# Sphericon, National Curve Bank, and Cardioid String Art

Welcome to this week’s Math Munch!

Behold the Sphericon!

What is that? Well, it rolls like a sphere, but is made of two cones attached with a twist– hence, the spheri-con! The one in the video is made out of pie (not sure why…), but you can make sphericons out of all kinds of materials.

It was developed by a few people at different times– like many brilliant new objects. But it entered the world of math when mathematician Ian Stewart wrote about it in his column in Scientific American. The wooden sphericon was made by Steve Mathias, an engineer from Sacramento, California, who read Ian’s article and thought sphericons would be fun to make. To learn more about how Steve made those beautiful wooden sphericons, check out his site!

Even if you’re not a woodworker, like Steve, you can still make your own sphericon. You can start with two cones and make one this way, by attaching the cones at their bases, slicing the whole thing in half, rotating one of the halves 90 degrees, and attaching again:

Or you can print out this image, cut it out, fold it up, and glue (click on the image for a larger printable size):

If you do make your own sphericon (which I recommend, because they’re really cool), watch the path it makes as it rolls. See how it wiggles? What shape do you think the path is?

I found out about the sphericon while browsing through an awesome website– the National Curve Bank. It’s just what it sounds like– an online bank full of curves! You can even make a deposit– though, unlike a real bank, you can take out as many curves as you like. The goal of the National Curve Bank is to provide great pictures and animations of curves that you’d never find in a normal math book. Think of how hard it would be to understand how a sphericon works if you couldn’t watch a video of it rolling?

There are lots of great animations of curves and other shapes in the National Curve Bank– like the sphericon! Another of my favorites is the “cycloid family.” A cycloid is the curve traced by a point on a circle as the circle rolls– like if you attached a pen to the wheel of your bike and rode it next to a wall, so that the pen drew on the wall. It’s a pretty cool curve– but there are lots of other related curves that are even cooler. The epicycloid (image on the right) is the curve made by the pen on your bike wheel if you rode the bike around a circle. Nice!

You should explore the National Curve Bank yourself, and find your own favorite curve! Let us know in the comments if you find one you like.

String art cardioid

Finally, to round out this week’s post on circle-y curves (pun intended), check out another of my favorite curves– the cardioid. A cardioid looks like a heart (hence the name). There are lots of ways to make a cardioid (some of which we posted about for Valentine’s Day a few years ago). But my favorite way is to make it out of string!

String art is really fun. If you’ve never done any string art, check out the images made by Julia Dweck’s class that we posted last year. Or, try making your own string art cardioid! This site shows you how to draw circles, ovals, cardioids, and spirals using just straight lines– you could follow the same instructions, replacing the straight lines you’d draw with pieces of string attached to tacks! If you’re not sure how the string part would work, check out this site for basic string art instructions.

Bon appetit!

# Pi Digit, Pi Patterns, and Pi Day Anthem

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.

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.

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!

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 & 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!

# Sequence Day, Penguins, and a little folding

Welcome to this week’s Math Munch! And… happy Sequence Day!

If you didn’t know that today was Sequence Day, don’t feel bad– I didn’t know until I ran across this article written by Aziz Inan, an electrical engineering professor at the University of Portland. Why is today Sequence Day? Well, because all of the digits in the sequence 0, 1, 2, 3, 4, 5 appear in today’s date– 3-4-2015!

This particular Sequence Day isn’t super special. There will be another one with the exact same sequence on April 3 (4-3-2015). But, according to Aziz, April 3 will be the last Sequence Day of this year– and the last until 2031! Aziz made this chart of all the Sequence Days that will happen this century. There are 48 all together– and if you look carefully at the chart, you may notice some interesting patterns.

See how the Sequence Days mostly occur at the beginnings of decades, in the first half of the month, and never later in the year than June? Why do you think that might be? Also, the last Sequence Day of the 21st century is in 2065. That means we’ll have to wait almost 40 years for the next Sequence Day after that– until 2103! But, in the scheme of things, this actually isn’t so bad– there were no Sequence Days at all in the last century. (Why might that be?)

We all know about days like Pi Day (coming up soon on 3-14-15 — and it’s a special one because we’ve got those two additional digits this year!), but, as Aziz likes to show, lots of days can be mathematical holidays– if you just look carefully enough. Maybe you’ll find a mathematical holiday of your own! If you do, let us know. We love any excuse to have a party!

Next up, you think penguins are cute, right? Well, take a look at this:

You may have heard the narrator say, “Something more organized is going on.” Well, several mathematicians wondered what that more organized thing was… and it turns out to be very mathematical!

How many penguins do you see?

Francois Blanchette, a mathematician at the University of California, Merced, had the idea to use math to study how penguins keep warm while watching penguin movies like this one. He studies the math of something called fluid dynamics, which, basically, is how things like water and air flow. Francois and several other mathematicians at the University of Erlangen-Nuremberg in Germany noticed that when one penguin in a huddle moves just a little bit, it triggers a chain reaction in which all of the other penguins move in an organized way to keep warm. Their tiny movements cause the huddle to organize into the best shape for all penguins to keep warm during the cold of winter.

Huddle up, little guy!

Scientists and mathematicians are only now realizing all of the amazing ways that math comes into play in the lives of animals, especially in large groups. It seems that penguins are only the beginning! To learn more about the organization of large groups of animals, I suggest you check out this awesome PBS documentary about animal swarms.

Finally, we haven’t heard from Vi Hart in a while. If you’ve been feeling the need for some math art fun in your life, check out this video I dug up from the archives. Origami meets Pythagorean Theorem– what could be better?

Stay warm, and bon appetit!