Tag Archives: pi

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!

4 Million Digits, Fifteen Furlongs, and 5 Eames Vids

Welcome to this week’s Math Munch!

We’ve written about Pi before, but when I found this new way of visualizing the number, Pi, I knew I’d have to share it with you. In 2011, Shigeru Kondo and Alex Yee concluded an incredible project – to design and execute a program to calculate digits in the decimal expansion of Pi. What makes their attempt so remarkable is that the program ran for over a year (371 days), during which time it calculated precisely the first 10 trillion digits of Pi! (1 with 13 zeroes!)

A New York design firm, called Two-N, built a wonderful website using the first 4 million digits to help us see the patterns in the digits (or lack thereof). Each digit was assigned a color, and included in the image as a single pixel. What we see is a long (really long) string of colored digits. You can drag across the screen to zoom in on rows. There’s even a search bar so that you can find where your birthday appears, or any other 6-digit string for that matter.

If you’re having a hard time wrapping your head around 4,000,000 digits, check out Fifteen Furlongs. It’s a website designed by Kevin Wang, a college student at the University of Chicago, and it’s designed to help us understand different sizes and units of measurements. Try it.

Fifteen Furlongs? – “That’s about two minutes on the highway.” Didn’t help me  much, but 1 Furlong? – “That’s just under one Empire State Building tall.” Which is really interesting. So, if we laid down several empire state buildings in a row to make a highway, then I could drive over 15 of them in about 2 minutes. Cool! How can I understand 4 million?

  • 4 million pounds is the weight of 1,000 cars.  hmmmm.
  • 4 million cups is about one Olympic-sized pool.  whoa.
  • 4 million seconds is just over forty-six day’s time.  so cool.

Maybe you can play around and figure out just how big 10 trillion is. After each answer there’s a place for you to say whether or not the information was useful, which I assume they use that to improve the responses. Have fun.

Kevin agreed to answer a few questions for us, which you can read in our Q&A section.  If you have ideas for how to improve the site, Kevin wants to hear them. Just leave it in the comments, and he’ll see what he can do.

Finally, some mathematical videos by the well-known 20th century design team of Charles and Ray Eames. In 1961 they worked on an exhibition for IBM called “Mathematica: A World of Numbers and Beyond,” which included a huge timeline with descriptions of famous mathematicians and mathematical discoveries from antiquity to modern times. It also included a “mathematics peepshow,” a collection of fantastic short math films, some of which can be seen on YouTube:

Actually my favorites aren’t even available online! There are 5 more videos available in a new fantastic, free iPad app called Minds of Modern Mathematics. If you donwload the app, check out “Symmetry” and “Exponents.” They’re simply stunning.

The best-known Eames vid is probably Powers of Ten, (embedded below) their 1977 film meant to illustrate the incredible scale of the universe, big and small, and how exponents can help us keep track of the different “levels.” It surely inspired the Huang Twins when they designed The Scale of the Universe.

You know, we typically feature at least one video a week, and they’re starting to pile up! Good news, though: we’ve been keeping track on a YouTube playlist of every video ever Featured on Math Munch. You can also use the Videos link at the top of any page.

Have a great week. Bon appetit!

Line Fractals, Knitting, and 3-D Design

Welcome to this week’s Math Munch!

Take a look at this beautiful line drawing:

Jason Padgett

This is called, “Towards Pi 3.141552779 Hand-Drawn,” and it’s by mathematician and artist Jason Padgett.  Jason wasn’t always a mathematician or an artist.  In fact, it was only after a severe head injury that Jason suddenly found that he “saw” fractals and other geometric images in mathematical and scientific ideas.  Jason is interested in limits.  The picture above, for example, is Jason’s artistic interpretation of a limit that approaches pi.  If you draw a circle with radius 1 and make polygons inside of it using secants for their sides, the areas of the polygons get closer and closer to pi as the number of sides increases – but always stay less than pi.  If you take that same circle and make polygons around it using tangents for their sides, the areas of the polygons also get closer and closer to pi as the number of sides increases – but always stay larger than pi.  Jason tried to draw the way that those sequences “trap pi” in this picture.

I think it’s really amazing that Jason draws these by hand.  Here’s some more of Jason’s artwork, and a video of Jason drawing “Towards Pi 3.141552779 Hand-Drawn.”

Space Time Sine Cosine and Tangent Waves

The Power of Pi

Wave Particle Duality

[youtube http://www.youtube.com/watch?v=uHqRTtnU8Wg&feature=fvwrel]

Next, did you like Sondra Eklund’s sweater from last week?  Did it inspire you to do some mathematical knitting of your own?  If so, check out the website Woolly Thoughts.

Woolly Thoughts is run by “mathekniticians” Pat Ashforth and Steve Plummer who love to do, teach, and share math with others through their knitting.  They’ve designed many beautiful and mathematical afghan and pillow patterns, and some patterns for interesting math toys.  Here are some of my favorites:

The “Finite Field” afghan is a color-coded addition table using binary.

The “Fibo-Optic” afghan is made to look like a flying cube using side-lengths based on the Fibonacci sequence.

Finally, one of the programs featured in the new Math Art Tools link is TinkerCAD.  TinkerCAD is a program you can use to make 3D designs – just because, or to print out with a 3D printer!

TinkerCAD has three parts: Discover, Learn, and Design.  In the Discover section, you can browse things that other tinkerers have made and download them to print yourself.  There are some really cool things out there, like this Father’s Day mug made by Fabricatis and this sail boat made by Klyver Boys.

Next, in the Learn section, you can play different “quests” to hone your TinkerCAD skills.  Finally, in the Design section, you can make your own thing!  TinkerCAD is really intuitive to use.  The TinkerCAD tutorial video is really helpful if you want to learn how to use TinkerCAD – as are the quests.

Stay tuned for pictures of some TinkerCAD things made by friends of Math Munch!

Bon appetit!