by Tim Chartier
[This article is cross posted from The Huffington Post] As the last days of April unfold, we head into May and the end of the school year. Many classes focus on testing and final grades. Teachers often must focus and ready their students for end-of-the-year testing. Math classes will be asked problem after problem and question after question. In all those classrooms, a thought probably, if not often, races through someone’s mind. Yes, the thought… the one that makes pencils heavier, word problems harder and students wish they were somewhere, anywhere but where they are. There are a lot of ways that thought turns into a question. A common one: “Why study math?” So let’s go and ask, particularly given that we are in April, which is Math Awareness Month. For some, math may be something to beware of rather than be aware of. In fact, that’s precisely the point of the month. Math has many applications, from theoretical to applied. Mathematicians continue to expand the boundaries of what we know mathematically. With the publication of each new issue of a journal, the field of math grows. NBA teams use mathematics to gain a competitive edge over their opponent. Will the better team with better mathematics win? It definitely helped the Oakland As in 2002 with the math that became known as Moneyball. Every day, credit card numbers are encrypted to allow for secure online transactions. Developing methods of encryption that simply cannot be broken with a faster computer comes from mathematics. Studying math enables one to appreciate and possibly understand its applications. Yet one does not need to study math just so the techniques can be used in theoretical or applied settings. Mathematics teaches a way of thinking. Returning to basketball, mathematical formulas won’t pop off the court. Someone must derive them and study them to ensure their usefulness. It can take time to gain such insight. The process toward such understanding is what probably draws many mathematicians to their field. I like to think of it as a path of wonder. For example, I’ve periodically been contacted by ESPN’s Sport Science program to aid in their analysis. They call when they are stuck. When the problem is first presented, my first thought is, “I have no idea how to do this.” And yes, every time I have found a way. Part of this stems from my awareness of that path of mathematical wonder. You don’t have to simply know the answer to a math problem to solve it. In fact, math is usually more interesting when you don’t know how to solve a problem. Would a jigsaw puzzle be fun if it had only two or three pieces? You never know exactly how to fit a 1,000-piece puzzle together when you start, and you won’t always try to fit connecting pieces. It’s a puzzle, so you explore and experiment. Math can be the same way. As such, there is a certain sense of mystery to math. You step into a question and simply stand in the unknown. Then you begin to explore, looking for pieces that fit together. This type of thinking is helpful for life, as it offers its unknowns. In life, you may be forced to stand in the unknown. What questions do you want to explore, and what pieces do you want to try to fit together? Some math ideas are developed through a similar process of exploration. For example, about 10 years ago, I learned how Robert Bosch, Adrianne Herman and Craig Kaplan were creating pictures like the one that I made (after learning their ideas) below. The image above is a portrait of Martin Gardner, who we’ll return to momentarily. Later, it occurred to me that I could make mazes with these images if I used a math formula developed by Leonhard Euler, who lived in the 1700s. Seeing that I could fit these two ideas together — one about a decade old and another hundreds of years old — enabled me to create mazes for my book Math Bytes. Returning again to the NBA, here is such a maze: Click here if you’d like a larger version of the maze. This creative edge of math engages me. It makes teaching math every day at Davidson College a great job. And it makes answering that question “Why learn math?” a question I look forward to being asked. But does this sound like the mathematics you know? If not, then you might want to spend some of these last days of April exploring the Mathematics Awareness Month website. The theme for April 2014 is Mathematics, Magic and Mystery. Each day of the month an engaging idea of mathematics has been unfolded. See the ones already shared and await those yet to come. Learn secrets of mental math, mathematics of juggling, optical illusions, and many more interesting ideas and the math behind them! Want to dig deeper? Note that the theme was chosen as 2014 marks what would have been the 100th birthday of Martin Gardner. Simply put, he engaged millions in his mathematical writing and made mathematicians and children alike aware of the wonders and mysteries of math. So be aware of math! It has many applications, from magic to sports to the theoretical to the historical. I often tell my students in class that if you don’t like math, it may simply be that you haven’t discovered the area of math that fits the way you think! Be careful of sampling from only one part of the math buffet and walking away. A great place to sample many engaging ideas of math is every April with Math Awareness Month. This April, you can learn math and soon engage friends and family with ideas in the mystery and magic of mathematics! So why study math? It develops your mathematical sense, which enables you to see life through that lens. In the process, you hone your ability to think in ways that can make you more aware of life itself. So enjoy these last days of April and be aware of math! Follow Tim Chartier on Twitter: www.twitter.com/timchartier |
Exclusive content from Princeton University Press Click on these thumbnails to open larger JPGs Who are these mathematicians? Leave your guesses in the comments. |
Don’t Beware of Math… Be Aware of It!
THIS IS MATH: Beautiful Geometry
Since this is still April, I will direct you back to the Math Awareness Month Calendar to the window marked The Beautiful Geometry of Crop Circles. You can use a compass and ruler to make beautiful geometric patterns and you can use other media as well. Many of you probably have already done this using a Spirograph.
To find out more about the connection between art and geometry, I will point you to Beautiful Geometry. Eli Maor, who is a mathematician, and Eugen Jost, who is an artist, teamed up to illustrate 51 geometric proofs and assorted mathematical curiosities.
Let’s start with one that most people know about—the Pythagorean theorem or a^{2 }+ b^{2} = c^{2}. No one knows exactly how many proofs there are but Elisha Loomis wrote a book that includes 367 of them. The following illustration is a graphical statement of the theorem that if you draw a square on each of the three sides of a triangle, you will find that the sum of the areas of the two small squares equals the area of the big one.
If you look at the colorful figure below by Eugen Jost, you will see something similar, but much more interesting to look at. The figure above is a 30, 60, 90 degree triangle whereas the one below is a 45, 45, 90 degree triangle.
25 + 25 = 49, Eugen Jost, Beautiful Geometry
Using the Pythagorean formula, we know that 5^{2} + 5^{2 } should equal 7^{2}. Now this means that
5^{2 } + 5^{2} = 7^{2}
25 + 25 = 49
I think we all know that is just not true, yet we know that the formula is correct. What is going on here? It seems that the artist is having a bit of fun with us. Mathematics must be precise but art is not bound by the laws of mathematics. See if you can figure out what happened here.
Where’s the Math?
We know that there are at least 367 different proofs for the Pythagorean theorem but the most famous of them is Euclid’s proof. Eli Maor will walk you through it below, and, he will not try to trick you.
Important Note: We are going to assume you agree that all triangles with the same base and top vertices that lie on a line parallel to the base have the same area. Euclid proved this in book I of the Elements (Proposition 38).
Before he gets to the heart of the proof, Euclid proves a lemma (a preliminary result): the square built on one side of a right triangle has the same area as the rectangle formed by the hypotenuse and the projection of that side on the hypotenuse. The figure above shows a right triangle ACB with its right angle at C. Consider the square ACHG built on side AC. Project this side on the hypotenuse AB, giving you segment AD. Now construct AF perpendicular to AB and equal to it in length. Euclid’s lemma says that area ACHG = area AFED.
To show this, divide AFED into two halves by the diagonal FD. By I 38, area FAD = area FAC, the two triangles having a common base AF and vertices D and C that lie on a line parallel to AF. Likewise, divide ACHG into two halves by diagonal GC. Again by I 38, area AGB = area AGC, AG serving as a common base and vertices B and C lying on a line parallel to it. But area FAD = 1⁄2 area AFED, and area AGC = 1⁄2 area ACHG. Thus, if we could only show that area FAC = area BAG, we would be done.
It is here that Euclid produces his trump card: triangles FAC and BAG are congruent because they have two pairs of equal sides (AF = AB and AG = AC) and equal angles ∠FAC and ∠BAG (each consisting of a right angle and the common angle ∠BAC). And as congruent triangles, they have the same area.
Now, what is true for one side of the right triangle is also true of the other side: area BMNC = area BDEK. Thus, area ACHG + area BMNC = area AFED + area BDEK = area AFKB: the Pythagorean theorem.
THIS IS MATH: Magic Squares, Circles, and Stars
If you have been following the opening of the windows in the Mathematical Awareness Month Poster, you might want to go back to window #1 and review Magic Squares. If you haven’t been there yet, please take a look at it. You will learn how to amaze your friends with your magical math abilities.
Magic squares come in many types, shapes, and sizes. Below you will see a magic square, a magic circle, and a magic star. If you would like to see hundreds more, you might want to check out The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions by Clifford Pickover.
Normal Magic Squares
This is a third-order normal magic square where all of the rows, columns, and diagonals add to 15.
Is this the only solution to this magic square? Can you find others?
You could also have a 4 x 4 square or a 5 x 5 square and so on. How big of a square can you solve?
Magic Circles
Below you will see a magic circle composed of eight circles of four numbers each and the numbers on each circle all add to 18. The thing that makes this magic circle special is that each number is at the intersection of four circles but no other point is common to the same four circles.
Magic Stars
The magic star below is one of the simplest. They can get extremely complicated and also quite beautiful.
So, where’s the math?
Well, you should have noticed already that there are numbers on this page. However, there is more to math than numbers. Let’s add at least one equation.
If we go back to the normal magic square you should know that all these magic squares have the same number of rows and columns, they are n^{2}. The constant that is the same for every column, row, and diagonal is called the magic sum and we will call it M. Now we can figure out what that constant should be. If we use our 3 x 3 square above, we know that n = 3. If we plug our n into the given formula below we will find what our constant has to be.
Since our n = 3, the formula says M = [3 (3^{2} + 1)]/2, which simplifies to 15. For normal magic squares of order n = 4, 5, and 6 the magic constants are, respectively: 34, 65, and 111. What would M be for n = 8? See if you can solve this square. (The figure for the normal square is from Wikipedia.)
THIS IS MATH!: Amaze your friends with The Baby Hummer card trick
Welcome to THIS IS MATH! a new series from math editor Vickie Kearn.
This is the first of a series of essays on interesting ways you can use math. You just may not have thought about it before but math is all around us. I hope that you will take away something from each of the forthcoming essays and that you will pass it on to someone you know.
April is Math Awareness Month and the theme this year is Mathematics, Magic, and Mystery. There is a wonderful website where you will find all kinds of videos, puzzles, games, and interesting facts about math. The homepage has a poster with 30 different images. Each day of the month, a new window will open and reveal all of the wonders for that day.
Today I am going to elaborate on something behind window 3 which is about math and card magic. You will find more magic behind another window later this month. This particular trick is from Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks by Persi Diaconis and Ron Graham. It is a great trick and it is easy to learn. You only need any four playing cards. Take a look at the bottom card of your pack of four cards. Now remember this card and follow the directions carefully:
- Put the top card on the bottom of the packet.
- Turn the current top card face up and place it back on the top of the pack.
- Now cut the cards by putting any amount you like on the bottom of the pack.
- Take off the top two cards (keeping them together) and turn them over and place them back on top.
- Cut the cards again and then turn the top two over and place them back on top.
- Give the cards another cut and turn the top two over together and put them back on top.
- Give the cards a final cut.
- Now turn the top card over and put it on the bottom of the pack.
- Put the current top card on the bottom of the pack without turning it over.
- Finally, turn the top card over and place it back on top of the pack.
- Spread out the cards in your pack. Three will be facing one way and one in the opposite way.
- Surprise! Your card will be the one facing the opposite way.
This trick is called the Baby Hummer and was invented by magician Charles Hudson. It is a variation on a trick invented by Bob Hummer.
So where’s the math?
The math behind this trick covers 16 pages in the book mentioned above.
THIS IS MATH! will be back next week with an article on Math-Pickover Magic Squares!
Celebrate Math Awareness Month with Us
April is Math Awareness month and this year the theme is Mathematics, Magic, and Mystery. To kick off the celebration, visit the Math Awareness web site where you can “open” the days on an advent calendar revealing wonderful math and magic tricks. Today, for example, you can learn a bit about Geometrical Vanishes which make everyday objects appear to … disappear. The videos show how to make everything from dollar bills to chocolate disappear. Tomorrow we’ll start a new series of posts called This Is Math! in which our acquisitions editor for math titles will explain the various ways we encounter math in our everyday lives…and perhaps even add a few tricks of her own!
In the meantime, here’s another Geometrical Vanish courtesy of Tim Chartier, author of Math Bytes:
Play along by printing and cutting out your own set of vanishing PUP Logos. Cut along the solid lines and reverse the top two sections to see a logo magically disappear and reappear. if you have a suggestion for something else you would like to make appear and disappear, leave a comment below and I’ll see if we can get more of these print outs made (keep it clean please!).
Click the smaller images above to open full size images.
New Mathematics Catalog!
Be among the first to browse and download our new mathematics catalog!
Of particular interest is Undiluted Hocus-Pocus: The Autobiography of Martin Gardner. Gardner takes readers from his childhood in Oklahoma to his college days at the University of Chicago, his service in the navy, and his varied and wide-ranging professional pursuits. Before becoming a columnist for Scientific American, he was a caseworker in Chicago during the Great Depression, a reporter for the Tulsa Tribune, an editor for Humpty Dumpty, and a short-story writer for Esquire, among other jobs. Gardner shares colorful anecdotes about the many fascinating people he met and mentored, and voices strong opinions on the subjects that matter to him most, from his love of mathematics to his uncompromising stance against pseudoscience. For Gardner, our mathematically structured universe is undiluted hocus-pocus—a marvelous enigma, in other words. Undiluted Hocus-Pocus offers a rare, intimate look at Gardner’s life and work, and the experiences that shaped both.
Also be sure to note Wizards, Aliens, and Starships: Physics and Math in Fantasy and Science Fiction by Charles L. Adler. From teleportation and space elevators to alien contact and interstellar travel, science fiction and fantasy writers have come up with some brilliant and innovative ideas. Yet how plausible are these ideas—for instance, could Mr. Weasley’s flying car in the Harry Potter books really exist? Which concepts might actually happen, and which ones wouldn’t work at all? Wizards, Aliens, and Starships delves into the most extraordinary details in science fiction and fantasy–such as time warps, shape changing, rocket launches, and illumination by floating candle—and shows readers the physics and math behind the phenomena.
And don’t miss out on Beautiful Geometry by Eli Maor and Eugen Jost. If you’ve ever thought that mathematics and art don’t mix, this stunning visual history of geometry will change your mind. As much a work of art as a book about mathematics, Beautiful Geometry presents more than sixty exquisite color plates illustrating a wide range of geometric patterns and theorems, accompanied by brief accounts of the fascinating history and people behind each. With artwork by Swiss artist Eugen Jost and text by acclaimed math historian Eli Maor, this unique celebration of geometry covers numerous subjects, from straightedge-and-compass constructions to intriguing configurations involving infinity. The result is a delightful and informative illustrated tour through the 2,500-year-old history of one of the most important and beautiful branches of mathematics.
Even more foremost titles in mathematics can be found in the catalog. You may also sign up with ease to be notified of forthcoming titles at http://press.princeton.edu/subscribe/. Your e-mail address will remain confidential!
If you’re heading to the Joint Mathematics Meeting in Baltimore, MD, January 15th-18th, come visit us at booth 407. We’ll be hosting the following book signings:
–The 5 Elements of Effective Thinking, Edward B. Burger and Michael Starbird
Wednesday, January 15th 4:30-5:30 p.m.
–Wizards, Aliens, and Starships: Physics and Math in Fantasy and Science Fiction, Charles L. Adler
Thursday, January 16th 11:00 a.m.-12:00 p.m.
Also stop by 629, the Martin Gardner Centennial Booth. Staffed by a team of enthusiasts who have long been inspired by Gardner, there will be interactive activities and different handouts and puzzles to enjoy each day. Don’t miss “Martin Gardner’s Outreach in His Centennial Year: Mathematics Awareness Month 2014,” a short talk by Colm Mulcahy, Bruce Torrence, and Eve Torrence, Saturday, January 18th at 1:00 p.m. in Convention Center room 346.
Follow @MGardner100th on Twitter for more updates throughout the year, and #JMM14 and @PrincetonUnivPress for updates and information on our new and forthcoming titles throughout the meeting. See you there!
How Mathematical Models Make Sense of Big Data
Tim Chartier, co-author with Anne Greenbaum of Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms, explains how to make sense of big data with numerical analysis. |
You submit a query to Google or watch football bowl games as we enter a new year. In either case, you benefit from mathematical methods that can garner meaningful information from large amounts of data. Such techniques fall in the field of data mining.
Massive datasets are available with every passing minute in our world. For example, during the Oscars in February, the Cirque du Soleil performance resulted in 18,718 tweets in one minute according to TweetReachBlog. While tweets cannot exceed 140 characters in length, their average length is 81.9 characters according to MediaFuturist. So, in one minute, approximately 1.5 million characters zoomed through Twitter. From Wikipedia, we’ll take the average length of a word (in English) to be 5.1 characters. Assuming these Oscar tweets are written in English and conform to the standard length of words, 300,000 words were tweeted in one minute. This is approximately the number of words contained in the entire Hunger Games Trilogy!
Mathematical models and numerical analysis play important roles in data mining. For example, a foundational part of Google’s search engine algorithm is a method called PageRank. In Anne Greenbaum and my book, Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms, published by Princeton University Press, we discuss the PageRank method– both its underlying mathematical model and how it is computed on a computer.
In an exercise in the text, you can develop a system of linear equations in a manner similar to that used by the Bowl Championship Series to rank college football teams (editor – or college basketball teams for March Madness). An important part of this problem is developing the linear system. Our text also discusses the computation challenges of such problems and what numerical methods result in the most accurate results.
Many techniques utilized to solve the large linear systems of data mining are also utilized in engineering and science. The book discusses how large linear systems (containing millions of rows) can derive from problems involving partial differential equations. Again, the book analyzes the efficiency and accuracy of the methods utilized to solve such systems. Such techniques led to the computed animated figures we enjoy in modern movies and aid in simulating the aerodynamics of a car created with computer-aided design.
As stated at the opening of Chapter 1 of the text, “Numerical methods play an important role in modern science. Scientific exploration is often conducted on computers rather than laboratory equipment. While it is rarely meant to completely replace work in the scientific laboratory, computer simulation often complements this work.” As such modern science demands the use and understanding of numerical methods.
Guesstimation #3
We have another Guesstimation special for you. As a reminder, we are posting these problems in support of Math Awareness Month which this year is celebrating Mathematics, Statistics, and the Data Deluge. One way anyone can deal with huge amounts of data is estimating — a skill that is examined and taught in much greater detail in Lawrence Weinstein and John Adam’s book, Guesstimation: Solving the World’s Problems on the Back of a Cocktail Napkin. |
Question #3
On average, how many people are airborne over the US at any given moment?
Hint: Don’t choose 3:00 AM, choose sometime during the day.
If you like this, try your hand at the other Guesstimation problems:
- If all the humans in the world were crammed together, how much area would we require? http://blog.press.princeton.edu/2012/04/12/what-would-you-do-with-all-that-data-guesstimation-1/
- How massive is a mole of cats? http://blog.press.princeton.edu/2012/04/17/guestimation-2/
Solution to Guesstimation #2
The second problem was how massive is a mole of cats. Here is the solution. |
Solution to Guesstimation #2:
[Apologies for the odd formatting on this solution, but math is hard to reproduce in the confines of a blog]
Large Data in the City
John Adam, author of X and the City: Modeling Aspect of Urban Life, explains the role of large data in city life. |
Every ten years a census is taken in this country. Each household is asked to provide pertinent information about the home and its occupants, and such data from across the nation is collected, collated and assembled into a huge database that is ultimately accessible to anyone with access to a computer, or even a smart phone. If you are considering relocating to another city, for example, you may wish to find out about commute times and driving conditions within that city, the cost of housing, crime rate statistics, employment opportunities, aspects of city infrastructure, frequency of cultural events, ease of access by rail and air, and a host of other things. In fact, there is so much statistical information ‘out there’ that it can appear to be overwhelming at times. Sometimes city planners, transportation engineers and others seek to develop mathematical models of such aspects of city life in order to reduce often exceedingly complex problems to their barest essentials. This is valuable because a basic model, even if inaccurate, can often give insight into some of the mechanisms ‘driving’ congestion, or city growth (or decay), and this in turn can lay the groundwork for a more sophisticated model, especially if the former has some predictive capability that can be compared with available data.
Some of these models— of traffic flow, city growth, air pollution, for example—have been addressed at different levels of mathematical sophistication in X and the City: Modeling Aspects of Urban Life. They are designed to give a taste of the kind of mathematical structures that undergird many of the things we take for granted about city life.
John Adam is Professor of Mathematics at Old Dominion University. He is the author of Mathematics in Nature: Modeling Patterns in the Natural World, Guesstimation: Solving the World’s Problems on the Back of a Cocktail Napkin (with Lawrence Weinstein), A Mathematical Nature Walk, and the forthcoming X and the City: Modeling Aspects of Urban Life (all Princeton University Press).
Guesstimation #2
We have another Guesstimation special for you. As a reminder, we are posting these problems in support of Math Awareness Month which this year is celebrating Mathematics, Statistics, and the Data Deluge. One way anyone can deal with huge amounts of data is estimating — a skill that is examined and taught in much greater detail in Lawrence Weinstein and John Adam’s book, Guesstimation: Solving the World’s Problems on the Back of a Cocktail Napkin. |
Question #2
How massive is a mole of cats?
A mole is the number of atoms that weigh that element’s atomic weight in grams. For example, a mole of hydrogen weighs 1 gram and a mole of carbon weighs 12 grams. It is used in chemistry to make sure that there are equivalent numbers of atoms for a chemical reaction.
Compare this to the mass of a mountain, a continent, the moon, the Earth.
If you like this, try your hand at Guesstimation #1: http://blog.press.princeton.edu/2012/04/12/what-would-you-do-with-all-that-data-guesstimation-1/
Solution to Guesstimation #1
As a reminder, the first problem was to estimate how much space all the people on Earth would take up if we were crammed into a small area. Here’s the solution. |
Solution to Guestimation #1:
[Apologies for the odd formatting on this solution, but math is hard to reproduce in the confines of a blog]