Concepts in Color: 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.

We’ve created this slideshow so that you can sample some of the beautiful images in this book, so please enjoy!

Plate 00
Plate 4
Plate 6
Plate 7
Plate 10
Plate 15.1
Plate 16
Plate 17
Plate 18
Plate 19
Plate 20
Plate 21
Plate 22
Plate 23
Plate 24.2
Plate 26.2
Plate 29.1
Plate 29.2
Plate 30
Plate 33
Plate 34.1
Plate 36
Plate 37
Plate 38
Plate 39
Plate 40.2
Plate 44
Plate 45
Plate 47
Plate 48
Plate 49
Plate 50
Plate 51

Beautiful Geometry by Eli Maior and Eugen Jost

"My artistic life revolves around patterns, numbers, and forms. I love to play with them, interpret them, and metamorphose them in endless variations." --Eugen Jost

Figurative Numbers

Plate 4, Figurative Numbers, is a playful meditation on ways of arranging 49 dots in different patterns of color and shape. Some of these arrangements hint at the number relations we mentioned previously, while others are artistic expressions of what a keen eye can discover in an assembly of dots. Note, in particular, the second panel in the top row: it illustrates the fact that the sum of eight identical triangular numbers, plus 1, is always a perfect square.

Pythagorean Metamorphosis

Pythagorean Metamorphosis shows a series of right triangles (in white) whose proportions change from one frame to the next, starting with the extreme case where one side has zero length and then going through several phases until the other side diminishes to zero.

The (3, 4, 5) Triangle and its Four Circles

The (3, 4, 5) Triangle and its Four Circles shows the (3, 4, 5) triangle (in red) with its incircle and three excircles (in blue), for which r = (3+4-5)/2 = 1, r = (5+3-4)/2 = 2, rb = (5+4-3)/2 = 3, and rc = (5+4+3)/2 = 6.

Mean Constructions

Mean Constructions (no pun intended!), is a color-coded guide showing how to construct all three means from two line segments of given lengths (shown in red and blue). The arithmetic, geometric, and harmonic means are colored in green, yellow, and purple, respectively, while all auxiliary elements are in white.

Prime and Prime Again

Plate 15.1, Prime and Prime Again, shows a curious number sequence: start with the top eight-digit number and keep peeling off the last digits one by one, until only 7 is left. For no apparent reason, each number in this sequence is a prime.

0.999... = 1

Celtic Motif 1

Our illustration (Plate 17) shows an intriguing lace pattern winding its way around 11 dots arranged in three rows; it is based on an old Celtic motif.

Seven Circles a Flower Maketh

Parquet

Plate 19, Parquet, seems at first to show a stack of identical cubes, arranged so that each layer is offset with respect to the one below it, forming the illusion of an infinite, three-dimensional staircase structure. But if you look carefully at the cubes, you will notice that each corner is the center of a regular hexagon.

Girasole

Plate 20, Girasole, shows a series of squares, each of which, when adjoined to its predecessor, forms a rectangle. Starting with a black square of unit length, adjoin to it its white twin, and you get a 2x1 rectangle. Adjoin to it the green square, and you get a 3x2 rectangle. Continuing in this manner, you get rectangles whose dimensions are exactly the Fibonacci numbers. The word Girasole ("turning to the sun" in Italian) refers to the presence of these numbers in the spiral arrangement of the seeds of a sunflower - a truly remarkable example of mathematics at work in nature.

The Golden Ratio

Plate 21 showcases a sample of the many occurrences of the golden ratio in art and nature.

Pentagons and Pentagrams

Homage to Carl Friedrich Gauss

Gauss's achievement is immortalized in his German hometown of Brunswick, where a large statue of him is decorated with an ornamental 17-pointed star (Plate 23 is an artistic rendition of the actual star on the pedestal, which has deteriorated over the years); reportedly the mason in charge of the job thought that a 17-sided polygon would look too much like a circle, so he opted for the star instead.

Celtic Motif 2

Plate 24.2 shows a laced pattern of 50 dots, based on an ancient Celtic motif. Note that the entire array can be crisscrossed with a single interlacing thread; compare this with the similar pattern of 11 dots (Plate 17), where two separate threads were necessary to cover the entire array. As we said before, every number has its own personality.

Metamorphosis of a Circle

Plate 26.2, 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 ins till constant. The entire sequence is thus a metamorphosis from square to circle and back.

Reflecting Parabola

Ellipses and Hyperbolas

When you throw two stones into a pond, each will create a disturbance that propagates outward from the point of impact in concentric circles. The two systems of circular waves eventually cross each other and form a pattern of ripples, alternating between crests and troughs. Because this interference pattern depends on the phase difference between the two oncoming waves, the ripples invariably form a system of confocal ellipses and hyperbolas, all sharing the same two foci. In this system, no two ellipses ever cross one another, nor do two hyperbolas, but every ellipse crosses every hyperbola at right angles. The two families form an orthogonal system of curves, as we see in plate 29.2.

3/3=4/4

Euler's e

Plate 33, Euler's e, gives the first 203 decimal places of this famous number - accurate enough for most practical applications, but still short of the exact value, which would require an infinite string of nonrepeating digits. In the margins there are several allusions to events that played a role in the history of e and the person most associated with it, Leonhard Euler: an owl ("Eule" in German); the Episcopal crosier on the flag of Euler's birthplace, the city of Basel; the latitude and longitude of Königsberg (now Kaliningrad in Russia), whose seven bridges inspired Euler to solve a famous problem that marked the birth of graph theory; and an assortment of formulas associated with e

Spira Mirabilis

Epicycloids

Plate 36 shows a five-looped epicycloid (in blue) and a prolate epicycloid (in red) similar to Ptolemy's planetary epicycles. In fact, this latter curve closely resembles the path of Venus against the backdrop of the fixed stars, as seen from Earth. This is due to an 8-year cycle during which Earth, Venus, and the Sun will be aligned almost perfectly five times. Surprisingly, 8 Earth years also coincide with 13 Venusian years, locking the two planets in an 8:13 celestial resonance and giving Fibonacci aficionados one more reason to celebrate!

Nine Points and Ten Lines

Our illustration Nine Points and Ten Lines (plate 37) shows the point-by-point construction of Euler's line, beginning with the three points of defining the triangle (marked in blue). The circumference O, the centroid G, and the orthocenter H are marked in green, red, and orange, respectively, and the Euler line, in yellow. We call this a construction without words, where the points and lines speak for themselves.

Inverted Circles

Steiner's Prism

Plate 39 illustrates several Steiner chains, each comprising five circles that touch an outer circle (alternately colored in blue and orange) and an inner black circle. The central panel shows this chain in its inverted, symmetric "ball-bearing" configuration.

Line Design

Plate 40.2 shows a Star of David-like design made of 21 line parabolas.

Gothic Rose

Plate 44, Gothic Rose, shows a rosette, a common motif on stained glass windows like those one can find at numerous places of worship. The circle at the center illustrates a fourfold rotation and reflection symmetry, while five of the remaining circles exhibit threefold rotation symmetries with or without reflection (if you disregard the inner details in some of them). The circle in the 10-o'clock position has the twofold rotation symmetry of the yin-yang icon.

Symmetry

Pick's Theorem

Plate 47 shows a lattice polygon with 28 grid points (in red) and 185 interior points (in yellow). Pick's formula gives us the area of this polygon as A = 185 + 28/2 - 1 = 198 square units.

Morley's Theorem

Variations on a Snowflake Curve

Plate 49 is an artistic interpretation of Koch's curve, starting at the center with an equilateral triangle and a hexagram (Star of David) design but approaching the actual curve as we move toward the periphery.

Sierpinski's Triangle

The Rationals Are Countable!

In a way, [Cantor] accomplished the vision of William Blake's famous verse in Auguries of Innocence:

To see the world in a grain of sand,
And heaven in a wild flower.
Hold infinitely in the palm of your hand,
And eternity in an hour.

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Click here to sample selections from the book.

Place Your Bets: Tim Chartier Develops FIFA Foe Fun to Predict World Cup Outcomes

Tim ChartierTim Chartier, author of Math Bytes: Google Bombs, Chocolate-Covered Pi, and Other Cool Bits in Computing has turned some mathematical tricks to help better predict the outcome of this year’s World Cup in Brazil.

Along with the help of fellow Davidson professor Michael Mossinghoff and Whittier professor Mark Kozek, Chartier developed FIFA Foe Fun, a program that enables us ordinary, algorithmically untalented folk to generate a slew of possible match outcomes. The tool weighs factors like penalty shoot-outs and the number of years of matches considered, all with the click of a couple buttons. Chartier used a similar strategy in his March Mathness project, which allowed students and basketball fans alike to create mathematically-produced brackets – many of which were overwhelmingly successful in their predictions.

Although the system usually places the most highly considered teams, like Brazil, Germany, and Argentina at the top, the gadget is still worth a look. Tinker around a bit, and let us know in the comments section how your results pan out over the course of the competition.

In the meantime, check out the video below to hear Chartier briefly spell out the logic of the formula.

Happy calculating!

June, summer, and Princeton University Press in the movies

Friends of Princeton University Press,

With June here, and summer finally upon us, our thoughts go to pleasant things—vacations, beaches, baseball, and the summer movie season.

ivory tower
Hodges_ImitationGame_Poster
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Princeton University Press has a special movie connection this summer–and beyond.

For starters, the soon-to-be-released documentary Ivory Tower, about the financial crisis in higher education, features prominently one of our authors, Andrew Delbanco, whose widely admired 2012 book, College: What It Was, Is, and Should Be, has been at the center of the debates over the future of higher education. Those who saw Page One, the acclaimed documentary about The New York Times and the challenges besetting newspapers, will be familiar with the work of Andrew Rossi, who made the film, Ivory Tower. Journalist Peter Coy reviews it in the current issue of Bloomberg Business Week, and mentions Andy Delbanco and our book.

Another PUP book forms the basis of the November 2014 release, The Imitation Game, the story of Alan Turing, the cryptologist who cracked the Enigma code during World War II and was later tortured for his homosexuality. The movie is based on our 2012 biography by Andrew Hodges, Alan Turing: The Enigma. The Imitation Game sports an all-star cast including Benedict Cumberbatch, Keira Knightly, and Charles Dance. We will be re-releasing Hodges’ biography under the title, The Imitation Game, in September. A related PUP book is Alan Turing’s Systems of Logic: The Princeton Thesis, edited in 2012 by Andrew Appel of the Princeton School of Engineering.  Our poster for The Imitation Game generated huge interest last week at Book Expo in New York.

Speaking of all-star casts, the third movie with a connection to a forthcoming PUP book is Interstellar, also to be released in November, and starring Matthew McConaughey, Anne Hathaway, Jessica Chastain, Matt Damon, and Michael Caine. The premise of Interstellar is based on the work of PUP author and Caltech theoretical physicist Kip Thorne, who is credited as a consultant and executive producer of the film. His forthcoming book, with Stanford’s Roger Blandford, is Modern Classical Physics. Kip Thorne has another PUP connection, serving as he does on the Executive Committee of the Einstein Papers Project.

See you at the movies,

Peter J Dougherty
Director

THIS IS MATH: Wake up and Smell the Functions

Have you ever heard anyone say “I need a cat nap” or “I feel so much better after a power nap”? Why is this true? And how long should such naps last? If you ask my cat, 1.5 hours is the magic number.* When he wakes up he stretches and gets right back to chasing a ball. But does the length of the nap  really matter?  The answer is yes, and here’s why.

In humans, the sleep cycle lasts 90 minutes. It begins with REM sleep–where you often dream–and then progresses into non-REM sleep.  Throughout the four stages of non-REM sleep our bodies repair themselves. If we awake at the end of one of these 90 minute cycles, we feel refreshed and ready to go. But if we wake up in the middle of the non-REM cycle (when we are in a deep sleep) we feel groggy.

catnon-REM sleep

cat2This one did not do the math!

So, where’s the math?

Because our REM/non-REM stages cycle every 1.5 hours, that tells us that we can model the sleep stage S by a periodic function f(t)—one whose values repeat after an interval of time T, called the period—and that the period T = 1.5 hours.  f(t)—one whose values repeat after an interval of time T, called the period—and that the period T = 1.5 hours. If we let the awake stage be S = 0 and assign each following stage to the next negative number, for example, stage 1 as -1, and so on, we can construct a trigonometric function that results in the following graph.

fernandez_fig1-1

The peaks at the top of the graph show that 1.5, 3, 4.5, 6, and 7.5 hours are the optimal amounts of time to sleep. Don’t worry if you go a bit over or under, but you might want to keep the snooze on for only 5 minutes.

If you want to see more of the math, you can find it in Everyday Calculus: Discovering the Hidden Math All around Us by Oscar Fernandez. This problem is in Chapter 1 which is available free here [PDF].

*According to the College of Veterinary Medicine at The Ohio State University, cats do not have the daily sleep-wake cycle that we and many other animals have. Rather, they sleep and wake frequently throughout the day and night. This is because cats in the wild need to hunt as many as 20 small prey each day; they must be able to rest between each hunt so they are ready to pounce quickly when prey approaches. Although their sleep cycle differs from ours, they do have a cycle and need to be ready to go as soon as they wake up.

March Mathness Postscript

march-mathness1-150x150[1]The ups and downs of March Madness are slipping into memory, but we have one final postscript to write. Who won the March Mathness challenge put forth by Tim Chartier to his students at Davidson College?

We are delighted to announce that Robin Malloch, a history major who is graduating this month, picked the best brackets out of Dr. Chartier’s class. She has joined Teach for America and will be teaching Middle School math in Charlotte, North Carolina this fall. Thankfully, before she heads off to do the good work of teaching algebra and geometry to eager (or truthfully, not so eager, we’re guessing) 7th and 8th graders, she has provided us with some insight into her bracket strategy during March Mathness:

I will describe my method as best I can. I made several brackets with different methodologies–one based on basketball gut, one on pure math, and others with a combination.  My purely mathematical bracket did not fair that well. My two combination brackets which were a bit more arbitrary fared the best (89th and 92nd percentile). For those, I used the math to inform my basketball knowledge. Any time my basketball bracket contradicted the math ranking or if the rankings were nearly tied, I would look into the game further to see if either team had key players injured, how player matchups, what experience/track record the coaches had, etc.

As for my actual math rankings, I used Colley. I tried to account for strength of schedule, so I broke the season into three parts. The last couple games of the season are the conference tournament, which I gave more value to than any other games of the season. Then I broke the rest of the season in half: the first half being primarily out-of-conference games and the second half being primarily in-conference ones. For teams in a competitive conference (more than 3 teams from the conference in the tournament), I weighted the second half more heavily. For teams in weaker conferences, I weighted the first half more heavily when teams would likely face tougher competition.  This method actually made my rankings closely resemble the 1-16 rankings produced by the NCAA selection committee.  I enjoyed applying math brackets and looking at basketball with a new lens.

For more on the various bracketology methods Tim Chartier teaches his students, please check out our March Mathness page.

What is the reality behind the race for scientific talent? Watch this EPI event with Michael Teitelbaum to find out

Also, in a related review of Michael Teitelbaum’s book Falling Behind? from Spectrum Magazine, published by the IEEE, they had this fun little quiz:

Okay, here are your choices: 1957, 1982, and 2014. Match each year to when the following statements were made:

a. “It is pretty generally realized that our country faces a serious scientific and engineering manpower shortage. We have at present about half the engineers which we need, and each year we are graduating only about half our annual needs.”

b. “Science, technology, engineering and math form the foundation of the global economy. Yet, … if educational trends continue, fewer qualified candidates will be available to support growth in these areas.”

c. “We appear to be raising a generation of Americans, many of whom lack the understanding and the skills necessary to participate fully in the technological world in which they live and work.”

To see the answers and to read their review, please visit http://spectrum.ieee.org/riskfactor/at-work/tech-careers/exposing-the-roots-of-the-perpetual-stem-crisis-

To learn more about the boom and bust cycles of STEM education, please read Falling Behind?

A Mathematical Flower Bouquet for Mother’s Day

This guest post is from Oscar Fernandez, author of Everyday Calculus: Discovering the Hidden Math All around Us. Looking for a last minute Mother’s Day gift idea? Consider sending her a beautiful bouquet of Mathematical Flowers–not only are they pretty and allergy free, but mom will surely be impressed by your math skills, too!

This post is cross-posted at Huffington Post.


Did you know that you can create beautiful flowers with an equation? Before I tell you what that equation is, let me convince you first. The picture below shows a particular graph of this equation (on the left) along with an actual flower (on the right).

Picture 1

Pretty neat huh? But wait, there’s more! Let’s change just one number in this equation and see what happens.

picture2

A fourth petal! (And yes, I’ve also changed the color.) Thus far there is one clear takeaway: these “mathematical flowers” are strikingly similar to the real thing. Let me give you one last example:

picture 3

Picture4

Picture5
Complete your Mathematical Bouquet with these exclusive flowers from Oscar Fernandez.

So, what’s the equation that can seemingly duplicate many of the pretty flowers we see around us, and what is the number that controls how many petals are generated? The equation is

equation

and by changing n we control the number of petals. But to generate recognizable flowers we need to choose n to be a natural number (1,2,3,…). In that case the equation above produces the “mathematical flowers” in the three pictures you saw. For example, in the first picture n=3, in the second picture n=2 (not 4), and in the third picture n=5. When n is not a natural number, you get curves that don’t look anything like flowers. For example, here’s the peanut-like curve you get for n=1/6:

loop

I produced all of these graphs on my computer, but you don’t have to have special software to do it yourself. You can now share the joy of receiving “mathematical flowers” with the moms in your life by using the widget I wrote that produces several of these virtual flowers. Here it is (it’s free to use):

http://www.wolframalpha.com/widgets/view.jsp?id=ef9a949ab675fac1828cc63c80a0202f

I’m sure mom will be as surprised as you were to see just how realistic these mathematical flowers look. And if your mom is a math enthusiast, or would just like to understand how the r-equation produces those flowers, read on and you’ll be able to explain it to her.

First, have a look at the diagram below.

graph

Here I’m showing you two different ways to plot a point, in this case (a,b). In the method we’re most familiar with you move a units to the right from the origin (the point where the two axes intersect) and b units up from there. That’s called graphing in Cartesian coordinates. But you can also graph the same point in polar coordinates. To do so, you first move r units away from the origin, and then rotate counter-clockwise by an angle q.

One quick way to generate lots of points is to graph an equation, like y = x2. This rule tells us that the y-value is the square of the x-value. For example, when x=1 we get y=1, and when x=2 we get y=4. Therefore, the points (1,1) and (2,4) (in Cartesian coordinates) are on the graph of y = x2.

Our “mathematical flowers” are just the graphs of the equation r = cos(nθ), but where each point is plotted in the polar coordinate system. For example, when q = 0 we get r=1, meaning that the point (1,0) is on the graph of every mathematical flower, regardless of what n is. As q ranges from 0 to 2p, the graph of r = cos(nθ) generates the flower-like curves I showed you in the first three pictures. (I took the extra steps of shading in the graphs and putting them on a nice background.)

Now that you know how those “mathematical flowers” were generated, let me leave you with an interesting thought. Instead of creating curves with an equation and then comparing the results to real flowers, we could reverse this reasoning and come to a much more interesting possibility: maybe flowers follow mathematical laws as they grow! For many flowers—and indeed many other examples of natural phenomena—this is indeed true. This surprising fact is just another example of hidden mathematics all around us.

Have a happy Mother’s Day,

Oscar E. Fernandez

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

052914Reimer

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.

Don’t Beware of Math… Be Aware of It!

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.

2014-04-20-gardnerTSP.png
 

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:

2014-04-20-bBallMaze.png
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

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mathMazes_Page_1

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Click on these thumbnails to open larger JPGs

Who are these mathematicians?

Leave your guesses in the comments.

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 a2 + b2 = c2. 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.

pythagorean 1

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.

Plate 5 NEW

25 + 25 = 49, Eugen Jost, Beautiful Geometry

Using the Pythagorean formula, we know that 52 + 52  should equal 72. Now this means that

52  + 52 = 72

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.

06-02_maor_fig

 

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.

 

squares

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.

circles

Magic Stars

The magic star below is one of the simplest. They can get extremely complicated and also quite beautiful.

star

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 n2. 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.

equation

 Since our n = 3, the formula says M = [3 (32 + 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.)

 

 

 

 

 

Princeton University Press’s best-selling audio books

The Five Elements of Effective ThinkingWe’re changing things up a bit. Each week we list the best-selling titles according to BookScan, but today we’re focusing on our audio titles. These are Princeton University Press’s best-selling audio books for the final quarter of 2013. Click through to listen to samples or to add them to your book queue.

  1. The 5 Elements of Effective Thinking by Edward Burger & Michael Starbird
  2. Women Don’t Ask by Linda Babcock and Sara Laschever
  3. Einstein and the Quantum by A. Douglas Stone
  4. Lost Enlightenment by S. Frederick Starr
  5. The Founders’ Dilemmas by Noam Wasserman