) that using an angular measure in which there were 2π angle units in a full circle would eliminate the need for a ‘fudge factor’ in the formula for the arc length of a circle subtended by a central angle. For instance, if one measured the angle D in degrees, the formula for the arc length of a circle of radius r subtended by a central angle would be s = (π/180)rD, and who wants to memorize that? The word ‘radian’ first appeared in an examination at Queen’s College in Belfast, Ireland, given by James Thomson, whose better-known brother William would later be known as Lord Kelvin.

The wisdom of this choice can be seen in its far-reaching consequences in the calculus of the trigonometric functions, and undoubtedly elsewhere. First semester calculus students learn that as long as one uses radian measure for angles, the derivative of sin x is cos x, and the derivative of cos x is – sin x. A standard problem in first-semester calculus, here left to the reader, is to compute what the derivative of sin x would be if the angle were measured in degrees rather than radians. Of course, the fudge factor π/180 would raise its ugly head, its square would appear in the formula for the second derivative of sin x, and instead of the elegant repeating pattern of the derivatives of sin x and cos x that are a highlight of the calculus of trigonometric functions, the ensuing formulas would be beyond ugly.

One of the simplest known formulas for the computation of π is via the infinite series ????4=1−13+15−17+⋯

This deliciously elegant formula arises from integrating the geometric series with ratio -x^2 in the equation 1/(1+????^2)=1−????2+????4−????6+⋯

The integral of the left side is the inverse tangent function tan-1 x, but only because we have been fortunate enough to emphasize the importance of π by utilizing an angle measurement system which is the essence of πraeteritio; the recognition of the importance of π by excluding it from the discussion.

So on π Day, let us take a moment to recognize not only the beauty of π when it makes all the memorable appearances which we know and love, but to acknowledge its supreme importance and value in those critical situations where, like a great character in a play, it exerts a profound dramatic influence even when offstage.

LA MathJames D. Stein is emeritus professor in the Department of Mathematics at California State University, Long Beach. His books include Cosmic Numbers (Basic) and How Math Explains the World (Smithsonian). His most recent book is L.A. Math: Romance, Crime, and Mathematics in the City of Angels.

Where would we be without Pi?

Pi Day, the annual celebration of the mathematical constant π (pi), is always an excuse for mathematical and culinary revelry in Princeton. Since 3, 1, and 4 are the first three significant digits of π, the day is typically celebrated on 3/14, which in a stroke of serendipity, also happens to be Albert Einstein’s birthday. Pi Day falls on Monday this year, but Princeton has been celebrating all weekend with many more festivities still to come, from a Nerd Herd smart phone pub crawl, to an Einstein inspired running event sponsored by the Princeton Running Company, to a cocktail making class inside Einstein’s first residence. We imagine the former Princeton resident would be duly impressed.

Einstein enjoying a birthday/ Pi Day cupcake

Einstein enjoying a birthday/ Pi Day cupcake

Pi Day in Princeton always includes plenty of activities for children, and tends to be heavy on, you guessed it, actual pie (throwing it, eating it, and everything in between). To author Paul Nahin, this is fitting. At age 10, his first “scientific” revelation was,  If pi wasn’t around, there would be no round pies! Which it turns out, is all too true. Nahin explains:

Everybody “knows’’ that pi is a number a bit larger than 3 (pretty close to 22/7, as Archimedes showed more than 2,000 years ago) and, more accurately, is 3.14159265… But how do we know the value of pi? It’s the ratio of the circumference of a circle to a diameter, yes, but how does that explain how we know pi to hundreds of millions, even trillions, of decimal digits? We can’t measure lengths with that precision. Well then, just how do we calculate the value of pi? The symbol π (for pi) occurs in countless formulas used by physicists and other scientists and engineers, and so this is an important question. The short answer is, through the use of an infinite series expansion.

NahinIn his book In Praise of Simple Physics, Nahin shows you how to derive such a series that converges very quickly; the sum of just the first 10 terms correctly gives the first five digits. The English astronomer Abraham Sharp (1651–1699) used the first 150 terms of the series (in 1699) to calculate the first 72 digits of pi. That’s more than enough for physicists (and for anybody making round pies)!

While celebrating Pi Day has become popular—some would even say fashionable in nerdy circles— PUP author Marc Chamberland points out that it’s good to remember Pi, the number. With a basic scientific calculator, Chamberland’s recent video “The Easiest Way to Calculate Pi” details a straightforward approach to getting accurate approximations for Pi without tables or a prodigious digital memory. Want even more Pi? Marc’s book Single Digits has more than enough Pi to gorge on.

Now that’s a sweet dessert.

If you’re looking for more information on the origin of Pi, this post gives an explanation extracted from Joseph Mazur’s fascinating history of mathematical notation, Enlightening Symbols.

You can find a complete list of Pi Day activities from the Princeton Tour Company here.

#PiDay Activity: Using chocolate chips to calculate the value of pi

Chartier_MathTry this fun Pi Day activity this year. Mathematician Tim Chartier has a recipe that is equal parts delicious and educational. Using chocolate chips and the handy print-outs below, mathematicians of all ages can calculate the value of pi. Start with the Simple as Pi recipe, then graduate to the Death by Chocolate Pi recipe. Take it to the next level by making larger grids at home. If you try this experiment, take a picture and send it in and we’ll post it here.

Download: Simple as Pi [Word document]
Download: Death by Chocolate Pi [Word document]

For details on the math behind this experiment please read the article below which is cross-posted from Tim’s personal blog. And if you like stuff like this, please check out his new book Math Bytes: Google Bombs, Chocolate-Covered Pi, and Other Cool Bits in Computing.

For more Pi Day features from Princeton University Press, please click here.


 

Chocolate Chip Pi

How can a kiss help us learn Calculus? If you sit and reflect on answers to this question, you are likely to journey down a mental road different than the one we will traverse. We will indeed use a kiss to motivate a central idea of Calculus, but it will be a Hershey kiss! In fact, we will have a small kiss, more like a peck on the cheek, as we will use white and milk chocolate chips. The math lies in how we choose which type of chip to use in our computation.

Let’s start with a simple chocolatey problem that will open a door to ideas of Calculus. A Hershey’s chocolate bar, as seen below, is 2.25 by 5.5 inches. We’ll ignore the depth of the bar and consider only a 2D projection. So, the area of the bar equals the product of 2.25 and 5.5 which is 12.375 square inches.

Note that twelve smaller rectangles comprise a Hershey bar. Suppose I eat 3 of them. How much area remains? We could find the area of each small rectangle. The total height of the bar is 2.25 inches. So, one smaller rectangle has a height of 2.25/3 = 0.75 inches. Similarly, a smaller rectangle has a width of 5.5/4 = 1.375. Thus, a rectangular piece of the bar has an area of 1.03125, which enables us to calculate the remaining uneaten bar to have an area of 9(1.03125) = 9.28125 square inches.

Let’s try another approach. Remember that the total area of the bar is 12.375. Nine of the twelve rectangular pieces remain. Therefore, 9/12ths of the bar remains. I can find the remaining area simply be computing 9/12*(12.375) = 9.28125. Notice how much easier this is than the first method. We’ll use this idea to estimate the value of π with chocolate, but this time we’ll use chocolate chips!

Let’s compute the area of a quarter circle of unit radius, which equals π/4 since the full circle has an area of π. Rather than find the exact area, let’s estimate. We’ll break our region into squares as seen below.

This is where the math enters. We will color the squares red or white. Let’s choose to color a square red if the upper right-hand corner of the square is in the shaded region and leave it white otherwise, which produces:

Notice, we could have made other choices. We could color a square red if the upper left-hand corner or even middle of the square is under the curve. Some choices will lead to more accurate estimates than others for a given curve. What choice would you make?

Again, the quarter circle had unit radius so our outer square is 1 by 1. Since eight of the 16 squares are filled, the total shaded area is 8/16.

How can such a grid of red and white squares yield an estimate of π? In the grid above, notice that 8/16 or 1/2 of the area is shaded red. This is also an approximation to the area of the quarter circle. So, 1/2 is our current approximation to π/4. So, π/4 ≈ 1/2. Solving for π we see that π ≈ 4*(1/2) = 2. Goodness, not a great estimate! Using more squares will lead to less error and a better estimate. For example, imagine using the grid below:

Where’s the chocolate? Rather than shading a square, we will place a milk chocolate chip on a square we would have colored red and a white chocolate chip on a region that would have been white. To begin, the six by six grid on the left becomes the chocolate chip mosaic we see on the right, which uses 14 white chocolate of the total 36 chips. So, our estimate of π is 2.4444. We are off by about 0.697.

Next, we move to an 11 by 11 grid of chocolate chips. If you count carefully, we use 83 milk chocolate chips of the 121 total. This gives us an estimate of 2.7438 for π, which correlates to an error of about 0.378.

Finally, with the help of public school teachers in my seminar Math through Popular Culture for the Charlotte Teachers Institute, we placed chocolate chips on a 54 by 54 grid. In the end, we used 2232 milk chocolate chips giving an estimate of 3.0617 having an error of 0.0799.

What do you notice is happening to the error as we reduce the size of the squares? Indeed, our estimates are converging to the exact area. Here lies a fundamental concept of Calculus. If we were able to construct such chocolate chip mosaics with grids of ever increasing size, then we would converge to the exact area. Said another way, as the area of the squares approaches zero, the limit of our estimates will converge to π. Keep in mind, we would need an infinite number of chocolate chips to estimate π exactly, which is a very irrational thing to do!

And finally, here is our group from the CTI seminar along with Austin Totty, a senior math major at Davidson College who helped present these ideas and lead the activity, with our chocolatey estimate for π.