## Praeteritio and the quiet importance of Pi

by 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 𝜋4=1−13+15−17+⋯

This deliciously elegant formula arises from integrating the geometric series with ratio -x^2 in the equation 1/(1+𝑥^2)=1−𝑥2+𝑥4−𝑥6+⋯

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.

James 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

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.

In 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.

## Pi Day Recipe: Apple Pie from Jim Henle’s The Proof and the Pudding

Tomorrow (March 14, 2015) is a very important Pi Day. This year’s local Princeton Pi Day Party and other global celebrations of Albert Einstein’s birthday look to be truly stellar, which is apt given this is arguably the closest we will get to 3.1415 in our lifetimes.

Leading up to the publication of the forthcoming The Proof and the Pudding: What Mathematicians, Cooks, and You Have in Common by Jim Henle, we’re celebrating the holiday with a recipe for a classic Apple Pie (an integral part of any Pi Day spread). Publicist Casey LaVela recreates and photographs the recipe below. Full text of the recipe follows. Happy Pi Day everyone!

## Notes on Jim Henle’s Apple Pie recipe from Publicist Casey LaVela

The Proof and the Pudding includes several recipes for pies or tarts that would fit the bill for Pi Day, but the story behind Henle’s Apple Pie recipe is especially charming, the recipe itself is straightforward, and the results are delicious. At the author’s suggestion, I used a mixture of baking apples (and delightfully indulgent amounts of butter and sugar).

#### Crust:

All of the crust ingredients (flour, butter, salt) ready to go:

After a few minutes of blending everything together with a pastry cutter, the crust begins to come together. A glorious marriage of flour and butter.

Once the butter and flour were better incorporated, I dribbled in the ice water and then turned the whole wonderful mess out between two sheets of plastic wrap in preparation for folding. The crust will look like it won’t come together, but somehow it always does in the end. Magical.

Now you need to roll out and fold over the dough a few times. This is an important step and makes for a light and flaky crust. (You use a similar process to make croissants or other viennoiserie from scratch.)

I cut the crust into two (for the top crust and bottom crust) using my handy bench scraper:

#### Apples:

The apples cored, peeled, and ready to be cut into slices. I broke out my mandolin slicer (not pictured) to make more even slices, but if you don’t own a slicer or prefer to practice your knife skills you can just as easily use your favorite sharp knife.

Beautiful (even) apple slices:

Action shot of me mixing the apple slices, sugar, and cinnamon together. I prefer to prepare my apple pie filling in a bowl rather than sprinkling the dry ingredients over the apple slices once they have been arranged in the bottom crust. I’m not sure if it has much impact on the flavor and it is much, much messier, but I find it more fun.

#### Assembly:

The bottom crust in the pie plate:

Arrange the apple slices in the bottom crust:

Top with the second crust, seal the top crust to the bottom with your fingers, and (using your sharp knife) make incisions in the top crust to allow steam to escape:

The apple pie before going into the oven (don’t forget to put a little extra sugar on top):

#### The finished product:

There was a little crust left over after cutting, so I shaped it into another pi symbol, covered it in cinnamon and sugar, and baked it until golden brown. I ate the baked pi symbol as soon as it had cooled (before thinking to take a picture), but it was delicious!

# Apple Pie

The story of why I started cooking is not inspiring. My motives weren’t pure. Indeed, they involved several important sins.

I really am a glutton. I love to eat. As a child, I ate well; my mother was a wonderful cook. But I always wanted more than I got, especially dessert. And of all desserts, it was apple pie I craved most. Not diner pies, not restaurant pies, and not bakery pies, but real, homemade apple pies.

When I was six, I had my first homemade apple pie. It was at my grandmother’s house. I don’t remember how it tasted, but I can still recall the gleam in my mother’s eye when she explained the secret of the pie. “I watched her make it. Before she put on the top crust, she dotted the whole thing with big pats of butter!”

Several times as I was growing up, my mother made apple pie. Each one was a gem. But they were too few—only three or four before I went off to college. They were amazing pies. The apples were tart and sweet. Fresh fall apples, so flavorful no cinnamon was needed. The crust was golden, light and crisp, dry when it first hit the tongue, then dissolving into butter.

I grew up. I got married. I started a family. All the while, I longed for that pie. Eventually I set out to make one.

Success came pretty quickly, and it’s not hard to see why. The fact is, despite apple pie’s storied place in American culture, most apple pies sold in this country are abysmal. A pie of fresh, tart apples and a crust homemade with butter or lard, no matter how badly it’s made, is guaranteed to surpass a commercial product.

That means that even if you’ve never made a pie before, you can’t go seriously wrong. The chief difficulty is the crust, but I’ve developed a reliable method. Except for this method, the recipe below is standard.

For the filling:
5 cooking apples (yielding about 5 cups of pieces)
1/4 to 1/3 cup sugar
2 Tb butter
1/2 to 1 tsp cinnamon
lemon juice, if necessary
1 tsp flour, maybe

For the crust:
2 cups flour
1 tsp salt
2/3 cup lard or unsalted butter (1 1/3 sticks)
water

The crust is crucial. I’ll discuss its preparation last. Assume for now that you’ve rolled out the bottom crust and placed it in the pie pan.

Core, peel, and slice the apples. Place them in the crust. Sprinkle with sugar and cinnamon. Dot with butter. Roll out the top crust and place it on top. Seal the edge however you like. In about six places, jab a knife into the crust and twist to leave a hole for steam to escape. Sprinkle the crust with the teaspoon of sugar.

Bake in a preheated oven for 15 minutes at 450° and then another 35 minutes at 350°. Allow to cool. Serve, if you like, with vanilla ice cream or a good aged cheddar.

Now, the crust:

Mix the flour and salt in a large bowl. Place the lard or butter or lard/butter in the bowl. Cut it in with a pastry cutter.

Next, the water. Turn the cold water on in the kitchen sink so that it dribbles out in a tiny trickle. Hold the bowl with the flour mixture in one hand and a knife in the other. Let the water dribble into the bowl while you stir with the knife. The object is to add just enough water so that the dough is transformed into small dusty lumps. Don’t be vigorous with the knife, but don’t allow the water to pool. If the water is dribbling too fast, take the bowl away from the faucet from time to time. When you’re done, the dough will still look pretty dry.

Recipes usually call for about 5 tablespoons of water. This method probably uses about that much.

Actually, the dough will look so dry that you’ll think it won’t stick together when it’s rolled out. In fact, it probably won’t stick together, but trust me. This is going to work.

Tear off a sheet of plastic wrap and lay it on the counter. Place a bit more than half the dough on the sheet and cover it with a second sheet of plastic.

With a rolling pin, roll the dough out between the two sheets. Roll it roughly in the shape of a rectangle.

It won’t look great and it probably would fall apart if you picked it up.

Don’t pick it up. Remove the top sheet of plastic wrap and fold the bottom third up, and fold the top third down, then do the same horizontally, right and left.

Now replace the top sheet of plastic wrap and roll the dough out gently into a disk.

This time it should look pretty decent. This time the dough will stick together.

You should be able to remove the top sheet of plastic and, using the bottom sheet, turn it over into the pie pan. The crust should settle in nicely without breaking.

Form the top crust the same way.

This method rolls each crust twice—usually not a good idea because working the dough makes it tough. But remarkably, crusts produced this way are tender and light. I’m not sure why but I suspect it’s because the dough is fairly dry.

Notes:
• Cooking apples are tart apples. The best I know is the Rhode Island Greening, but they’re hard to find. Baldwins and Jonathans are decent, but they’re hard to find too. The British Bramleys are terrific. I’ve made good pies from the French Calville Blanc d’Hiver. But we’re not living in good apple times. Most stores don’t sell apples for cooking. When in doubt, use a mixture.
• The lemon juice and the larger quantity of cinnamon are for when you have tired apples with no oomph. The cheese also serves this purpose. It should be a respectable old cheddar and it should be at room temperature.
• Consumption of too many commercial pies makes me loath to add flour or cornstarch to pie filling. The flour is here in case you fear your apples will be too juicy. I don’t mind juice in a pie, in moderation. If adding flour, mix the apples, sugar, cinnamon, and flour in a bowl before pouring into the crust.
• Lard is best. Its melting point is higher than butter’s. It successfully separates the flour into layers for a light, crispy crust. Butter is more likely to saturate the flour and produce a heavy crust. Some like half butter/half lard, preferring butter for its flavor. But the flavor of lard is nice too, and its porkiness is wonderful with apple.

This recipe is taken from:

### What Mathematicians, Cooks, and You Have in Common

#### Jim Henle

“If you’re a fan of Julia Child or Martin Gardner—who respectively proved that anyone can have fun preparing fancy food and doing real mathematics—you’ll enjoy this playful yet passionate romp from Jim Henle. It’s stuffed with tasty treats and ingenious ideas for further explorations, both in the kitchen and with pencil and paper, and draws many thought-providing parallels between two fields not often considered in the same mouthful.”—Colm Mulcahy, author of Mathematical Card Magic: Fifty-Two New Effects

## Pi Day: Where did π come from anyway?

When one sees π in an equation, the savvy reader automatically knows that something circular is lurking behind. So the symbol (a relatively modern one, of course) does not fool the mathematician who is familiar with its many disguises that unintentionally drag along in the mind to play into imagination long after the symbol was read.

Here is another disguise of π: Consider a river flowing in uniformly erodible sand under the influence of a gentle slope. Theory predicts that over time the river’s actual length divided by the straight-line distance between its beginning and end will tend toward π. If you guessed that the circle might be a cause, you would be right.

The physicist Eugene Wigner gives an apt story in his celebrated essay, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” A statistician tries to explain the meaning of the symbols in a reprint about population trends that used the Gaussian distribution. “And what is this symbol here?” the friend asked.

“Oh,” said the statistician. “This is pi.”

“What is that?”

“The ratio of the circumference of the circle to its diameter.”

“Well, now, surely the population has nothing to do with the circumference of the circle.”

Wigner’s point in telling this story is to show us that mathematical concepts turn up in surprisingly unexpected circumstances such as river lengths and population trends. Of course, he was more concerned with understanding the reasons for the unexpected connections between mathematics and the physical world, but his story also points to the question of why such concepts turn up in unexpected ways within pure mathematics itself.

# The Good Symbol

The first appearance of the symbol π came in 1706. William Jones (how many of us have ever heard of him?) used the Greek letter π to denote the ratio of the circumference to the diameter of a circle. How simple. “No lengthy introduction prepares the reader for the bringing upon the stage of mathematical history this distinguished visitor from the field of Greek letters. It simply came, unheralded.” But for the next thirty years, it was not used again until Euler used it in his correspondence with Stirling.

We could accuse π of not being a real symbol. It is, after all, just the first letter of the word “periphery.” True, but like i, it evokes notions that might not surface with symbols carrying too much baggage. Certain questions such as “what is ii?” might pass our thoughts without a contemplating pause. Pure mathematics asks such questions because it is not just engaged with symbolic definitions and rules, but with how far the boundaries can be pushed by asking questions that everyday words could ignore. You might think that ii makes no sense, that it’s nothing at all, or maybe a complex number. Surprise: it turns out to be a real number!

It seems that number has a far broader meaning than it once had when we first started counting sheep in the meadow. We have extended the idea to include collections of conceptual things that include the usual members of the number family that still obey the rules of numerical operations. Like many of the words we use, number has a far broader meaning than it once had.

Think only of the so-called imaginary quantities with which mathematicians long operated, and from which they even obtained important results ere they were in a position to assign to them a perfectly determinate and withal visualizable meaning.

It is not the job of mathematics to stick with earthly relevance. Yet the world seems to eventually pick up on mathematics abstractions and generalizations and apply them to something relevant to Earth’s existence. Almost a whole century passed with mathematicians using imaginary exponents while a new concept germinated. And then, from the symbol i that once stood for that one-time peculiar abhorrence √−1, there emerged a new notion: that magnitude, direction, rotation may be embodied in the symbol itself. It is as if symbols have some intelligence of their own.

What is good mathematical notation? As it is with most excellent questions, the answer is not so simple. Whatever a symbol is, it must function as a revealer of patterns, a pointer to generalizations. It must have an intelligence of its own, or at least it must support our own intelligence and help us think for ourselves. It must be an indicator of things to come, a signaler of fresh thoughts, a clarifier of puzzling concepts, a help to overcome the mental fatigues of confusion that would otherwise come from rhetoric or shorthand. It must be a guide to our own intelligence. Here is Mach again:

In algebra we perform, as far as possible, all numerical operations which are identical in form once for all, so that only a remnant of work is left for the individual case. The use of the signs of algebra and analysis, which are merely symbols of operations to be performed, is due to the observation that we can materially disburden the mind in this way and spare its powers for more important and more difficult duties, by imposing all mechanical operations upon the hand.

The student of mathematics often finds it hard to throw off the uncomfortable feeling that his science, in the person of his pencil, surpasses him in intelligence—an impression which the great Euler confessed he often could not get rid of.

A single symbol can tell a whole story.

There was no single moment when xn was first used to indicate the nth power of x. A half century separated Bombelli’s , from Descartes’s xn. It may seem like a clear-cut idea to us, but the idea of symbolically labeling the number of copies of x in the product was a huge step forward. The reader no longer had to count the number of x’s, which paused contemplation, interrupted the smoothness of reading, and hindered any broad insights of associations and similarities that could extend ideas. The laws xnxm = xn+m and (xn)m = xnm, where n and m are integers, were almost immediately suggested from the indexing symbol. Not far behind was the idea to let x½ denote √x, inspired by extending the law xnxm = xn+m to include fractions, so x½ x½  = x1.

Further speculation on what nx might be would surely have inspired questions such as what x might be for a given y in an equation such as y = 10x. Answer that and we would have a way of performing multiplication by addition. But Napier, the inventor of logarithms, already knew the answer long before mathematics had any symbols at all!

Symbols acquire meanings that they originally didn’t have. But symbolic representation has, likewise, the disadvantage that the object represented is very easily lost sight of, and that operations are continued with the symbols to which frequently no object whatever corresponds.

Ernst Mach once again:

A symbolical representation of a method of calculation has the same significance for a mathematician as a model or a visualisable working hypothesis has for a physicist. The symbol, the model, the hypothesis runs parallel with the thing to be represented. But the parallelism may extend farther, or be extended farther, than was originally intended on the adoption of the symbol. Since the thing represented and the device representing are after all different, what would be concealed in the one is apparent in the other.

## Pi Day Recipe: Brandy Alexander Pie from Cooking for Crowds

This recipe is presented as part of our Pi Day celebration. For more Pi Day features from Princeton University Press, please click here.

# Brandy Alexander Pie

This pie is as sweet and delicious as the drink for which it is named, and a great deal less alcoholic. It is light and fluffy, but very filling.

 6 12 20 50 unflavored gelatin envelopes 1 2 4 8 cold water ½ c 1 c 2 c 4 c granulated sugar ⅔ c 1⅓ c 2⅔ c 2 lbs salt ⅛ tsp ¼ tsp ½ tsp 1 tsp eggs, separated 3 6 12 24 Cognac ¼ c ½ c 1 c 2 c Grand Marnieror ¼ c ½ c 1 c 2 c creme de cacao ¼ c ½ c 1 c 2 c heavy cream 2 c 4 c 4 pts 8 pts graham cracker crust 1 2 4 8 Garnish 4-ounce bars semisweet chocolate 1 2 2½ 3 heavy cream 1 c 2 c 3½ c 6 c

Sprinkle the gelatin over the cold water in a saucepan. Add ⅓ cup [⅔ cup, 1⅓ cups, 2⅔ cups] of the sugar, the salt, and egg yolks. Stir to blend, then heat over low heat, stirring, until the gelatin dissolves and the mixture thickens. Do not boil. Remove from the heat and stir in the Cognac and Grand Marnier (or creme de cacao). Chill in the refrigerator until the mixture mounds slightly and is thick.

Beat the egg whites until stiff (use a portable electric mixer in a large kettle). Gradually beat in the remaining sugar and fold into the thickened mixture. Whip half of the cream until it holds peaks. Fold in the whipped cream, and turn into the crusts. Chill several hours, or overnight. To serve, garnish with the remaining cream, whipped. Using a vegetable peeler, make chocolate curls from the chocolate bars and let drop onto the cream.

For additional recipes for feeding the masses, please check out Cooking for Crowds by Merry “Corky” White.

## #PiDay Art from Beautiful Geometry – “Squaring the Circle”

This article has been extracted from Beautiful Geometry by Eli Maor and Eugen Jost. For more Pi Day features from Princeton University Press, please click here.

At first glance, the circle may seem to be the simplest of all geometric shapes and the easiest to draw: take a string, hold down one end on a sheet of paper, tie a pencil to the other end, and swing it around—a simplified version of the compass. But first impressions can be misleading: the circle has proved to be one of the most intriguing shapes in all of geometry, if not the most intriguing of them all.

How do you find the area of a circle, when its radius is given? You instantly think of the formula A = πr2. But what exactly is that mysterious symbol π? We learn in school that it is approximately 3.14, but its exact value calls for an endless string of digits that never repeat in the same order. So it is impossible to find the exact area of a circle numerically. But perhaps we can do the next best thing—construct, using only straightedge and compass, a square equal in area to that of a circle?

This problem became known as squaring the circle— or simply the quadrature problem—and its solution eluded mathematicians for well over two thousand years. The ancient Egyptians came pretty close: In the Rhind Papyrus, a collection of 84 mathematical problems dating back to around 1800 BCE, there is a statement that the area of a circle is equal to the area of a square of side  of the circle’s diameter. Taking the diameter to be 1 and equating the circle’s area to that of the square, we get π(½)2 = ( )2, from which we derive a value of π equal to ≈ 3.16049—within 0.6 percent of the true value. However, as remarkable as this achievement is, it was based on “eyeballing,” not on an exact geometric construction.

Numerous attempts have been made over the centuries to solve the quadrature problem. Many careers were spent on this task—all in vain. The definitive solution—a negative one—came only in 1882, when Carl Louis Ferdinand von Lindemann (1852– 1939) proved that the task cannot be done—it is impossible to square a circle with Euclidean tools. Actually, Lindemann proved something different: that the number π, the constant at the heart of the quadrature problem, is transcendental. A transcendental number is a number that is not the solution of a polynomial equation with integer coefficients. A number that is not transcendental is called algebraic. All rational numbers are algebraic; for example,  is the solution of the equation 5x − 3 = 0. So are all square roots, cubic roots, and so on; for example,  is the positive solution of x2 − 2 = 0, and  is one solution of x6 − 4x3 − 1 = 0. The name transcendental has nothing mysterious about it; it simply implies that such numbers transcend the realm of algebraic (polynomial) equations.

Now it had already been known that if π turned out to be transcendental, this would at once establish that the quadrature problem cannot be solved. Lindemann’s proof of the transcendence of π therefore settled the issue once and for all. But settling the issue is not the same as putting it to rest; being the most famous of the three classical problems, we can rest assured that the “circle squarers” will pursue their pipe dream with unabated zeal, ensuring that the subject will be kept alive forever.

Metamorphosis of a Circle

Metamorphosis of a Circle, shows four large panels. The panel on the upper left contains nine smaller frames, each with a square (in blue) and a circular disk (in red) centered on it. As the squares decrease in size, the circles expand, yet the sum of their areas remains constant. In the central frame, the square and circle have the same area, thus offering a computer-generated “solution” to the quadrature problem. In the panel on the lower right, the squares and circles reverse their roles, but the sum of their areas is still constant. The entire sequence is thus a metamorphosis from square to circle and back.

Of course, Euclid would not have approved of such a solution to the quadrature problem, because it does not employ the Euclidean tools—a straightedge and compass. It does, instead, employ a tool of far greater power—the computer. But this power comes at a price: the circles, being generated pixel by pixel like a pointillist painting, are in reality not true circles, only simulations of circles.1 As the old saying goes, “there’s no free lunch”—not even in geometry.

Note:

1. The very first of the 23 definitions that open Euclid’s Elements defines a point as “that which has no part.” And since all objects of classical geometry—lines, circles, and so on—are made of points, they rest on the subtle assumption that Euclidean space is continuous. This, of course, is not the case with computer space, where Euclid’s dimensionless point is replaced by a pixel—small, yet of finite size—and space between adjacent pixels is empty, containing no points.

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

Try 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: 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.

## Pi Day: “Was Einstein Right?” Chuck Adler on the twin paradox of relativity in science fiction

This post is extracted from Wizards, Aliens, and Starships by Charles Adler. Dr. Adler will kick off Princeton’s Pi Day festivities tonight with a talk at the Princeton Public Library starting at 7:00 PM. We hope you can join the fun!

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

Robert A. Heinlein’s novel Time for the Stars is essentially one long in-joke for physicists. The central characters of the novel are Tom and Pat Bartlett, two identical twins who can communicate with each other telepathically. In the novel, telepathy has a speed much faster than light. Linked telepaths, usually pairs of identical twins, are used to maintain communications between the starship Lewis and Clark and Earth. Tom goes on the spacecraft while Pat stays home; the ship visits a number of distant star systems, exploring and finding new Earth-like worlds. On Tom’s return, nearly seventy years have elapsed on Earth, but Tom has only aged by five.

I call this a physicist’s in-joke because Heinlein is illustrating what is referred to as the twin paradox of relativity: take two identical twins, fly one around the universe at nearly the speed of light, and leave the other at home. On the traveler’s return, he or she will be younger than the stay-at- home, even though the two started out the same age. This is because according to Einstein’s special theory of relativity, time runs at different rates in different reference frames.

This is another common theme in science fiction: the fact that time slows down when one “approaches the speed of light.” It’s a subtle issue, however, and is very easy to get wrong. In fact, Heinlein made some mistakes in his book when dealing with the subject, but more on that later. First, I want to list a few of the many books written using this theme:

• The Forever War, by Joe W. Haldeman. This story of a long-drawn-out conflict between humanity and an alien race has starships that move at speeds near light speed to travel between “collapsars” (black holes), which are used for faster-than-light travel. Alas, this doesn’t work. The hero’s girlfriend keeps herself young for him by shuttling back and forth at near light speeds between Earth and a distant colony world.
• Poul Anderson’s novel, Tau Zero. In this work, mentioned in the last chapter, the crew of a doomed Bussard ramship is able to explore essentially the entire universe by traveling at speeds ever closer to the speed of light.
• The Fifth Head of Cerberus, by Gene Wolfe. In this novel an anthropologist travels from Earth to the double planets of St. Croix and St. Anne. It isn’t a big part of the novel, but the anthropologist John Marsch mentions that eighty years have passed on Earth since he left it, a large part of his choice to stay rather than return home.
• Larris Niven’s novel A World out of Time. The rammer Jerome Corbell travels to the galactic core and back, aging some 90 years, while three million years pass on Earth.

There are many, many others, and for good reason: relativity is good for the science fiction writer because it brings the stars closer to home, at least for the astronaut venturing out to them. It’s not so simple for her stay-at-home relatives. The point is that the distance between Earth and other planets in the Solar System ranges from tens of millions of kilometers to billions of kilometers. These are large distances, to be sure, but ones that can be traversed in times ranging from a few years to a decade or so by chemical propulsion. We can imagine sending people to the planets in times commensurate with human life. If we imagine more advanced propulsion systems, the times become that much shorter.

Unfortunately, it seems there is no other intelligent life in the Solar System apart from humans, and no other habitable place apart from Earth. If we want to invoke the themes of contact or conflict with aliens or finding and settling Earth-like planets, the narratives must involve travel to other stars because there’s nothing like that close to us. But the stars are a lot farther away than the planets in the Solar System: the nearest star system to our Solar System, the triple star system Alpha Centauri, is 4.3 light-years away: that is, it is so far that it takes light 4.3 years to get from there to here, a distance of 40 trillion km. Other stars are much farther away. Our own galaxy, the group of 200 billion stars of which our Sun is a part, is a great spiral 100,000 light-years across. Other galaxies are distances of millions of light-years away.

From our best knowledge of physics today, nothing can go faster than the speed of light. That means that it takes at least 4.3 years for a traveler (I’ll call him Tom) to go from Earth to Alpha Centauri and another 4.3 years to return. But if Tom travels at a speed close to that of light, he doesn’t experience 4.3 years spent on ship; it can take only a small fraction of the time. In principle, Tom can explore the universe in his lifetime as long as he is willing to come back to a world that has aged millions or billions of years in the meantime.

# Was Einstein Right?

This weird prediction—that clocks run more slowly when traveling close to light speed—has made many people question Einstein’s results. The weirdness isn’t limited to time dilation; there is also relativistic length contraction. A spacecraft traveling close to the speed of light shrinks in the direction of motion. The formulas are actually quite simple. Let’s say that Tom is in a spacecraft traveling along at some speed v, while Pat is standing still, watching him fly by. We’ll put Pat in a space suit floating in empty space so we don’t have to worry about the complication of gravity. Let’s say the following: Pat has a stopwatch in his hand, as does Tom. As Tom speeds by him, both start their stopwatches at the same time and Pat measures a certain amount of time on his watch (say, 10 seconds) while simultaneously watching Tom’s watch through the window of his spacecraft. If Pat measures time ∆t0 go by on his watch, he will see Tom’s watch tick through less time. Letting ∆t be the amount of time on Tom’s watch, the two times are related by the formula

where the all-important “gamma factor” is

The gamma factor is always greater than 1, meaning Pat will see less time go by on Tom’s watch than on his. Table 12.1 shows how gamma varies with velocity.

Note that this is only really appreciable for times greater than about 10% of the speed of light. The length of Tom’s ship as measured by Pat (and the length of any object in it, including Tom) shrinks in the direction of motion by the same factor.

Even though the gamma factor isn’t large for low speeds, it is still measurable. To quote Edward Purcell, “Personally, I believe in special relativity. If it were not reliable, some expensive machines around here would be in very deep trouble”. The time dilation effect has been measured directly, and is measured directly almost every second of every day in particle accelerators around the world. Unstable particles have characteristic lifetimes, after which they decay into other particles. For example, the muon is a particle with mass 206 times the mass of the electron. It is unstable and decays via the reaction

It decays with a characteristic time of 2.22 μs; this is the decay time one finds for muons generated in lab experiments. However, muons generated by cosmic ray showers in Earth’s atmosphere travel at speeds over 99% of the speed of light, and measurements on these muons show that their decay lifetime is more than seven times longer than what is measured in the lab, exactly as predicted by relativity theory. This is an experiment I did as a graduate student and our undergraduates at St. Mary’s College do as part of their third-year advanced lab course. Experiments with particles in particle accelerators show the same results: particle lifetimes are extended by the gamma factor, and no matter how much energy we put into the particles, they never travel faster than the speed of light. This is remarkable because in the highest-energy accelerators, particles end up traveling at speeds within 1 cm/s of light speed. Everything works out exactly as the theory of relativity says, to a precision of much better than 1%.

How about experiments done with real clocks? Yes, they have been done as well. The problems of doing such experiments are substantial: at speeds of a few hundred meters per second, a typical speed for an airplane, the gamma factor deviates from 1 by only about 1013. To measure the effect, you would have to run the experiment for a long time, because the accuracy of atomic clocks is only about one part in 1011 or 1012; the experiments would have to run a long time because the difference between the readings on the clocks increases with time. In the 1970s tests were performed with atomic clocks carried on two airplanes that flew around the world, which were compared to clocks remaining stationary on the ground. Einstein passed with flying colors. The one subtlety here is that you have to take the rotation of the Earth into account as part of the speed of the airplane. For this reason, two planes were used: one going around the world from East to West, the other from West to East. This may seem rather abstract, but today it is extremely important for our technology. Relativity lies at the cornerstone of a multi-billion-dollar industry, the global positioning system (GPS).

GPS determines the positions of objects on the Earth by triangulation: satellites in orbit around the Earth send radio signals with time stamps on them. By comparing the time stamps to the time on the ground, it is possible to determine the distance to the satellite, which is the speed of light multiplied by the time difference between the two. Using signals from at least four satellites and their known positions, one can triangulate a position on the ground. However, the clocks on the satellites run at different rates as clocks on the ground, in keeping with the theory of relativity. There are actually two different effects: one is relativistic time dilation owing to motion and the other is an effect we haven’t considered yet, gravitational time dilation. Gravitational time dilation means that time slows down the further you are in a gravitational potential well. On the satellites, the gravitational time dilation speeds up clock rates as compared to those on the ground, and the motion effect slows them down. The gravitational effect is twice as big as the motion effect, but both must be included to calculate the total amount by which the clock rate changes. The effect is small, only about three parts in a billion, but if relativity weren’t accounted for, the GPS system would stop functioning in less than an hour. To quote from Alfred Heick’s textbook GPS Satellite Surveying,

Relativistic effects are important in GPS surveying but fortunately can be accurately calculated. . . . [The difference in clock rates] corresponds to an increase in time of 38.3 μsec per day; the clocks in orbit appear to run faster. . . . [This effect] is corrected by adjusting the frequency of the satellite clocks in the factory before launch to 10.22999999543 MHz [from their fundamental frequency of 10.23 MHz].

This statement says two things: first, in the dry language of an engineering handbook, it is made quite clear that these relativistic effects are so commonplace that engineers routinely take them into account in a system that hundreds of millions of people use every day and that contributes billions of dollars to the world’s commerce. Second, it tells you the phenomenal accuracy of radio and microwave engineering. So the next time someone tells you that Einstein was crazy, you can quote chapter and verse back at him!

## Fantasy Physics: Should Einstein Have Won Seven Nobel Prizes?

This guest post from A. Douglas Stone is part of our celebration of all things Einstein, pi, and, of course, pie this week. For more articles, please click here. Please join Prof. Stone at the Princeton Public Library on March 14 at 6 PM for a lecture about Einstein’s quantum breakthroughs.

Cross-posted with the Huffington Post.

Albert Einstein never cared too much about receiving awards and honors, and that included the Nobel Prizes, which were established in 1901, at roughly the same time as Einstein was beginning his research career in physics. In 1905, at the age of 25, Einstein began his ascent to scientific pre-eminence and world-wide fame with his proposal of the Special Theory of Relativity, as well as a “revolutionary” paper on the particulate properties of light, his foundational work on molecular (“Brownian”) motion, and finally his famous equation, E = mc2. In 1910, he was first nominated for the Prize and was nominated many times subsequently, usually by multiple physicists, until he finally won the 1921 Prize (awarded in 1922). Surprisingly, he did not win for his most famous achievement, Relativity Theory, which was still deemed too speculative and uncertain to endorse with the Prize. Instead, he won for his 1905 proposal of the law of the photoelectric effect—empirically verified in the following decade by Robert Millikan—and for general “services to theoretical physics.” It was a political decision by the Nobel committee; Einstein was so renowned that their failure to select him had become an embarrassment to the Nobel institution. But this highly conservative organization could find no part of his brilliant portfolio that they either understood or trusted sufficiently to name specifically, except for this relatively minor implication of his 1905 paper on particles of light. The final irony in this selection was that, among the many controversial theories that Einstein had proposed in the previous seventeen years, the only one not accepted by almost all of the leading theoretical physicists of the time was precisely his theory of light quanta (or photons), which he had used to find the law of the photoelectric effect!

In keeping with his relative indifference to such honors, Einstein declined to attend the award ceremony, because he had previously committed to a lengthy trip to Japan at that time and didn’t feel it was fair to his hosts to cancel it. Moreover, when the Prize was officially announced and the news reached him during his long voyage to Japan, he neglected to even mention the Prize in the travel diary he was keeping. He had taken one practical note of it however, in advance. When he divorced his first wife, Mileva Maric in 1919, he agreed to transfer to her the full prize money, a substantial sum, in the form of a Trust for the benefit of her and his sons, should he eventually win.

However, while Einstein himself barely dwelt at all on this honor, it is an interesting exercise to ask how many distinct breakthroughs Einstein made during his productive research career, spanning primarily the years 1905 to 1925, that could be judged of Nobel caliber, when placed in historical context and evaluated by the standards of subsequent Nobel Prize awards. Admittedly, this analysis has a bit in common with fantasy sports, in which athletes are judged and ranked by their statistical achievements and arguments are made about who was the GOAT (“greatest of all time”). Well, why not spend a few pages on this guilty pleasure, at least partly in the service of illuminating the achievements of this historic genius, even if Einstein would not have approved?

Let’s start with the Prize he did receive, which was absolutely deserved, if the committee had had the courage to write the citation, “for his proposal of the existence of light quanta.” The law of the photoelectric effect, which they cited, only makes sense if light behaves like a particle in some important respects, and that is what he proposed in 1905. This proposal came at a time when the wave theory of light was absolutely triumphant and was even enshrined in a critical technology: radio. Not a single physicist in the world was thinking along similar lines as Einstein, nor were all of the important theorists convinced by his arguments for two more decades. Nonetheless, the photon concept was unambiguously confirmed in experiments by 1925, and now is considered the paradigm for our modern quantum theory of force-carrying particles. It is the first in a family of particles known as bosons, most recently augmented by the (Nobel-winning) discovery of the Higgs particle. So the photon is a Nobel slam dunk.

We can move next to two more “no-brainers,” the two theories of relativity, the Special Theory, proposed in 1905, and the General Theory, germinated in 1907 and completed in 1915. These are quite distinct contributions. The Special Theory introduced the Principle of Relativity, that the law of physics must all be the same for bodies in uniform relative motion. An amazing implication of this statement is that time does not elapse uniformly, independent of the motion of observers, but rather that the time interval between events depends on the state of relative motion of the observer. Einstein was the first to understand and explain this radical notion, which is now well-verified by direct experiments. Moreover, Einstein’s concept of “relativistic invariance” is built into our theory of the elementary particles, and so it has had a profound impact on fundamental physics. However, here it must be noted that the equations of Special Relativity were first written down by Hendrik Lorentz, the great Dutch physicist whom Einstein admired the most of all his contemporaries. Lorentz just failed to give them the radical interpretation with which Einstein endowed them; he also failed to notice that they implied that energy and mass were interchangeable: E = mc2. There are also a few votes out there for the French mathematician, Henri Poincare, who enunciated the Principle of Relativity before Einstein, but I can’t put him in the same category as Lorentz with regard to this debate. Einstein would have been happy to share Special Relativity with Lorentz, so let’s split this one 50-50 between the two.

General Relativity on the other hand is all Albert. Like the photon, no one on the planet even had an inkling of this idea before Einstein. Einstein realized that the question of the relativity of motion was tied up with the theory of Gravity: that uniform acceleration (e.g. in an elevator in empty space) was indistinguishable from the effect of gravity on the surface of a planet. It gave one the same sense of weight. From this simple seed of an idea arose arguably the most beautiful and mathematically profound theory in all of physics, Einstein’s Field Equations, which predict that matter curves space and that the geometry of our universe is non-Euclidean in general. The theory underlies modern cosmology and has been verified in great detail by multiple heroic and diverse experiments. The first big experiment, which measured the deflection of starlight as it passed by the sun during a total eclipse, is what made Einstein a worldwide celebrity. This one is probably worth two Nobel prizes, but let’s just mark it down for one.

Here we exhaust what most working physicists would immediately recognize as Einstein’s works of genius, and we’re only at 2.5 Nobels. But it is a remarkable fact that Einstein’s work on early atomic theory, what we now call quantum theory, is vastly under-rated. This is partially because Einstein himself downplayed it due to his rejection of the final version of the theory, which he dismissed with the famous phrase, “God does not play dice.” But if one looks at what he actually did, the Nobels keep piling up.

The modern theory of the atom, quantum theory, began in 1900 with the work of the German physicist, Max Planck, who, in what he called “an act of desperation,” introduced into physics a radical notion, quantization of energy. Or so the textbooks say. This is the idea that when energy is exchanged between atoms and radiation (e.g. light), it can only happen in discrete chunks, like a parking meter that only accepts quarters. This idea turns out to be central to modern atomic physics, but Planck didn’t really say this in his work. He said something much more provisional and ambiguous. It was Einstein in his 1905 paper—but then much more clearly in a follow-up paper on the vibrations of atoms in solids in 1907—who really stated the modern principle. It is not clear if Planck himself accepted it fully even a decade after his seminal work (although he was given credit for it by the Nobel Prize committee in 1918). In contrast, Einstein boldly applied it to the mechanical motion of atoms, even when they are not exchanging energy with radiation, and stated clearly the need for a quantized mechanics. So despite the textbooks, Einstein clearly should have shared Planck’s Nobel Prize for the principle of quantization of energy. We are up to 3.0 Nobels for Big Al.

The next one in line is rarely mentioned. After Einstein proposed his particulate theory of light in 1905, he did not adopt the view that light was simply made of particles in the ordinary sense of a localized chunk of matter, like a grain of sand. Instead, he was well aware that light interfered with itself in a similar manner to water waves (a peak can cancel a trough, leading to no wave). In 1909, he came up with a mathematical proof that the particle and wave properties were present in one formula that described the fluctuations of the intensity of light. Hence, he announced that the next era of theoretical physics would see a “fusion” of the particle and wave pictures into a unified theory. This is exactly what happened, but it took fourteen years for the next advance and another three (1926) for it all to fall into place. In 1923, the French physicist Louis de Broglie hypothesized that electrons, which have mass (unlike light) and were always previously conceived of as particles, actually had wavelike properties similar to light. He freely admitted his debt to Einstein for this idea, but when he got the Nobel Prize for “wave-particle” duality in 1929, it was not shared. But it should have been. Another half for Albert, at 3.5 and counting.

From 1911 to 1915 Einstein took a vacation from the quantum to invent General Relativity, which we have already counted, so his next big thing was in 1916 (he didn’t leave a lot of dead time in those days). That was three years after Niels Bohr introduced his “solar system” model of the atom, where the electrons could only travel in certain “allowed orbits” with quantized energy. Einstein went back to thinking about how atoms would absorb light, with the benefit of Bohr’s picture. He realized that once an atom had absorbed some light, it would eventually give that light energy back by a process called spontaneous emission. Without any particular event to cause it, the electron would jump down to a lower energy orbit, emitting a photon. This was the first time that it was proposed that the theory of atoms had such random, uncaused events, a notion that became a second pillar of quantum theory. In addition, he stated that sometimes there was causal emission, that the imposition of more light could cause the atom to release its absorbed light energy in a process called stimulated emission. Forty-four years later, physicists invented a device that uses this principle to produce the purest and most powerful light sources in nature, the LASER (Light Amplified by Stimulated Emission of Radiation). The principles of spontaneous and stimulated emission introduced by Einstein underlie the modern quantum theory of light. One full Prize please—now at 4.5.

After that 1916-1917 work, Einstein had some health problems and became involved in political and social issues for a while, leading to a Nobel batting slump for a few years. (He did still collect some hits, like the prediction of gravitational waves (a double) and the first paper on cosmology and the geometry of the Universe using General Relativity (a triple)). But he came out of his slump with a vengeance in 1924 when he received a paper out of the blue from an unknown Indian, physicist Satyendranath Bose. It was yet another paper about particles of light, and although Bose did not state his revolutionary idea very clearly, reading between the lines, Einstein detected a completely new principle of quantum theory, the idea that all fundamental particles are indistinguishable. This is the standard terminology in physics, but it is actually very misleading. Here, indistinguishability is not the idea that humans can’t tell two photons apart (like identical twins); it is the idea that Nature can’t tell them apart, and in a real sense interchanging the two photons doesn’t count as a different state of light.

When Bose applied this principle to light he didn’t get anything radically new; it was just a different way of thinking about Planck’s original discovery in 1900. But Einstein then took the principle and applied it to atoms for the very first time, with amazing results. He discovered that a simple gas of atoms, if cooled down sufficiently, would cease to obey all the laws that physicists and chemist had discovered for gases over the centuries, and to which no exception had ever been found. Instead, all gases should behave like a weird liquid or super-molecule known as a Bose-Einstein condensate. But remember, Bose had no clue this would happen; he didn’t even try to apply his principle to atoms. It turns out that Einstein condensation underlies some of the most dramatic quantum effects, such as superconductivity, which is needed to make the magnets in MRI machines and has been the basis for five Nobel Prizes. No knowledgeable physicist would dispute that Einstein deserved a full Nobel Prize for this discovery, but I am sure that Einstein would have wanted to share it with Bose (who never did receive the Prize).

So we are at 5.0 “units” of Nobel Prize but seven trips to Stockholm. And this leaves out other arguably Nobel-caliber achievements (Brownian motion as well as the Einstein-Podolsky-Rosen effect, which underlies modern quantum information physics). And wait a minute—when someone shares the Nobel Prize do we refer to them as a “half- Laureate”? No way. Even scientists who get a “measly” third of a Prize are Nobel Laureates for life. Thus by the standard we apply to normal humans, Einstein deserved at least seven Nobel Prizes. So next time you make your fantasy scientist draft, you know who to take at number one.

A. Douglas Stone is author of Einstein and the Quantum: The Quest of The Valiant Swabian.

## The complete line up for Princeton’s Pi Day Celebration

As noted earlier, we are partnering with the Princeton Public Library and the Princeton Tour Company on some author presentations this week. In fact, Chuck Adler is the kick-off for the entire weekend with a talk on Wizards, Aliens, and Starships at the Princeton Public Library on Thursday evening. Physicist Doug Stone will then present about Einstein’s under acknowledged contributions to quantum theory and quantum mechanics on Pi Day proper. We hope you will join the library in welcoming our authors and that you will check out the other fantastic, fun events scheduled over the weekend.

To really give you a sense of what to expect, read this excellent preview from the Princeton Packet.

An Infinitely Delightful Number of Events Planned for the 2014 Pi Day Princeton & Einstein Birthday Party Celebrations!

Thursday, 3.13.14             PI DAY EVE

7:00 p.m.

Academic Celebrity Pi Day Event with Charles Adler at Princeton Public Library

Friday, 3.14.14                 PI DAY & EINSTEIN’S BIRTHDAY

11:00 a.m.

Walking Tour of Einstein’s Neighborhood begins at 116 Nassau Street (the U-Store)

1:59 p.m.

Deadline to submit International Pi Day Princeton Video Contest

3:14 p.m.

Walk a Pi Event at YMCA

3:14 p.m.

Pizza Pi Competition at Princeton Pi – Mayor & Superintendent of Princeton Schools are judges!   Winner receives free pizza for a year!  (Email here to register your middle school aged competitor.)

3:14 p.m.

6:00 p.m.

Academic Celebrity Pi Day Event with famed physicist A. Douglas Stone at Princeton Library

8:00 p.m.

Princeton Light Up The Night Event – Courtesy of Princeton University, Princeton Township and Princeton Pedestrian/Bicyclist Advisory Committee

8:00 p.m.

Outerbridge Ensemble, led by pianist, Steve Hudson at Arts Council of Princeton

Saturday, 3.15.14            OUR UNREAL CELEBRATION DAY

9:00 a.m.

Pie Eating Contest at McCaffrey’s at Princeton Shopping Center and moderated by Princeton comedic celebrity, Adam Bierman. Winner gets bragging rights and all the pie they can eat first thing in the morning!

10:00 a.m.

Kids’ Violin Exhibition at Princeton Library by Princeton Symphony Orchestra (Email here to register your 3yr – 6yr old child)

11:00 a.m.

Einstein Look A Like Contest at Princeton Library. Winner of 13yrs and younger category receives \$314.15  (Email here to register your child.)

11:00 a.m.

“Happy Birthday Einstein!” party at Historical Society of Princeton (Email here to register your child)

12:00 p.m.

International Puzzle Celebrity Guest: Tetsuya Miyamoto, inventor of KENKEN at Princeton Library

12:00 p.m.

Dinky Rides with Einstein at Dinky Station

12:00 p.m

Academic Celebrity Book Signing with Jennifer Berne at Jazams

1:00 p.m.

KENKEN Tournament for Teens (and other teen-spirited humans) at Princeton Library

1:00 p.m.

Pi Recitation Contest at Princeton Library. Winner of Youth Category (aged 7yrs – 13 yrs) receives \$314.15  (Email here to register your child.)

1:30 p.m.

Finding Pi – hands on activities for children 5yrs and up at Princeton Library

2:00 p.m.

Celebrity Book Party with Laura Overdeck at Labyrinth Books

2:15 p.m.

Rubik’s Cube Interactive Demonstration at Princeton Library

2:45 p.m.

Pie Judging Event at Nassau Inn Yankee Doodle Tap Room by Real Possibilities Accounting Firm  First 50 participants to arrive will decide the Best Apple Pie among select Princeton bakeries!

3:14 p.m.

Pie Throwing Event at Palmer Square Green

3:14 p.m.

World Premiere & Announcement of International Video Contest Winner on Facebook . Winning Middle School receives \$314.15

3:30 p.m.

Guided Einstein Tour with Mimi Omiecinski of Princeton Tour Company  begins at Library

4:00 p.m.

“Happy Birthday Einstein!” party at Historical Society of Princeton (Email here to register your child)

4:00 p.m.

Mega Chess Champion Demo & Free Style Play featuring chess champion David Hua

5:00 p.m.

Pi Social & Concert at Princeton Library

ADVANCED REGISTRATION for Pi Day Competitions and EARLY ARRIVAL are preferred to guarantee participation.  All contests are free and open to the public.  Arts Council Performance and Historical Society Birthday Parties require a nominal fee.  See website for additional details.

A detailed description, rules and addresses for Pi Day 2014 Events can be found here!

## Pi Day and Princeton as perfect as…well…pie

As you can well imagine, Einstein is kind of a big deal in Princeton. So, it’s not too surprising that Pi Day, the annual celebration of Einstein’s actual birthday on March 14 (3.14!) that has morphed into a celebration of all things scientific and mathematical, is practically a town-wide holiday. Princeton University Press is partnering with Princeton Public Library on some very exciting events with our authors.

Chuck Adler will kick things off at 7 PM on Pi Day Eve (yes, I may have just invented a new holiday) at the Princeton Public Library with a discussion of his new book Wizards, Aliens, and Starships. Chuck’s specialty is looking at the mathematical underpinnings of some of our favorite works of science fiction and fantasy literature. Why is Hogwart’s always so dark? Could the Weasleys’ flying car really exist? How much longer do we have to wait for Star Trek-style teleportation and/or space elevators? Chuck answers these questions and more with fun, accessible math.

The following day, Doug Stone headlines the Pi Day festivities with a talk about Einstein and the Quantum: The Quest of the Valiant Swabian, a new book that argues that Einstein’s contributions to science have not been fully realized. While we acknowledge Einstein as the father of relativity, we haven’t really understood the scope of the work he did on quantum theory and why he ultimately turned his back on this area of inquiry. Join Doug at the Princeton Public Library at 6 PM as he fills in the gaps and presents a more complete portrait of Einstein’s career than ever available before.

For a complete list of PiDay events in Princeton, including a mysterious pizza pi competition and Einstein walking tours, please visit the official Pi Day Princeton web site.

## Celebrate Pi Day with Princeton University Press

Happy Pi Day, everyone!  In honor of the day, we’ve come up with a reading list that includes some of our favorite Einstein books at Princeton University Press, along with some free chapter excerpts. Celebrate Pi Day and Einstein’s birthday with a great book — we’ve got plenty to choose from!

Einstein’s Jury: The Race to Test Relativity
Jeffrey Crelinsten
This book tells the dramatic story of how astronomers in Germany, England, and America competed to test Einstein’s developing theory of relativity.

The Ultimate Quotable Einstein
Collected and edited by Alice Calaprice
“Without the belief that it is possible to grasp reality with our theoretical constructions, without the belief in the inner harmony of our world, there could be no science. This belief is and will remain the fundamental motive for all scientific creation.” 1938; p. 390
Want more quotes? Check out The Ultimate Quotable Einstein’s Facebook page.

The Meaning of Relativity, Fifth Edition: Including the Relativistic Theory of the Non-Symmetric Field
by Albert Einstein, with a new introduction by Brian Greene

The Collected Papers of Albert Einstein, Volume 13: The Berlin Years: Writings & Correspondence, January 1922 – March 1923 (Documentary Edition)
Edited by Diana Kormos Buchwald, József Illy, Ze’ev Rosenkranz, & Tilman Sauer
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Here’s all of the Collected Papers of Albert Einstein

Einstein’s Miraculous Year: Five Papers That Changed the Face of Physics
Edited and introduced by John Stachel
Far more than just a collection of scientific articles, this book presents work that is among the high points of human achievement and marks a watershed in the history of science.

Albert Einstein, Mileva Maric: The Love Letters
Edited by Jürgen Renn & Robert Schulmann, Translated by Shawn Smith
Informative, entertaining, and often very moving, this collection of letters captures for scientists and general readers alike a little known yet crucial period in Einstein’s life.

The Curious History of Relativity: How Einstein’s Theory of Gravity Was Lost and Found Again
Jean Eisenstaedt
Written with flair, this book poses – and answers – the difficult questions raised by Einstein’s magnificent intellectual feat.

Einstein for the 21st Century: His Legacy in Science, Art, and Modern Culture
Edited by Peter L. Galison, Gerald Holton & Silvan S. Schweber
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In this wide-ranging collection, eminent artists, historians, scientists, and social scientists describe Einstein’s influence on their work, and consider his relevance for the future.

The Little Book of String Theory
Steven S. Gubser
A short, accessible, and entertaining introduction to one of the most talked-about areas of physics today.

The Nature of Space and Time
Stephen Hawking & Roger Penrose
Einstein said that the most incomprehensible thing about the universe is that it is comprehensible. But was he right? On this issue, two of the world’s most famous physicists – Stephen Hawking and Roger Penrose – disagree. Here they explain their positions in a work based on six lectures with a final debate.

Traveling at the Speed of Thought: Einstein and the Quest for Gravitational Waves
Daniel Kennefick
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Daniel Kennefick’s landmark book takes readers through the theoretical controversies and thorny debates that raged around the subject of gravitational waves after the publication of Einstein’s theory.

The Extravagant Universe: Exploding Stars, Dark Energy, and the Accelerating Cosmos
by Robert P. Kirshner
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One of the world’s leading astronomers, Robert Kirshner, takes readers inside a lively research team on the quest that led them to an extraordinary cosmological discovery: the expansion of the universe is accelerating under the influence of a dark energy that makes space itself expand.

Quantum Generations: A History of Physics in the Twentieth Century
Helge Kragh
Combining a mastery of detail with a sure sense of the broad contours of historical change, Kragh has written a fitting tribute to the scientists who have played such a decisive role in the making of the modern world.

Philosophy of Physics: Space and Time
Tim Maudlin
Tim Maudlin’s broad historical overview examines Aristotelian and Newtonian accounts of space and time, and traces how Galileo’s conceptions of relativity and space-time led to Einstein’s special and general theories of relativity.

It’s About Time: Understanding Einstein’s Relativity
by N. David Mermin
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The book reveals that some of our most intuitive notions about time are shockingly wrong, and that the real nature of time discovered by Einstein can be rigorously explained without advanced mathematics.

Dynamics and Evolution of Galactic Nuclei
David Merritt
Deep within galaxies like the Milky Way, astronomers have found a fascinating legacy of Einstein’s general theory of relativity: supermassive black holes. This is the first comprehensive introduction to dynamical processes occurring in the vicinity of supermassive black holes in their galactic environment.

Einstein Before Israel: Zionist Icon or Iconoclast?
by Ze’ev Rosenkranz
Rosenkranz explores a host of fascinating questions, such as whether Zionists sought to silence Einstein’s criticism of their movement, whether Einstein was the real manipulator, and whether this Zionist icon was indeed a committed believer in Zionism or an iconoclast beholden to no one.

Einstein on Politics: His Private Thoughts and Public Stands on Nationalism, Zionism, War, Peace, and the Bomb
Edited by David E. Rowe & Robert Schulmann
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A vivid firsthand view of how one of the twentieth century’s greatest minds responded to the greatest political challenges of his day, this work will forever change our picture of Einstein’s public activism and private motivations.

Einstein’s German World
Fritz Stern