Looking forward to spring warblers? Join The Warbler Guide at these events in Philadelphia

We’re looking forward to spring with three fantastic warbler events this weekend at John Heinz National Wildlife Refuge at Tinicum. Tom Stephenson and Scott Whittle, co-authors of The Warbler Guide, will be on-hand to give workshops on warbler ID and guide a few walks.

Capture

 

Click here to download a PDF flyer for these events.

2014 Lawrence Stone Lecture Series to Feature Lorraine Daston

This year’s Lawrence Stone Lecture Series, featuring Lorraine Daston, will be held April 29 thru May 1. Entitled “Rules: A Short History of What We Live By,” the lecture will feature three different sessions:

April 29 — Rules of Iron, Rules of Lead: A Prehistory of an Indispensable and Impossible Genre

April 30 — Rules Go Rigid: Natural Laws, Calculations, and Algorithms

May 1 — Rules, Rationality, and Reasonableness

The events will be held in 010 East Pyne Building at 4:30 p.m.

The lecture series is co-sponsored by Princeton University Press, Princeton University’s History Department, and the Shelby Cullom Davis Center for Historical Studies. The Center was founded by former chair of the History Department, Lawrence Stone (1919-91). Each year, the lecture series features Princeton’s Lawrence Stone Visiting Professor, and the professor’s three lectures are then included in a book published by Princeton University Press.

Lorraine Daston is the executive director of the Max Planck Institute for the History of Science in Berlin as well as a visiting professor on the Committee on Social Thought at the University of Chicago.

april 15 lecture

Show Me the Money: PUP Authors on the Role of Wealth in Politics

How much can your buck get you in politics today? A forthcoming paper by PUP author Martin Gilens and Benjamin Page puts a finer point on the idea that money can enhance your influence on political policy. In fact, the authors give us an actual number for gauging that influence. Fifteen times — that is how much more important the collective preferences of “economic elites” are than those of other citizens, Gilens and Page found. Yes, you read that correctly.

Gilens and Page’s paper, which will run in Perspectives on Politics, explains how they came to this conclusion, studying “1,779 instances between 1981 and 2002 in which a national survey of the general public asked a favor/oppose question about a proposed policy change.” They write:

Multivariate analysis indicates that economic elites and organized groups representing business interests have substantial independent impacts on U.S. government policy, while average citizens and mass-based interest groups have little or no independent influence.

In a recent article on the Washington Post‘s Monkey Cage blog, PUP author and co-director of the Center for the Study of Democratic Institution, Larry Bartels, examines Gilens and Page’s findings and other research that contributes to what we know about the effects of money on political influence. Check out the article for Bartels’ take on this issue.

In this midterm election year, following the McCutcheon v. FEC ruling, money is on everyone’s minds. Looking to brush up on the theories and research behind these issues? You can read more from Bartels and Gilens — we invite you to read the sample chapters and other supplementary materials from their award-winning Princeton University Press books. We have also included a peek at political scientists Kay Lehman Schlozman, Sidney Verba and Henry E. Brady’s systematic examination of political voice in America.

 

 bookjacketRead Chapter One here. Using a vast swath of data spanning the past six decades, Unequal Democracy debunks many myths about politics in contemporary America, using the widening gap between the rich and the poor to shed disturbing light on the workings of American democracy. Larry Bartels shows the gap between the rich and poor has increased greatly under Republican administrations and decreased slightly under Democrats, leaving America grossly unequal. This is not simply the result of economic forces, but the product of broad-reaching policy choices in a political system dominated by partisan ideologies and the interests of the wealthy. In this interview, Bartels answers tough questions about the effect of money in America.

 

bookjacket “We are the 99%” has quickly become the slogan of our political era as growing numbers of Americans express concern about the disappearing middle class and the ever-widening gap between the super-rich and everyone else. Has America really entered a New Gilded Age? What are the political consequences of the growing income gap? Can democracy survive such vast economic inequality? These questions dominate our political moment–and Larry Bartels provides answers backed by sobering data.Princeton Shorts are brief selections taken from influential Princeton University Press books and produced exclusively in ebook format. Providing unmatched insight into important contemporary issues or timeless passages from classic works of the past, Princeton Shorts enable you to be an instant expert in a world where information is everywhere but quality is at a premium.

 

 bookjacketPreview the introduction here. Can a country be a democracy if its government only responds to the preferences of the rich? Affluence and Influence definitively explores how political inequality in the United States has evolved over the last several decades and how this growing disparity has been shaped by interest groups, parties, and elections.With sharp analysis and an impressive range of data, Martin Gilens looks at thousands of proposed policy changes, and the degree of support for each among poor, middle-class, and affluent Americans. His findings are staggering: when preferences of low- or middle-income Americans diverge from those of the affluent, there is virtually no relationship between policy outcomes and the desires of less advantaged groups. In contrast, affluent Americans’ preferences exhibit a substantial relationship with policy outcomes whether their preferences are shared by lower-income groups or not. Yet Gilens also shows that under specific circumstances the preferences of the middle class and, to a lesser extent, the poor, do seem to matter. In particular, impending elections–especially presidential elections–and an even partisan division in Congress mitigate representational inequality and boost responsiveness to the preferences of the broader public.

 

bookjacketRead Chapter One here. Politically active individuals and organizations make huge investments of time, energy, and money to influence everything from election outcomes to congressional subcommittee hearings to local school politics, while other groups and individual citizens seem woefully underrepresented in our political system.Drawing on numerous in-depth surveys of members of the public as well as the largest database of interest organizations ever created–representing more than thirty-five thousand organizations over a twenty-five-year period — The Unheavenly Chorus conclusively demonstrates that American democracy is marred by deeply ingrained and persistent class-based political inequality. The well educated and affluent are active in many ways to make their voices heard, while the less advantaged are not. This book reveals how the political voices of organized interests are even less representative than those of individuals, how political advantage is handed down across generations, how recruitment to political activity perpetuates and exaggerates existing biases, how political voice on the Internet replicates these inequalities–and more.

 

A miracle in Arcata, CA – report from the our stalwart sales rep Steve Ballinger

k10185We knew 1177 B.C. by Eric Cline would be a big book for us, but it has become a run-away seller (even appearing on the Canadian best-seller list the week of its release)  since its release in late March and we have just ordered a third printing! Not only is it topping the archaeology charts on Amazon, but it’s also seeing some great sell through in independent bookstores. Case in point, check out this terrific story from our West coast sales representative, Steve Ballinger.

Greetings – I’m back after a long haul through California. The rain was Weather Channel worthy at times. The orders were great. Somehow I didn’t get one good meal out of the trip. Carl’s Jr and Jack in Box were the top spots for cuisine last week. The route of the sales calls took me over the mountain ranges and forests, past the vast orchards of angry farmers, up and down the dreaded Grapevine. Yet, a miracle happened.

Friday, after visiting the bookstores in Ukiah, I drove on up to Northtown Books in Arcata. Northtown Bookstore could easily fit in the hipster haven of Brooklyn. The parking is better though. It was one of those situations of selling the list at the front counter and pausing while Dante, the store owner, handled customers. We got the first two books in the order, The Extreme Life of the Sea and The Transformation of the World. But then he passed on the ancient history titles even after much whimpering and trying on my part. I could just visualize the 1177 B.C. title in the store.

Well, a few customers came and went, a twenty-something mom with tousled hair asked about #7 and #8 of the Unicorn series and he said he could get it in for her. We got to Lost Animals and then the phone rang. I could hear him say, “As it turns out I am having a meeting with the publisher’s representative right now.” He came back shaking his head in amazement, the customer on the phone, had called to see if he could order 1177 B.C. Princeton’s new book. We went from zero to 2. A miracle. It was some guy named Darius.

A miracle in Arcata, CA.

Bob Geddes to Give Talk, Tour, and Book Signing at the Institute for Advanced Study

Calling all Princeton-area architecture fans: Bob Geddes will be giving a lecture, tour, and book signing of Fit: An Architect’s Manifesto, at the Institute for Advanced Study in Princeton, NJ, on Saturday, April 5th, from 10:00 AM to 1:30 PM (EDT), sponsored by DOCOMOMO Philadelphia and DOCOMOMO NY/Tri-State.

Tickets and full event details are available via Eventbrite ($20 for DOCOMOMO members / $25 for non-members / FREE for IAS faculty, scholars, and staff).

Photo: Amy Ramsey, Courtesy of Institute for Advanced StudyMake it New, Make it Fit

The architecture of Geddes, Brecher, Qualls, and Cunningham (GBQC) has been largely overlooked in recent years—despite a remarkable and influential body of work beginning with their runner-up submission for the Sydney Opera House (1956). As significant contributors (along with Louis Kahn) to the “Philadelphia School,” GBQC’s efforts challenged modernist conceptions of space, functional relationships, technology, and—with an urbanist’s eye—the reality of change over time.

To explore the thinking behind the work, founding partner Robert Geddes, FAIA, will speak about his recent publication, Fit: An Architect’s Manifesto. In addition, Geddes will guide a tour through the venue for his talk, the Institute of Advanced Study’s Simmons Hall—a GBQC masterwork of 1971. Geddes will also participate in an informal discussion with participants during lunch at the IAS Cafeteria.

Schedule
10:00-10:30am      Dilworth Room. Event check in. Coffee served.
10:30-11:15am        Make it New, Make it Fit Lecture by Bob Geddes
11:15-11:50am        Building Tour
11:50-12:10pm       Lunch at cafeteria where discussion continues
12:10-1:00pm         Lunch and discussion
1:00-1:30pm           Wrap up and book signing.

Parking
LOT ‘B’ enter through West Building. When you arrive at the site, please bring a copy of your tickets, either printed or displayed on your mobile phone.

About the speaker
Robert Geddes is dean emeritus of the Princeton School of Architecture and founding partner of GBQC—recipient of the AIA’s Firm of the Year Award in 1979. Educated under Walter Gropius at Harvard’s Graduate School of Design, Geddes returned to his native Philadelphia in 1950 where he began his work as an educator at the University of Pennsylvania.

Pi Day: Where did π come from anyway?

This article is extracted from Joseph Mazur’s fascinating history of mathematical notation, Enlightening Symbols. For more Pi Day features from Princeton University Press, please click here.

 


 

k10204[1]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.

Ernst Mach mused:

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 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.


cookingFor 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.

 pi maor

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

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.

Free #PiDay E-Cards from The Ultimate Quotable Einstein

Send #PiDay Greetings with these free ecards featuring Einstein’s thoughts on birthdays as found in The Ultimate Quotable Einstein, edited by Alice Calaprice.


einstein birthday 2 web



einstein birthday web



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.


Tfts56[1]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!

Princeton University Press’s Best-selling Books for the Past Week

These are the best-selling books for the past week.

 

The Son Also Rises: Surnames and the History of Social Mobility by Gregory Clark
The Limits of Partnership: U.S.-Russian Relations in the Twenty-First Century by Angela E. Stent
Fragile by Design: The Political Origins of Banking Crises and Scarce Credit by Charles W. Calomiris & Stephen H. Haber
The Box: How the Shipping Container Made the World Smaller and the World Economy Bigger by Marc Levinson
GDP: A Brief but Affectionate History by Diane Coyle
Tesla: Inventor of the Electrical Age by W. Bernard Carlson Tesla: Inventor of the Electrical Age by W. Bernard Carlson
The Founder’s Dilemmas: Anticipating and Avoiding the Pitfalls That Can Sink a Startup by Noam Wasserman
The 5 Elements of Effective Thinking by Edward Burger and Michael Starbird
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How to Solve It: A New Aspect of Mathematical Method
by G. Polya
Rare Birds of North America by Steve Howell, Ian Lewington, and Will Russell