Tag Archives: obituary

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.

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.

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!

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.

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 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!

Posted by Justin Lanier. Categories: Math Munch. Tags: applied math, art, computers, contest, fractions, games, mazes, measurement, obituary, pi, polyominoes, puzzles, recreational mathematics, rulers. 1 Comment

Nov. ’14

Grothendieck, Circle Packing, and String Art

Welcome to this week’s Math Munch!

This week brought some sad news to the mathematical world. Alexander Grothendieck, known by many as the greatest mathematician of the past century, passed away on November 13th. You may not have heard of him, but many mathematicians say that the work he did in math was as influential as the work Albert Einstein did in physics.

One of the things that make Grothendieck so interesting is, of course, the math he did. Grothendieck was always very creative. When he was in high school, he preferred to do math problems he made up on his own over the problems assigned by his teacher. “These were the book’s problems, and not my problems,” he said.

When he was young, inspired by some gaps he found in definitions in his geometry book about measuring lengths and areas, Grothendieck re-created some of the most important mathematical ideas of the beginning of the twentieth century. Maybe this sounds silly to you– why re-invent something that’s already been done? But, to Grothendieck, the most important part was that he’d done the whole thing by himself. He’d figured out something in his own way. He later wrote that this experience showed him what being a mathematician was like:

Without having been told, I nevertheless knew ‘in
my gut’ that I was a mathematician: someone who
‘does’ math, in the fullest sense of the word…

During his years as a mathematician, Grothendieck worked on connecting different parts of math (a project requiring a lot of creativity)– algebra, geometry, topology, and calculus, among others.

Alexander Grothendieck as a kid

The other thing that makes Grothendieck so interesting is his life story. As a kid, Grothendieck and his parents fled from Germany to France to escape the Nazis. As an adult, Grothendieck spoke out strongly for peace. He used his fame to take a stand against the wars of the second half of the twentieth century. This eventually led him to step away from the world of mathematicians– which many regretted. But he left behind work that changed all of mathematics for the better.

If you’d like to learn more about Grothendieck’s fascinating life and work, check out this great (but long) article from the American Mathematical Society. This article provides a shorter history, including a great statement Grothendieck made about his feelings on creativity in mathematics. Grothendieck was a very private person, so many of his mathematical writings aren’t available online– but the Grothendieck Circle has done their best to collect everything written about him.

A pretty circle packing

Next up, a little something for you to play with! We were studying circle packing problems in one of my classes this week. Did you know that you can fit exactly six circles snugly around another circle of the same size? But, if you try to fit circles snugly around a circle twice as large, it doesn’t work? I wonder why that is…

I did it!

Anyway, my class inspired me to look for a circle packing game– and I found one! In this game, simply called Circle Packing, you have to fit all of the smaller circles into the larger circle– without any of them touching! It’s pretty tricky, and really fun.

Finally, the Math Munch team got something wonderful in the mail (email, I guess) this week! Math art made by Julia Dweck’s 5th grade math class! Julia’s class has been working hard to make parabolic curve string art– curves made by drawing (or stringing, in this case) many, many straight lines. They plotted each curve precisely before stringing it, to make sure it was both mathematically and artistically perfect. The pieces they made are so creative and beautiful. We’re proud to feature them on our site!

You can see the whole collection of string art pieces made by Julia’s class on our Readers’ Gallery String Art page. And, want to know more about how the 5th graders made their String Art? Have any questions for Julia and her students about their love of math and the connections they see between math and art? Write your questions here and we’ll send them to Julia’s students!

Have any math art of your own? Send it to mathmunchteam@gmail.com, and we’ll post it in the Readers’ Gallery!

Bon appetit!

Posted by Anna Weltman. Categories: Math Munch. Tags: Alexander Grothendieck, art, circle packing, circles, Grothendieck, history, interview, marth, math art, math history, obituary, packing problems, parabolic curves, q&a, string art. 4 Comments

Jul. ’14

Zippergons, High Fashion, and Really Big Numbers

Welcome to this week’s Math Munch!

Bill Thurston

Recently I attended a conference in memory of Bill Thurston. Bill was one of the most imaginative and influential mathematicians of the second half of the twentieth century. He worked with many mathematicians on projects and had many students before he passed away in the fall of 2012 at the age of 65. You can read Bill’s obituary in the New York Times here.

Bill worked where geometry and topology meet. In fact, Bill throughout his career showed that there are rich connections between the two fields that no one thought was possible. For instance, it’s an amazing fact that every surface—no matter how bumpy or holey or twisted—can be given a nice, symmetric curvature. A uniform geometry, it’s called. This was proven by Henri Poincaré in 1907. It was thought that 3D spaces would be far too complicated to be behave according to a similar rule. But Bill had a vision and a conjecture—that every 3D space can be divided into parts that can be given uniform geometries. To give you a flavor of these ideas, here’s a video of Bill describing some unusual and fabulous 3D spaces.

Any surface can be given a nice, symmetric geometry.

Any surface can be given a uniform geometry. Even a bunny. Another video.

As you can probably tell, visualizing and experiencing math was very important to Bill. He even taught a course with John Conway called Geometry and the Imagination. Bill often used computers to help himself see the math he was thinking about, and he enjoyed making hands-on models as well. Beginning in spring of 2010, Bill and Kelly Delp of Ithaca College worked out an idea. Usually all of the curving or turning of a polyhedron is concentrated at the vertices. Most of a cube is flat, but there’s a whole lot of pinch at the corners. What if you could spread that pinching out along the edges? And if you could, wouldn’t longer and perhaps wiggly edges help spread it even better? Yes and yes! You can see some examples of these “zippergons” that Bill and Kelly imagined and made in this gallery and read about them in their Bridges article.

A paper octahedron zippergon.

A foam icosadodecahedron zippergon.

One of Bill’s last collaborations happened not with a mathematician but with a fashion designer. Dai Fujiwara, a noted creator of high fashion in Tokyo, got inspired by some of Bill’s illustrations. In collaboration with Bill, Dai created eight outfits. Each one was based on one of the eight Thurston geometries. You can see the result of their work together in this video and read more about it in this article.

Isn’t it amazing how creative minds in very different fields can learn from each other and create something together?

Richard Evan Schwartz (self-portrait)

Richard Evan Schwartz was one of the speakers at the conference honoring Bill. Rich studied with Bill at Princeton and now is a math professor at Brown University.

Like Bill, Rich’s work can be highly visual and playful, and he often taps the power of computers to visualize and analyze mathematical structures. There’s lots to explore on Rich’s website. Check out these applets he has made, including ones on Poncelet’s Porism, the Euclidean algorithm (previously), and a game called Lucy & Lily (JAVA required). I love how Rich shares some of his earliest applet-making efforts, like Click On A Triangle To Change Its Color. It’s motivating to see that even an accomplished mathematician like Rich began with the basics of programming—a place where any of us can start!

On Rich’s site you’ll also find information about his project “Counting on Monsters“. And you should definitely make time to read some of the conversations that Rich has had with his five-year-old daughter Lucy.

Recently Rich published a wonderful new book for kids called “Really Big Numbers“. It is a colorful romp through larger and larger numbers and layers of abstraction, with evocative images to light the way. Check out the trailer for “Really Big Numbers” below!

Do you have a question for Rich—about his book, or about the math that he does, or about his life, or about Bill? Then send it to us in the form below and we’ll try to include it in our interview with him!

EDIT: Thanks for all your questions! Our Q&A with Rich will be posted soon.

Diana and Rich

Diana and Bill

Bill taught Rich, and Rich in turn taught Diana Davis, whose Dance Your PhD video we featured a while back. In fact, Bill’s influence on mathematics can be seen throughout many of our posts on Math Munch. Bill collaborated with Daina Taimina on hyperbolic crochet projects. He taught Jeff Weeks and helped inspire the games and software Jeff created. Bill oversaw the production of the film Outside In about the eversion of a sphere. He even coined the mathematical term “pair of pants.”

Bill’s vision of mathematics will live on in many people. That could include you, if you’d like. It’s just as Bill wrote:

“In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new.”

Bon appetit!

Posted by Justin Lanier. Categories: Math Munch. Tags: applet, art, big numbers, children's books, fashion, games, geometry, mathematical art, obituary, q&a, quotes, topology, visualization. 10 Comments