## Tim Chartier and the Mega Menger

How many business cards are needed to complete a level 3 Menger Sponge? What is a level 3 Menger Sponge? Tim Chartier, author of Math Bytes: Google Bombs, Chocolate Covered Pi, and Other Cool Bits in Computing, explains.

In a Huffington Post article, A Million Business Cards Presents a Math Challenge, Chartier asks readers to go in their wallet and check for  business cards. If there aren’t any, “they may be part of a worldwide math challenge. Over the past month, people around the world have been building a mathematical structure out of more than a million business cards.” That mathematical structure is a Mega Menger, but before we get there, let’s discuss what a level 1 Menger Sponge is.

A level 1 Menger Sponge is a fractal consisting of twenty cubes. Each cube is made up of 6 business cards, so a level 1 Menger needs 120 business cards. A level 2 Menger is created using 20 level 1 Mengers, a level 3 Menger is made with 20 level 2 Mengers, so on and so on.

Here at the Press, we don’t have a million business cards to complete a Mega Menger, but we do have a lot of books. Using Chartier’s code (which you can play with here: http://lifeislinear.davidson.edu/sierpinski.html), we were able to create 2-D versions of a Menger Sponge called a Sierpinski Carpet using our jacket images. Below, see if you can figure out which books are featured in the fractal images below! Click the images to see the related title.

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## Invisible in the Storm wins the 2015 Louis J. Battan Author’s Award, American Meteorological Society

Congratulations to Ian Roulstone & John Norbury, co-authors of Invisible in the Storm: The Role of Mathematics in Understanding Weather, on winning the 2015 Louis J. Battan Author’s Award given by the American Meteorological Society.

The prize is “presented to the author(s) of an outstanding, newly published book on the atmospheric and related sciences of a technical or non-technical nature, with consideration to those books that foster public understanding of meteorology in adult audiences.” In the announcement of the prize, the committee said Invisible in the Storm “illuminates the mathematical foundation of weather prediction with lucid prose that provides a bridge between meteorologists and the public.”

 Invisible in the Storm The Role of Mathematics in Understanding Weather Ian Roulstone & John Norbury

## This is how you survive the zombie apocalypse

Williams College math professor Colin Adams risks life and limb to record these survival guide videos. Ready your gear–armor, baseball bat, calculus textbook–and prepare for the onslaught.

# Part 2: Escaping zombies in hot pursuit.

Credit: PBS’s NOVA and director Ari Daniel.

## Grab your M&Ms and ace math this year with Math Bytes

In this segment from WCCB in Charlotte, NC, Tim Chartier shows how math can be both educational and delicious! This experiment is taken directly from his recent book Math Bytes: Google Bombs, Chocolate-Covered Pi, and Other Cool Bits in Computing. There are lots of other hands-on experiments that are suitable for aspirational mathematicians of all ages in the book.

## Paying It Forward, Using Math: Oscar Fernandez’s ‘Everyday Calculus’ Donated to Libraries in Franklin County, PA

What a week!

It was recently announced that one of our books, Everyday Calculus by Oscar Fernandez, is to be donated by the United Way of Franklin County, in partnership with the Franklin County Library System, to public libraries all throughout Franklin County. The decision recognizes the 2013 Campaign Chair, Jim Zeger, who has demonstrated a dedication to service and a “willingness to teach others” during the course of his four-year tenure on the board of directors.

But the choice of text was far from random; Everyday Calculus was selected “because of the need for materials that support financial and mathematical literacy within our library systems,” says Mr. Zeger. He’s one to know; before coming to United Way, Zeger studied math at Juniata College and taught mathematics at the Maryland Correctional Institute. He also served for a number of years as part of the Tuscarora School District school board, and “is very supportive and understanding of the value of relating and connecting applied math to students.”

Bernice Crouse, executive director of the Franklin County Library System, accepted the books and has found them a place in each County library, including the bookmobile, in order to make them more accessible to readers. According to Crouse, this book fits perfectly with Pennsylvania Library Association’s PA Forward initiative, which “highlights Financial Literacy as a key to economic vitality in Pennsylvania.”

Mr. Fernandez is reportedly “delighted” and “honored” by the decision, and looks forward to further collaborating with United Way.

## Cue up The Bangles and join us as we Count Like an Egyptian with Fox News

Interested in learning more about how to do math like an ancient Egyptian, check out David Reimer’s book Count Like an Egyptian.

## Save the Date — David Reimer, “Count Like an Egyptian” at the Princeton Public Library on May 29

Join the fun on May 29 at 7:00 PM as the Princeton Public Library and Princeton University Press welcome David Reimer, professor of mathematics and statistics at The College of New Jersey, for an exploration of the world of ancient Egyptian math and the lessons it holds for mathematicians of all levels today.

Prof. Reimer will present a fun introduction to the intuitive and often-surprising art of ancient Egyptian math. Learn how to solve math problems with ancient Egyptian methods of addition, subtraction, multiplication and division and discover key differences between Egyptian math and modern day calculations (for example, in spite of their rather robust and effective mathematics, Egyptians did not possess the concept of fractions).

Following the lecture, Prof. Reimer will sign copies of his new book, Count Like an Egyptian. Copies of will be available for purchase at the lecture or you can pick up a copy ahead of time at Labyrinth Books.

## #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.

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.

## 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.

## Q&A: Eli Maor and Eugen Jost reveal the surprising inspirations and process of Beautiful Geometry

In January, we will publish Beautiful Geometry by Eli Maor and with illustrations by Eugen Jost. The book is equal parts beauty and mathematics and we were grateful that both authors took time to answer some questions for our readers. We hope you enjoy this interview.

Look for a slideshow previewing the art from the book in the New Year.

PUP: How did this book come to be? Where did you get the idea to create a book of Beautiful Geometry?

Eli Maor [EM]: It all started some five years ago through a mutual acquaintance of us by the name Reny Montandon, who made me aware of Eugen’s beautiful geometric artwork. Then, in 2010, I met Eugen in the Swiss town of Aarau, where I was invited to give a talk at their famous Cantonal high school where young Albert Einstein spent two of his happiest years. We instantly bonded, and soon thereafter decided to work together on Beautiful Geometry. To our deep regret, Reny passed away just a few months before we completed the project that he helped to launch. We dearly miss him.

PUP: Eugen, where did your interest in geometric artwork come from? Are you a mathematician or an artist by training?

Eugen Jost [EJ]: I have always loved geometry. For me, mathematics is an endless field in which I can play as an artist and I am also intrigued by questions that arise between geometry and philosophy.

For example, Euclid said: A point is that which has no part, and a line is what has no widths. Which raises questions for me as an artist—Do geometric objects really exist, then? Are the points and lines that I produce really geometric objects? With Adobe Illustrator you can transform points and lines into “non-existing” paths, which brings you deeper into geometry.

But, ultimately, it’s not my first purpose to illustrate mathematics—I just want to play and I’m happy if the onlooker of my pictures starts to play as well.

PUP: How did you decide which equations to include?

EM: It’s not so much about equations as about theorems. Of course, geometry has hundreds, if not thousands of theorems to choose from, so we had to be selective. We didn’t have any particular rules to guide us; sometimes Eugen would choose a particular theorem for which he had some artistic design in mind; in other cases we based our selection on theorems with an interesting history, or just for their simplicity. But we always had the artistic point of view in mind: our goal was to showcase the beauty of geometry and make it known to a wide public.

EJ: I think I’m seldom looking for mathematical topics with which I can make pictures. It’s the other way round: mathematics and geometry comes to me. A medieval town, the soles of your shoes, wheel rims, textile printing, patterns in pine cones: wherever I look I see mathematics and beautiful geometry. Euclid’s books among many others provide me with ideas, too.

Often I develop ideas when I’m walking in the woods with our dog or I’m scribbling in my small black diaries while I’m sitting in trains. At home I transform the sketches into pictures.

PUP: Were there any theorems you didn’t get to include but would have liked to?

EM: Yes, there were many theorems we would have liked to have included, but for practical reasons we decided to limit ourselves to about 50 chapters. That leaves us plenty of subjects for Beautiful Geometry II 🙂

PUP: Are there some theorems that simply didn’t lend themselves to artistic depictions?

EJ: In our book you won’t find many three-dimensional objects depicted. In my art I tend to create flat objects (circles, triangles, squares …) on surfaces and three-dimensional objects in space. Therefore we avoided theorems that have to do with space—with few exceptions.

PUP: Can you describe the layout of the book? How is it organized?

EM: We followed a more or less chronological sequence, but occasionally we grouped together subjects that are logically related to one another, so as to make the flow of ideas easier to follow.

PUP: The collaboration between you and Eugen Jost reminds me of a lyricist and musician—how did the two of you work together? Did you write and he created art alongside or did he have art already done and you wrote for it?

EM: Yes, that comparison between a lyricist (in opera we call it librettist) and a musician is very apt. As I mentioned earlier, we didn’t have a rigid guideline to follow; we just played with many ideas and decided which ones to include. We exchanged over a thousand emails between us (yes, Eugen actually counted them!) and often talked on Skype, so this aspect of our collaboration was easy. I can’t imagine having done that twenty years ago…

EJ: Communicating with Eli was big fun. He has so many stories to tell and very few of them are restricted to geometry. In 2012, when we thought our manuscript was finished, Eli and his wife came to Switzerland. For many days we travelled and hiked around lakes, cities and mountains with our manuscripts in our book sacks. We discussed all the chapters at great length. In some chapters, the relationship between Eli’s text and my pictures is very close and the art helps readers understand the text. In others, the connection is looser. Readers are invited to get the idea of a picture more or less independently—sort of like solving a riddle.

PUP: Most of the art in the book are original pieces by Eugen, right? Where did you find the other illustrations?

EM: Most of the artwork is Eugen’s work. He also took excellent photographs of sites with interesting geometric patterns or a historical significance related to our book. I have in mind, for example, his image of the famous headstone on the grave of Jakob Bernoulii in the town of Basel, Switzerland, which has the wrong spiral engraved on it—a linear spiral instead of a logarithmic one!

PUP: That is fascinating and also hints at What Eugen mentioned earlier–math and the beauty of math is hidden in plain sight, all around us. Are there other sites that stand out to you, Eugen?

EJ: Yesterday I went to Zurich. While I was walking through the streets I tried to find answers to your question. Within half an hour I found over a dozen examples of the mathematics that surrounds us:

• The clock face of the church St. Peter is the biggest one in Europe. This type of clock face links our concept of time with ancient Babylonians who invented a time system based on the numbers 12, 24, 12×30, 3600.
• In the Bahnhofstrasse of Zurich there is a sculpture by Max Bill in which many big cuboids form a wonderful ensemble. Max Bill was the outstanding artist of the so called “Zürcher Konkrete”; his oeuvre is full of mathematics.
• I saw a fountain and the jet of water formed wonderful parabolas in the air. Where the water entered the pool, it produced concentric circles.
• There are literally hundreds of ellipses on any street in Zurich, or any other town for that matter. Every wheel you see is an ellipse—unless you look at it at a precise angle of 90 degrees.
• Even under our feet, you can find mathematics. Manhole covers very often have wonderful patterns that you can interpret mathematically.

Mathematics and Beautiful Geometry is everywhere around us—we just have to open our eyes.

PUP: What equation does the artwork on the cover of the book illustrate? Can you give us a quick “reading” of it?

EM: The front cover shows the Sierpinski Triangle, named after Polish mathematician Waclaw Sierpinski (1882-1969). It is a bizarre construction, a triangle-like shape that has zero area but an infinite perimeter. This is but one of many fractal-type patterns that have become popular thanks to the ability to create them with modern computers, often adding dazzling color to make them into exquisite works of art.

PUP: Eugen, what is your process to create a piece of art like this?

EJ: Unfortunately, the Sierpinski Triangle was not my own idea, but I was awe-struck by the idea of a shape with an infinitely little area and an infinite perimeter, so I started to think about how it could be depicted. Like most of the pictures in the book, I created this piece of art on a computer. At the same time, I was immersed in other mathematical ideas like the Menger sponge, the Hilbert curve, the Koch snowflake. Of course Sierpinski himself and countless others must have sketched similar triangles, but that was the challenge for our book: to take theorems and to transform them into independent pieces of art that transcend mere geometric drawings.

PUP: Eli, do you have a favorite piece of art in the book?

EM: Truth be told, every piece of art in our book is my favorite! But if I must choose one, I’ll go for the logarithmic spiral that Eugen realized so beautifully in Plate 34.1; it is named spira mirabilis (“The miraculous spiral”), the name that Bernoulli himself used to describe his favorite curve.

PUP: We asked the mathematician to pick his favorite piece of art, so it is only fair that we should ask the artist to pick his favorite theorem. Do you have one, Eugen?

EJ: Being Eli’s first reader for the last three years has been a joy because he tells history and stories in our book. I like the chapter on the surprising theorem of George Alexander Pick, in part because of the biographical details. Eli describes how Pick bonded with Albert Einstein in Prague—imagine Einstein and Pick playing the violin and the viola together! Sadly, Pick ended his life in the concentration camp at Theresienstadt, but he left behind this wonderful contribution to mathematics.

PUP: Are there any particularly surprising pieces of art in the book that might have a good backstory or illustrate a particularly memorable equation?

EM: Again, this puts me in a difficult position – to choose among so many interesting subjects that are covered in our book. But if I have to pick my choice, I’d go for Plate 26.1, entitled PI = 3. The title refers to what has been called the “Biblical value of PI” and refers to a verse in the Bible (I Kings vii 23) which, if taken literally, would lead to a value of PI = 3. Our plate shows this value surrounded by the famous verse in its original Hebrew.

PUP: Writing a book is a long process filled with countless hours of hard work. Do any moments from this period stand out in particular?

EJ: I remember sitting in a boat on the lake of Thun with Eli and his wife Dalia in the spring of 2012. Eli and I were pondering on the chapter doubling the cube. As I mentioned before, I do not like to draw three-dimensional objects on a flat surface, so I didn’t want to depict a traditional cube. I was playing instead with the unrolling of two cubes, one having the double volume of the other. Eli was not sure this would work, but on the boat we thoroughly discussed it and all of a sudden Eli said, “Eugen, you have sold me on it.” I hadn’t heard that expression before. I then had a queasy conscience because I didn’t know whether I should have been flexible enough to leave my own idea for a better one. Ultimately, the art came out quite well and really illustrates the Delian problem.

A chance encounter gave me a sense of the broad appeal of the book. I was sitting at a table with a highly trained engineer and I told him about our book. I then tried to explain the theorem of Morley: In any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle. His response was “You don’t suppose that you could solve any statical problem with this, do you?”

PUP: Math is often quite visual, but where did the idea of making it both visual and beautiful come from?

EM: We are not the first, of course, to point out the visual beauty of many geometric theorems or patterns, but usually these gems are depicted in stark, black-and-white designs of lines and curves. Adding colors to these designs – and sometimes a bit of humor and imagination – makes all the difference between strictly mathematical beauty and a true work of art. This is what has really inspired us in writing our book.

PUP: Who do you hope reads Beautiful Geometry?

EM: We aim at a broad audience of students, teachers and instructors at all levels, and above all, laypersons who enjoy the beauty of patterns and are not afraid of a simple math equation here and there. We hope not to disappoint them!

##### Wassim Haddad, Winner of the 2014 Pendray Aerospace Literature Award, American Institute of Aeronautics and Astronautics

Professor Wassim Haddad of the School of Aerospace Engineering and chair of the Flight Mechanics and Control Discipline at Georgia Institute of Technology “has been selected to receive the 2014 Pendray Aerospace Literature Award. This is the highest honor in literature bestowed by the American Institute of Aeronautics and Astronautics (AIAA). The award is presented for an outstanding contribution or contributions to aeronautical and astronautical literature in the relatively recent past.”

The citation of Prof. Haddad’s award reads “Paramount and fundamental contributions to the literature of dynamical systems and control for large-scale aerospace systems.”

Prof. Haddad’s award is given in part for the research in his book, co-authored with Sergey G. Nersesov and published by PUP in 2011: Stability and Control of Large-Scale Dynamical Systems: A Vector Dissipative Systems Approach

Modern complex large-scale dynamical systems exist in virtually every aspect of science and engineering, and are associated with a wide variety of physical, technological, environmental, and social phenomena, including aerospace, power, communications, and network systems, to name just a few. This book develops a general stability analysis and control design framework for nonlinear large-scale interconnected dynamical systems, and presents the most complete treatment on vector Lyapunov function methods, vector dissipativity theory, and decentralized control architectures.

Wassim M. Haddad is a professor in the School of Aerospace Engineering and chair of the Flight Mechanics and Control Discipline at Georgia Institute of Technology.

## Join MoMath in New York City on October 26 for a celebration of Martin Gardner

Martin Gardner, an acclaimed popular mathematics and science writer and author of Undiluted Hocus-Pocus: The Autobiography of Martin Gardner, would have had his 99th birthday this month. In honor of this special occasion, the mathematical community is putting together a number of birthday celebrations.

MoMath joins the fun on October 26th from 10:00 – 5:00 with a Celebration of the Mind.

At this family-friendly event, math fans of all ages will enjoy some close-up magic tricks, explore favorite Gardner puzzles, and make their own hexaflexagon to take home (how many people can say they have their own hexaflexagon?!). As an added challenge, try to spot the two exhibits that Gardner asked Museum directors to include in MoMath.

Later that evening, MoMath will welcome Martin Gardner’s son James Gardner and a panel of experts for a discussion:

 Event: Who is Martin Gardner? A Conversation with Friends, Colleagues, and Family Date and Time: Saturday, October 26, 6:30 pm What is it? A panel of people who knew Martin Gardner well will share their favorite stories about him and reveal just how important his contributions have been to mathematics and to math lovers around the world. Ask questions, talk with the presenters, and share your own memories and stories. Who is participating? James Gardner (University of Oklahoma, Martin Gardner’s son) John Conway (Emeritus Professor of Mathematics, Princeton University) Mark Setteducati (President, Gathering 4 Gardner) Neil Sloane (The OEIS Foundation and Rutgers University) Colm Mulcahy (Spelman College and Author of Mathematical Card Magic: Fifty-Two New Effects) Location: National Museum of Mathematics 11 East 26th Street, New York, NY 10010 Contact: (212) 542-0566 | info@momath.org

Space will fill up for this event, so please pre-register here: http://momath.org/about/upcoming-events/)

There are many Celebration of Mind events taking place around the world. Check out the map (http://celebrationofmind.org/) to find events close to you.

Come and celebrate the joy of math!

## Martin Gardner Celebration At Princeton

Martin Gardner, an acclaimed popular mathematics and science writer and author of Undiluted Hocus-Pocus: The Autobiography of Martin Gardner, would have had his 99th birthday this month. In honor of this special occasion, the mathematical community is putting together a number of events celebrating this Gardner.

At Princeton University on October 25th from 6:30 – 8:30 PM in the Friend Center, Room 101, there is a free public lecture by Tadashi Tokieda on toy Models. He will share with you some unique toys he has made and collected, and show you the mathematics and physics behind them. Following the lecture, a panel of people who knew Martin Gardner well will share their favorite stories about him. You will have time to ask questions and talk with the presenters and share your memories as well.

Mark Setteducati (President, Gathering 4 Gardner) Panel Moderator
James Gardner (University of Oklahoma, Martin Gardner’s son)
John Conway (Emeritus Professor of Mathematics, Princeton University)
Colm Mulcahy (Spelman College and Author of Mathematical Card Magic: Fifty-Two New Effects)

There are many Celebration of Mind events taking place around the world. Check out the map (http://celebrationofmind.org/) and you can find event close to you.

Come and celebrate the joy of math!