Tag Archives: Vi Hart

Pi Digit, Pi Patterns, and Pi Day Anthem

pivolant1

Painting by Renée Othot for Simon Plouffe’s birthday.

Welcome to this week’s Math Munch!

It’s here—the Pi Day of the Century happens on Saturday: 3-14-15!

How will you celebrate? You might check to see if there are any festivities happening in your area. There might be an event at a library, museum, school, or university near you.

(Here are some pi day events in NYC, Baltimore, San Francisco, Philadelphia, Houston, and Charlotte.)

 

John Conway at the pi recitation contest in Princeton.

John Conway at the pi recitation contest in Princeton.

There’s a huge celebration here in Princeton—in part because Pi Day is also Albert Einstein’s birthday, and Albert lived in Princeton for the last 22 years of his life. One event involves kids reciting digits of pi and and is hosted by John Conway and his son, a two-time winner of the contest. I’m looking forward to attending! But as has been noted, memorizing digits of pi isn’t the most mathematical of activities. As Evelyn Lamb relays,

I do feel compelled to point out that besides base 10 being an arbitrary way of representing pi, one of the reasons I’m not fond of digit reciting contests is that, to steal an analogy I read somewhere, memorizing digits of pi is to math as memorizing the order of letters in Robert Frost’s poems is to literature. It’s not an intellectually meaningful activity.

I haven’t memorized very many digits of pi, but I have memorized a digit of pi that no one else has. Ever. In the history of the world. Probably no one has ever even thought about this digit of pi.

And you can have your own secret digit, too—all thanks to Simon Plouffe‘s amazing formula.

plouffe

Simon’s formula shows that pi can be calculated chunk by chunk in base 16 (or hexadecimal). A single digit of pi can be plucked out of the number without calculating the ones that come before it.

Wikipedia observes:

The discovery of this formula came as a surprise. For centuries it had been assumed that there was no way to compute the nth digit of π without calculating all of the preceding n − 1 digits.

Check out some of Simon's math art!

Check out some of Simon’s math art!

Simon is a mathematician who was born in Quebec. In addition to his work on the digits of irrational numbers, he also helped Neil Sloane with his Encyclopedia of Integer Sequences, which soon online and became the OEIS (previously). Simon is currently a Trustee of the OEIS Foundation.

There is a wonderful article by Simon and his colleagues David Bailey, Jonathan Borwein, and Peter Borwein called The Quest for Pi. They describe the history of the computation of digits of pi, as well as a description of the discovery of their digit-plucking formula.

According to the Guinness Book of World Records, the most digits that someone has memorized and recited is 67,890. Unofficial records go up to 100,000 digit. So just to be safe, I’ve used an algorithm by Fabrice Bellard based on Simon’s formula to calculate the 314159th digit of pi. (Details here and here.) No one in the world has this digit of pi memorized except for me.

Ready to hear my secret digit of pi? Lean in and I’ll whisper it to you.

The 314159th digit of pi is…7. But let’s keep that just between you and me!

And just to be sure, I used this website to verify the 314159th digit. You can use the site to try to find any digit sequence in the first 200 million digits of pi.

Aziz and Peter's patterns.

Aziz & Peter’s patterns.

Next up: we met Aziz Inan in last week’s post. This week, in honor of Pi Day, check out some of the numerical coincidences Aziz has discovered in the early digits in pi. Aziz and his colleague Peter Osterberg wrote an article about their findings. By themselves, these observations are nifty little patterns. Maybe you’ll find some more of your own. (This kind of thing reminds me of the Strong Law of Small Numbers.) As Aziz and Peter note at the end of the article, perhaps the study of such little patterns will one day help to show that pi is a normal number.

And last up this week, to get your jam on as Saturday approaches, here’s the brand new Pi Day Anthem by the recently featured John Sims and the inimitable Vi Hart.

Bon appetit!

Math Meets Art, Quarto, and Snow!

Welcome to this week’s Math Munch!

article-0-19F9E81700000578-263_634x286… And, if you happen to write the date in the European way (day/month/year), happy Noughts and Crosses Day! (That’s British English for Tic-Tac-Toe Day.) In Europe, today’s date is 11/12/13– and it’s the last time that the date will be three consecutive numbers in this century! We in America are lucky. Our last Noughts and Crosses Day was November 12, 2013 (11/12/13), and we get another one next year on December 13 (12/13/14). To learn more about Noughts and Crosses Day and find out about an interesting contest, check out this site. And, to our European readers, happy Noughts and Crosses Day!

p3p13Speaking of Noughts and Crosses (or Tic-Tac-Toe), I have a new favorite game– Quarto! It’s a mix of Tic-Tac-Toe and another favorite game of mine, SET, and it was introduced to me by a friend of mine. It’s quite tricky– you’ll need the full power of your brain to tackle it. Luckily, there are levels, since it can take a while to develop a strategy. Give it a try, and let us know if you like it!

BRUCKER-ICS-DARKRYE-SQUARE

Looking to learn about some new mathematical artists? Check out this article, “When Math Meets Art,” from the online magazine Dark Rye. It profiles seven mathematical artists– some of whom we’ve written about (such as Erik and Martin Demaine, of origami fame, and Henry Segerman), and some of whom I’ve never heard of. The work of string art shown above is by artist Adam Brucker, who specializes in making “unexpected” curves from straight line segments.

gauss17_smallAnother of my favorites from this article is the work of Robert Bosch. One of his specialities is making mosaics of faces out of tiles, such as dominoes. The article features his portrait of the mathematician Father Sebastien Truchet made out of the tiles he invented, the Truchet tiles. Clever, right? The mosaic to the left is of the great mathematician Gauss, made out of dominoes. Check out Robert’s website to see more of his awesome art.

Finally, it snowed in New York City yesterday. I love when it snows for the first time in winter… and that got me wanting to make some paper snowflakes to celebrate! Here’s a video by Vi Hart that will teach you to make some of the most beautiful paper snowflakes.

Hang them on your windows, on the walls, or from the ceiling, and have a very happy wintery day! Bon appetit!

Digital Art, Mastermind, and Pythagoras

Welcome to this week’s Math Munch… on (approximately) Math Munch’s second birthday! Hooray!

Check out this video of mathematical art made by artist Nathan Selikoff:

Cool, right? This piece is called “Beautiful Chaos.” The curves on the screen are made from equations (if you’ve ever graphed a line or a parabola you’ll know what I mean). As the viewer waves her hands around, the equations change– and as the equations change, so do the curves! The result is something that might remind you of the images your computer makes when you play music on it or maybe of something you’d make using a spirograph. All in all, a beautiful and interactive piece of mathematical art.

nathanNathan lives and works as a mathematical artist in Orlando, Florida. As he writes on his website, Nathan uses computer code along with other materials to make art that plays with the mathematical ideas of space, motion, and interaction between objects. To see more of how Nathan does this, check out his giant, interactive marionette or this song that explores the first, second, third, and fourth dimensions:

My school is really lucky to be hosting Nathan this week! We didn’t want any of you, dear readers, to miss out on the excitement, though– so Nathan has kindly agreed to answer your interview questions! Got a question for Nathan? Write it in the box below. He’ll answer seven of your best questions in two weeks!

565px-MastermindNext up, who doesn’t love to play Mastermind? It’s a great combination of logic, patterns, and trickery… but I just hate having to use all those tiny pegs. Well, guess what? You can play it online— no pegs (or opponent) necessary!

As I was playing Mastermind, I started wondering about strategy. What’s the best first guess to make? If I were as smart as a computer, is there a number of guesses in which I could guess any Mastermind code? (This kind of question reminds me of God’s Number and the Rubik’s cube…)

Well, it turns out there is a God’s Number for Mastermind – and that number is five. Just five. If you played perfectly and followed the strategy demonstrated by recreational mathematician Toby Nelson on his website, you could guess ANY Mastermind code in five guesses or less. Toby shares many more interesting questions about Mastermind on his website— I suggest you check it out.

What ARE those irrational numbers, so weird that they get their own bubble??

What ARE those irrational numbers, so weird that they get their own bubble??

Finally, sometime in your mathematical past you may have heard of irrational numbers. These are numbers like the square-root of 2 or pi or e that can’t be written as a fraction– or so people claim. When you start thinking about this claim, however, it may seem strange. There are A LOT of fractions– and none of them equal the square-root of 2? Really? What kind of number is that? It seems like only an irrational person would believe that, at least without proof.

Vi Hart to the rescue! Irrational numbers were encountered long, long ago by the ancient Greek mathematician (and cult leader) Pythagoras– and he didn’t like them much. In this great video, Vi tells all about Pythagoras and the controversial discovery of numbers that aren’t fractions.

If you didn’t follow her explanation of why the square-root of 2 is irrational on your first watch, don’t worry– it’s a complicated idea that’s worth a second (or third or fourth) run-through.

Thanks for a great two years of Math Munch! Bon appetit!

Maths Ninja, Folding Fractals, and Pi Fun

Welcome to this week’s Math Munch!

ninjaFirst up, have you ever been stuck on a gnarly math problem and wished that a math ninja would swoop in and solve the problem before it knew what hit it?  Have you ever wished that you had a math dojo who would impart wisdom to you in cryptic but, ultimately, extremely timely and useful ways?  Well, meet Colin Beverige, a math (or, as he would say, maths) tutor from England who writes a fun blog called Flying Colours Maths.  On his blog, he publishes a weekly series called, “Secrets of the Mathematical Ninja,” in which the mathematical ninja (maybe Colin himself?  He’s too stealthy to tell)  imparts nuggets of sneaky wisdom to help you take down your staunchest math opponent.

colin_bridgeFor example, you probably know the trick for multiplying by 9 using your fingers – but did you know that there’s a simple trick for dividing by 9, too?  Ever wondered how to express thirteenths as decimals, in your head?  (Probably not, but maybe you’re wondering now!)  Want to know how to simplify fractions like a ninja?  Well, the mathematical ninja has the answers – and some cute stories, too.  Check it out!

A picture of the Julia set.

A picture of a Julia set.

Next, I find fractals fascinating, but – I’ll admit it – I don’t know much about them.  I do know a little about the number line and graphing, though.  And that was enough to learn a lot more about fractals from this excellent post on the blog Hackery, Math, and Design by Steven Wittens.  In the post How to Fold a Julia Fractal, Steven describes how the key to understanding fractals is understanding complex numbers, which are the numbers we get when we combine our normal numbers with imaginary numbers.

complex multiplicationNow, I think imaginary numbers are some of the most interesting numbers in mathematics – not only because they have the enticing name “imaginary,” but because they do really cool things and have some fascinating history behind them.  Steven does a really great job of telling their history and showing the cool things they do in this post.  One of the awesome things that imaginary numbers do is rotate.  Normal numbers can be drawn on a line – and multiplying by a negative number can be thought of as changing directions along the number line.  Steven uses pictures and videos to show how multiplying by an imaginary number can be thought of as rotating around a point on a plane.

here comes the julia set

A Julia set in the making.

The Julia set fractal is generated by taking complex number points and applying a function to them that squares each point and adds some number to it.  The fractal is the set of points that don’t get infinitely larger and larger as the function is applied again and again.  Steven shows how this works in a series of images.  You can watch the complex plane twist around on itself to make the cool curves and figures of the Julia set fractal.

Steven’s blog has many more interesting posts.  Check out another of my favorites, To Infinity… and Beyond! for an exploration of another fascinating, but confusing, topic – infinity.

Finally, a Pi Day doesn’t go by without the mathematicians and mathematical artists of the world putting out some new Pi Day videos!  Pi Day was last Thursday (3/14, of course).  Here’s a video from Numberphile in which Matt Parker calculates pi using pies!

In this video, also from Numberphile, shows how you only need 39 digits of pi to make really, really accurate measurements for the circumference of the observable universe:

Finally, it wouldn’t be Pi Day without a pi video from Vi Hart.  Here’s her contribution for this year:

Bon appetit!

Harmonious Sum, Continuous Life, and Pumpkins

Welcome to this week’s Math Munch!

We’ve posted a lot about pi on Math Munch – because it’s such a mathematically fascinating little number.  But here’s something remarkable about pi that we haven’t yet talked about. Did you know that pi is equal to four times this? Yup.  If you were to add and subtract fractions like this, for ever and ever, you’d get pi divided by 4.  This remarkable fact was uncovered by the great mathematician Gottfried Wilhelm Leibniz, who is most famous for developing the calculus.  Check out this interactive demonstration from the Wolfram Demonstrations Project to see how adding more and more terms moves the sum closer to pi divided by four.  (We’ve written about Wolfram before.)

I think this is amazing for a couple of reasons.  First of all, how can an infinite number of numbers add together to make something that isn’t infinite???  Infinitely long sums, or series, that add to a finite number have a special name in mathematics: convergent series.  Another famous convergent series is this one:

The second reason why I think this sum is amazing is that it adds to pi divided by four.  Pi is an irrational number – meaning it cannot be written as a fraction, with whole numbers in the numerator and denominator.  And yet, it’s the sum of an infinite number of rational numbers.

In this video, mathematician Keith Devlin talks about this amazing series and a group of mathematical musicians (or mathemusicians) puts the mathematics to music.

This video is part of a larger work called Harmonious Equations written by Keith and the vocal group Zambra.  Watch the rest of them, if you have the chance – they’re both interesting and beautiful.

Next up, Conway’s Game of Life is a cellular automaton created by mathematician John Conway.  (It’s pretty fun: check out this to download the game, and this Munch where we introduce it.)  It’s discrete – each little unit of life is represented by a tiny square.  What if the rules that determine whether a new cell is formed or the cell dies were applied to a continuous domain?  Then, it would look like this:

Looks like a bunch of cells under a microscope, doesn’t it?  Well, it’s also a cellular automaton, devised by mathematician Stephan Rafler from Nurnberg, Germany.  In this paper, Stephan describes the mathematics behind the model.  If you’re curious about how it works, check out these slides that compare the new continuous version to Conway’s model.

Finally, I just got a pumpkin.  What should I carve in it?  I spent some time browsing the web for great mathematical pumpkin carvings.  Here’s what I found.

A pumpkin carved with a portion of Escher’s Circle Limit.

A pumpkin tiled with a portion of Penrose tiling.

A dodecapumpkin from Vi Hart.

I’d love to hear any suggestions you have for how I should make my own mathematical pumpkin carving!  And, if you carve a pumpkin in a cool math-y way, send a picture over to MathMunchTeam@gmail.com!

Bon appetit!

Rectangles, Explosions, and Surreals

Welcome to this week’s Math Munch!

What is 3 x 4?   3 x 4 is 12.

Well, yes. That’s true. But something that’s wonderful about mathematics is that seemingly simple objects and problems can contain immense and surprising wonders.

How many squares can you find in this diagram?

As I’ve mentioned before, the part of mathematics that works on counting problems is called combinatorics. Here are a few examples for you to chew on: How many ways can you scramble up the letters of SILENT? (LISTEN?) How many ways can you place two rooks on a chessboard so that they don’t attack each other? And how many squares can you count in a 3×4 grid?

Here’s one combinatorics problem that I ran across a while ago that results in some wonderful images. Instead of asking about squares in a 3×4 grid, a team at the Dubberly Design Office in San Francisco investigated the question: how many of ways can a 3×4 grid can be partitioned—or broken up—into rectangles? Here are a few examples:

How many different ways to do this do you think there are? Here’s the poster that they designed to show the answer that they found! You can also check out this video of their solution.

In their explanation of their project, the team states that “Design tools are becoming more computation-based; designers are working more closely with programmers; and designers are taking up programming.” Designing the layout of a magazine or website requires both structural and creative thinking. It’s useful to have an idea of what all the possible layouts are so that you can pick just the right one—and math can help you to do it!

If you’d like to try creating a few 3×4 rectangle partitions of your own, you can check out www.3x4grid.com. [Sadly, this page no longer works. See an archive of it here. -JL, 10/2016]

Next up, explosions! I could tell you about the math of the game Minesweeper (you can play it here), or about exploding dice. But the kind of explosion I want to share with you today is what’s called a “combinatorial explosion.” Sometimes a problem that appears to be an only slightly harder variation of an easy problem turns out to be way, way harder. Just how BIG and complicated even simple combinatorics problems can get is the subject of this compelling and also somewhat haunting video.

Donald Knuth

Finally, all of this counting got me thinking about big numbers. Previously we’ve linked to Math Cats, and Wendy has a page where you can learn how to say some really big numbers. But thinking about counting also made me remember an experience I had in middle school where I found out just how big numbers could be! I was in seventh grade when I read this article from the December 1995 issue of Discover Magazine. It’s called “Infinity Plus One, and Other Surreal Numbers” and was written by Polly Shulman. I remember my mind being blown by all of the talk of infinitely-spined aliens and up-arrow notation for naming numbers. Here’s an excerpt:

Mathematicians and precocious five-year-olds have long been fascinated by the endlessness of numbers, and they’ve named the endlessness infinity. Infinity isn’t a number like 1, 2, or 3; it’s hard to say what it is, exactly. It’s even harder to imagine what would happen if you tried to manipulate it using the arithmetic operations that work on numbers. For example, what if you divide it in half? What if you multiply it by 2? Is 1 plus infinity greater than, less than, or the same size as infinity plus 1? What happens if you subtract 1 from it?

After I read this article, John Conway and Donald Knuth became heros of mine. (In college, I had the amazing fortune to have breakfast with Conway one day when he was visiting to give a lecture!) Knuth has a book about surreals that’s the friendliest introduction to the surreal numbers that I know of, and in this video, Vi Hart briefly touches on surreal numbers in discussing proofs that .9 = 1. Boy, would I love to see a great video or online resource that simply and beautifully lays out the surreal numbers in all their glory!

It was fun for me to remember that Discover article. I hope that you, too, run across some mathematics that leaves a seventeen-year impression on you!

Bon appetit!

Bridges, Meander Patterns, and Water Sports

This past week the Math Munch team got to attend the Bridges 2012. Bridges is a mathematical art conference, the largest one in the world. This year it was held at Towson University outside of Baltimore, Maryland. The idea of the conference is to build bridges between math and the arts.

Participants gave lectures about their artwork and the math that inspired or informed it. There were workshop sessions about mathematical poetry and chances to make baskets and bead bracelets involving intricate patterns. There was even a dance workshop about imagining negative-dimensional space! There were also some performances, including two music nights (which included a piece that explored a Fibonacci-like sequence called Narayana’s Cows) and a short film festival (here are last year’s films). Vi Hart and George Hart talked about the videos they make and world-premiered some new ones. And at the center of it all was an art exhibition with pieces from around the world.

The Zen of the Z-Pentomino by Margaret Kepner

Does this piece by Bernhard Rietzl
remind you of a certain sweater?

5 Rhombic Screens by Alexandru Usineviciu

Pythagorean Proof by Donna Loraine

To see more, you should really just browse the Bridges online gallery.

A shot of the gallery exhibition

I know that Paul, Anna, and I will be sharing things with you that we picked up at Bridges for months to come. It was so much fun!

David Chappell

One person whose work and presentation I loved at Bridges is David Chappell. David is a professor of astronomy at the University of La Verne in California.

David shared some thinking and artwork that involve meander patterns. “Meander” means to wander around and is used to describe how rivers squiggle and flow across a landscape. David uses some simple and elegant math to create curve patterns.

Instead of saying where curves sit in the plane using x and y coordinates, David describes them using more natural coordinates, where the direction that the curve is headed in depends on how far along the curve you’ve gone. This relationship is encoded in what’s called a Whewell equation. For example, as you walk along a circle at a steady rate, the direction that you face changes at a contant rate, too. That means the Whewell equation of a circle might look like angle=distance. A smaller circle, where the turning happens faster, could be written down as angle=2(distance).

Look at how the Cauto River “meanders” across the Cuban landscape.

In his artwork, David explores curves whose equations are more complicated—ones that involve multiple sine functions. The interactions of the components of his equations allow for complex but rhythmic behavior. You can create meander patterns of your own by tinkering with an applet that David designed. You can find both the applet and more information about the math of meander patterns on David’s website.

David Chappell’s Meander #6
Make your own here!

When I asked David about how being a scientist affects his approach to making art, and vice versa, he said:

My research focuses on nonlinear dynamics and pattern formation in fluid systems. That is, I study the spatial patterns that arise when fluids are agitated (i.e. shaken or stirred). I think I was attracted to this area because of my interest in the visual arts. I’ve always been interested in patterns. The science allows me to study the underlying physical systems that generate the patterns, and the art allows me to think about how and why we respond to different patterns the way we do.  Is there a connection between how we respond to a visual image and the underlying “rules” that produced the image?  Why to some patterns look interesting, but others not so much?

For more of my Q&A with David, click here. In addition, David will be answering questions in the comments below, so ask away!

Since bridges and meandering rivers are both water-related, I thought I’d round out this post with a couple of interesting links about water sports and the Olympics. My springboard was a site called Maths and Sport: Countdown to the Games.

No wiggle rigs

Arrangements of rowers that are “wiggle-less”

Here’s an article that explores different arrangements of rowers in a boat, focusing on finding ones where the boat doesn’t “wiggle” as the rowers row. It’s called Rowing has its Moments.

Next, here’s an article about the swimming arena at the 2008 Beijing games, titled Swimming in Mathematics.


Paul used to be a competitive diver, and he says there’s an interesting code for the way dives are numbered.  For example, the “Forward 1 ½ Somersaults in Tuck Position” is dive number 103C.  How does that work?  You can read all about it here.  (Degree of difficulty is explained as well.)

Finally, enjoy these geometric patterns inspired by synchronized swimming!

Stay cool, and bon appetit!