Tag Archives: numbers

roTopo, de Gua, and Bibi-binary

Welcome to this week’s Math Munch!

Today we’re going to look at a few examples of going “up a dimension”. Our first example is what got me thinking about this theme. It’s a game called roTopo. (If you have trouble getting it to load, try using a different browser.)

 rotopo1.png  rotopo2

Maybe you have played the game B-Cubed. RoTopo is similar—trace through a sequence of squares as they get eliminated one by one. I like B-Cubed because it combines spatial thinking with strategic thinking—planning ahead. Rotopo, with its twists and turns in 3D, stretches a player’s spatial thinking even further. I hope you enjoy giving it a try! Maybe you could design a roTopo level of your own with a drawing or with some blocks.

What else can we find when we look “up a dimension”? Maybe the most famous theorem in all of mathematics is the Pythagorean theorem. There are several ways we might try to take a^2+b^2=c^2 up a dimension. If we start to increase the numbers in the exponents, like a^3+b^3=c^3, we head in the direction of Fermat’s Last Theorem. If we add more terms, like a^2+b^2+c^2=d^2, we can find distances in 3D instead of 2D.


A right tetrahedron—the kind needed for de Gua’s Theorem.

And if those aren’t enough to make you go “wow”, then you need to hear about De Gua’s Theorem. The Pythagorean Theorem relates the sides of a right triangle. De Gua’s Theorem relates the faces of a right tetrahedron. The sum of the squares of the areas of the the three “leg” faces is equal to the square of the area of the “hypotenuse” face. So wild! You can read a proof de Gua’s Theorem here. The theorem is named for the 18th-century French mathematician who presented it to the Paris Academy of Sciences in 1783 (although it was known to others before him). De Gua’s Theorem in turn is a special case of a still more general theorem. Once mathematicians start upping dimensions, the sky is the limit!

Last up: Bibi-binary. No, that’s not the way that Justin Timberlake counts—although that funny thought is why I Googled “bibibinary” in the first place. But when I did, this totally silly number system popped up!


How to count in Bibi-binary.

Well, I guess it’s not the number system that’s silly so much, since it’s actually just hexadecimal. Hexadecimal is like binary, but up a couple of dimensions. The system uses sixteen symbols to represent numbers, just as the decimal system uses ten symbols and binary uses two. What makes Bibi-binary silly, then, is not its logical structure but how it sounds.

There are sixteen syllables in Bibi-binary, which are made from combinations of four consonants and four vowels. Three is “hi” and eight is “ko”. If you want to have three 16’s and eight more—56—that would be “hiko”. As another example, 66319344 is “hidihidihidiho”. Bibi-binary was invented in 1971 by a French singer and actor named Boby Lapointe.

I think it would be fun to learn to count in Bibi-binary. Can you believe that I could find zero (“ho”) videos online of people counting in Bibi-binary? I wonder if any of our readers might enjoy making one…

img_colormapHexadecimal is not just fun and games. It’s also used for making codes to stand for colors, especially in making webpages. Most of Math Munch is either 683D29 or 6AB690, would you believe. You can explore using hexadecimal to name colors in this applet.

You can learn lots more about Bibi-binary on the great website dCode, and you’ll also find an applet there that can convert between decimal and Bibi-binary. DCode has lots of tools related to cryptography (get it?) and other math topics, too.

Do you have any favorite examples of math that goes “up a dimension”? We’d love to hear about them in the comments.

Bibi-bi for now! 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.

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!

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!

Fields Medal, Favorite Numbers, and The Grapes of Math

Welcome to this week’s Math Munch! And, if you’re a student or teacher, welcome to a new school year!

fieldsOne of the most exciting events in the world of math happened this August– the awarding of the Fields Medal! This award honors young mathematicians who have already done awesome mathematical work and who show great promise for the future. It also only happens every four years, at the beginning of an important math conference called the International Congress of Mathematicians, so it’s a very special occasion when it does!


Maryam Mirzakhani, first woman ever to win a Fields Medal

Maryam Mirzakhani, first woman ever to win a Fields Medal

This year’s award was even more special than usual, though. Not only were there four winners (more than the usual two or three), but one of the winners was a woman!

Now, if you’re like me, you probably heard about the Fields Medal and thought, “There’s no way I’ll understand the math that these Field Medalists do.” But this couldn’t be more wrong! Thanks to these great articles from Quanta Magazine, you can learn a lot about the super-interesting math that the Fields Medalists study– and why they study it.


Manjul Bhargava

One thing you’ll immediately notice is that each Fields Medalist has non-math interests that inspire their mathematical work. Take Manjul, for instance. When he was a kid, his grandfather introduced him to Sanskrit poetry. He was fascinated by the patterns in the rhythms of the poems, and the number patterns that he found inspired him to study the mathematics of number patterns– number theory!

But, don’t just take my word for it– you can read all about Manjul and the others in these great articles! And did I mention that they come with videos about each mathematician? 

Want to read more about this year’s Fields Medallists? Check out Alex Bellos’s article in The Guardian. Which brings me to…

download… What’s your favorite number? Is it 7? If it is, then you’re in good company! Alex polled more than 30,000 people about their favorite number, and the most popular was 7. But why? What’s so special about 7? Here’s why Alex thinks 7 is such a favorite:

grapes-of-mathWhy do you like your favorite number? People gave Alex all kinds of different reasons. One woman said about 3, her favorite number, “3 wishes. On the count of 3. 3 little pigs… great triumvirates!” Alex made these questions the topic of the first chapter of his new book, The Grapes of Math. (Get the reference?) In this book, Alex shares many curious ways that math appears in our world. Did you know that a weird pattern in numbers can be used to catch criminals? Or that the Game of Life, a simple computer program, shares surprisingly many characteristics with real life? These are only a few of the hundreds of topics Alex covers in his book. Whether you’re a math whiz or a newbie, you’ll learn something new on every page.

Alex currently writes about math for The Guardian in a blog called, “Alex’s Adventures in Numberland”— but he also loves and writes about soccer (or futbol, as it’s called in his native Brazil)! He even wrote a few articles for his blog about math and soccer. 

Do you have any questions for Alex? (About math, soccer, or their intersection?) Write them here and you might find them featured in our interview with Alex!

Good writing about math is hard to find. If you’ve ever picked up a standard math textbook, you’ll know what I mean. But reading something fascinating, that grabs your interest from the first page and leads you through the most complex ideas like they’re as natural as anything you’ve observed, is a great way to learn. The Grapes of Math and “Alex’s Adventures in Numberland” do just that. Give them a go!

Bon appetit!