Tag Archives: art

Solomon Golomb, Rulers, and 52 Master Pieces

Welcome to this week’s Math Munch.

I was saddened to learn this week of the passing of Solomon Golomb.

Solomon Golomb.

Solomon Golomb.

Can you imagine the world without Tetris? What about the world without GPS or cell phones?

Here at Math Munch we are big fans of pentominoes and polyominoes—we’ve written about them often and enjoy sharing them and tinkering with them. While collections of glued-together squares have been around since ancient times, Solomon invented the term “polyominoes” in 1953, investigated them, wrote about them—including this book—and popularized them with puzzle enthusiasts. But one of Solomon’s outstanding qualities as a mathematician is that he pursued a range of projects that blurred the easy and often-used distinction between “pure” and “applied” mathematics. While polyominoes might seem like just a cute plaything, Solomon’s work with discrete structures helped to pave the way for our digital world. Solomon compiled the first book on digital communications and his work led to such technologies as radio telescopes. You can hear him talk about the applications that came from his work and more in this video:

Here is another video, one that surveys Solomon’s work and life. It’s fast-paced and charming and features Solomon in a USC Trojan football uniform! Here is a wonderful short biography of Solomon written by Elwyn Berlekamp. And how about a tutorial on a 16-bit Fibonacci linear feedback shift register—which Solomon mentions as the work he’s most proud of—in Minecraft!

Another kind of mathematical object that Solomon invented is a Golomb ruler. If you think about it, an ordinary 12-inch ruler is kind of inefficient. I mean, do we really need all of those markings? It seems like we could just do away with the 7″ mark, since if we wanted to measure something 7 inches long, we could just measure from the 1″ mark to the 8″ mark. (Or from 2″ to 9″.) So what would happen if we got rid of redundancies of this kind? How many marks do you actually need in order to measure every length from 1″ to 12″?

An optimal Golomb ruler of order 4.

An optimal Golomb ruler of order 4.

Portrait of Solomon by Ken Knowlton.

Portrait of Solomon by Ken Knowlton.

I was pleased to find that there’s actually a distributed computing project at distributed.net to help find new Golomb Rulers, just like the GIMPS project to find new Mersenne primes. It’s called OGR for “Optimal Golomb Ruler.” Maybe signing up to participate would be a nice way to honor Solomon’s memory. It’s hard to know what to do when someone passionate and talented and inspiring dies. Impossible, even. We can hope, though, to keep a great person’s memory and spirit alive and to help continue their good work. Maybe this week you’ll share a pentomino puzzle with a friend, or check out the sequences on the OEIS that have Solomon’s name attached to them, or host a Tetris or Blokus party—whatever you’re moved to do.

Thinking about Golomb rulers got me to wondering about what other kinds of nifty rulers might exist. Not long ago, at Gathering for Gardner, Matt Parker spoke about a kind of ruler that foresters use to measure the diameter of tree. Now, that sounds like quite the trick—seeing how the diameter is inside of the tree! But the ruler has a clever work-around: marking things off in multiples of pi! You can read more about this kind of ruler in a blog post by Dave Richeson. I love how Dave got inspired and took this “roundabout ruler” idea to the next level to make rulers that can measure area and volume as well. Generalizing—it’s what mathematicians do!

 img_3975  measuringtapes1

I was also intrigued by an image that popped up as I was poking around for interesting rulers. It’s called a seam allowance curve ruler. Some patterns for clothing don’t have a little extra material planned out around the edges so that the clothes can be sewn up. (Bummer, right?) To pad the edges of the pattern is easy along straight parts, but what about curved parts like armholes? Wouldn’t it be nice to have a curved ruler? Ta-da!

A seam allowance curve ruler.

A seam allowance curve ruler.

David Cohen

David Cohen

Speaking of Gathering for Gardner: it was announced recently that G4G is helping to sponsor an online puzzle challenge called 52 Master Pieces. It’s an “armchair puzzle hunt” created by David Cohen, a physician in Atlanta. It will all happen online and it’s free to participate. There will be lots of puzzle to solve, and each one is built around the theme of a “master” of some occupation, like an architect or a physician. Here are a couple of examples:


Notice that both of these puzzles involve pentominoes!

The official start date to the contest hasn’t been announced yet, but you can get a sneak peek of the site—for a price! What’s the price, you ask? You have to solve a puzzle, of course! Actually, you have your choice of two, and each one is a maze. Which one will you pick to solve? Head on over and give it a go!

Maze A

Maze A

Maze B

Maze B

And one last thing before I go: if you’re intrigued by that medicine puzzle, you might really like checking out 100 different ways this shape can be 1/4 shaded. They were designed by David Butler, who teaches in the Maths Learning Centre at the University of Adelaide. Which one do you like best? Can you figure out why each one is a quarter shaded? It’s like art and a puzzle all at once! Can you come up with some quarter-shaded creations of your own? If you do, send them our way! We’d love to see them.

Six ways to quarter the cross pentomino. 94 more await you!

Eight ways to quarter the cross pentomino. 92 more await you!

Bon appetit!

Web Applets, Space Fillers, and Sisters

Welcome to this week’s Math Munch!

Recently I’ve been running across tons of neat, slick math applets. I feel like they all go together. What do they have in common? Maybe you’ll be able to tell me.

First up, you can tinker with some planetary gears. Then try out these chorded polygons. And then how about some threaded lines?

plantearygears chords shapes

Ready for some more? Because with these sorts of visualizations, Dan Anderson has been on fire lately. Dan is a high school math teacher in New York state. He and his students had fifteen minutes of fame last year when they investigated whether or not Double Stuf Oreos really have double the stuf.

Here is Dan’s page on OpenProcessing. (Processing is the computer language in which Dan programs his applets.) And check out the images and gifs on Dan’s Tumblr. Here’s a sampling!

tumblr_nm56rdMlvl1uppablo1_r3_400 tumblr_noqxoi8EsC1uppablo1_400 tumblr_nolvf9dSt61uppablo1_400

Dan also coordinates Daily Desmos, which we’ve feature previously. Check out the latest periodic and “obfuscation” challenges!

That’s a chunk of math to chew on already, but we’re just getting started! Next up, check out the space-filling artwork of John Shier.

doublecircles eyes
 fish  hearts

John’s artwork places onto the canvas shapes of smaller and smaller sizes. Notice that the circles below fill in gaps, but they don’t touch each other, they way circles do in an Apollonian gasket.

circle_prog_1B_AnimeYou can learn more about John’s space-filling shapes on this page and find further details in this paper.

Thanks for making us this sweet banner, John!

Thanks for making us this sweet banner, John!

Last up this week, head to this site to watch an awesome trailer of a film about Julia Robinson. The short clip focuses on Julia’s work on Hilbert’s tenth problem. It includes interviews with a number of people who knew Julia, including her sister Constance Reid. Constance wrote extensively about mathematics and mathematicians. I’ve read her biography of Hilbert and can highly recommend it. You can read more about Julia and Constance here and here.

Julia Robinson

Julia Robinson

Julia's sister, Constance Reid

Julia’s sister, Constance Reid

Julia and Constance as young girls.

Julia and Constance as young girls.

You might enjoy visiting the site of the Julia Robinson Mathematics Festival. Check to see if a festival will be hosted in your area sometime soon, or find out how you can run one yourself!

With May wrapped up and June getting started, I hope you have a lot of math to look forward to this summer. 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!