Author Archives: Paul Salomon

Havel-Hakimi, Temari, and more GIFS

Welcome to this week’s Math Munch!  We’ve got another great game for you, a followup with Temari artist Carolyn Yackel, and some mind-blowing math gifs.

Havel-Hakimi

Havel-Hakimi

First up, a nice little graph theory game created by Jacopo Notarstefano.  The game is about whether or not sets of numbers meet the conditions for being “graphical.”  Maybe the best way to understand what that means is to start playing.  If you can beat a level, then the starting number set is graphical.  Go play Havel-Hakimi.

In 1960, mathematicians Paul Erdös and Tibor Gallai proved a theorem about what number sets were graphical.  The name of the game refers to an algorithm you can use to solve the game.  You might figure it out just by playing the game, but here’s a (pretty dry) video explaining how the Havel-Hakimi algorithm works.

Jacopo’s website has a few other nice projects.  See if you can figure out Who Killed the Duke of Densmore, or try Four-Coloring the Dodecahedron.

Temari 1 Temari 2 Temari 3
Carolyn Yackel

Carolyn Yackel

Up next, remember Carolyn Yackel.  We wrote about Carolyn and her mathematical art a while back.  Well we finally got around to doing a little Q&A.  Give it a read to learn about Carolyn and her love of math.

Carolyn’s art (which can be seen here) is called temari, the japanese art of embroidered spheres. Since our post about Carolyn we found out that a now 93-year old grandmother posted a lifetime of temari on flickr.  These beautiful objects have symmetry that mimic various polyhedra, which I just love.  Read Carolyn’s Q&A to hear about how you make them.

Grandmother's temari work

A Grandmother’s Temari Work

Finally, a while back we shared some mathematical gif animations created by Bees and Bombs.  It’s time once again to look at some amazing animations.  This time they’re created by David Pope.  Here’s the complete archive of animations.  I’ll post some of my very favorites below, but there are dozens of dozens of good animations. (A dozen dozens is gross!)

Have a great week.  Bon appetit!

Full gallery of mathematical gifs

Pyramid

Pyramid

Rolling Prisms

Rolling Prisms

Sphere

Sphere

Spinning Octahedra

Spinning Octahedra

2048, 2584, and variations on a theme

Welcome to this week’s Math Munch! It’s a week of mathematical games, including a devilish little game and variations on the theme.

2048

2048

First up, check out this simple little game called 2048. Really, you must go try that game before reading on.

Gabriele Cirulli

Gabriele Cirulli

2048 was created by Gabriele Cirulli, a 20-year old who lives in northern Italy. He was inspired by a couple of very similar games called 1024 and threes, and he wanted to see if he could code a game from scratch. Nice work, Gabriele! (Stay tuned for a Q&A with Gabriele. Coming soon.)

The first time I played, I thought randomly moving the pieces around would work as well as anything, but wow was I wrong. Give it a try and see how far you get. Now watch how this AI (artificial intelligence) computer program plays 2048. You’ll probably notice some patterns that will help you play on your own.

A beautiful chain of powers of two.

A beautiful chain of powers of two.  Can you solve from here?

Did you notice that the smallest tiles are 2′s, and you can only combine matching tiles to create their double? This makes all of the tile values powers of two! (e.g. 2048=2^11) These are the place values for the binary number system! (Did you see our recent post binary?) This has something to do with the long chains that are so useful in solving the game. It’s just like this moment in the marble calculator video.

4, a silly, but interesting little variation

4, a silly, but interesting little variation

If you’re finding 2048 a bit too hard, here’s an easier version.  It’s called 4. It’s a little silly, but it’s also quite interesting. After you make the 4 tile (tying the world record for fewest moves), click “keep going” and see how far you can get. I’ve never been able to get past the 16 tile. Can anyone make the 32? What’s the largest possible tile that can be made in the original 2048 game? Amazingly, someone actually made a 16384 tile!!!

2584, the Fibonacci variant of 2048

2584, the Fibonacci version of 2048

Silly versions aside, there are lots and lots of ways you could alter 2048 to make an interesting game. I wondered about a version where three tiles combined instead of two, but I couldn’t quite figure out how it would work. Can you? (See below.) When I thought about different types of numbers that could combine, I thought of the perfect thing. The Fibonacci numbers!!! 1, 1, 2, 3, 5, 8, 13, 21, … The great thing is that someone else had the same idea, and the game already exists! Take some time now to play 2584, the Fibonacci version of 2048.

2048 and 2584 might seem like very similar games at first, (they’re only 536 apart), but there are some really sneaky and important differences. In the Fibonacci version, a tile doesn’t combine with itself. It has two different kinds of tiles it can match with. I think this makes the game a little easier, but the website says 2584 is more difficult than the original. What do you think?

I have a few more 2048 variations to share with you, as if you didn’t have enough already. These are my favorites:

I hope you dig into some of these games this week. Really think and analyze. If you come up with clever strategies or methods to solve these puzzles, please let us know in the comments. Have a great week, and bon appetit!

Byrne’s Euclid, Helen Friel, and PolygonJazz

Welcome to this week’s Math Munch! We’ve got geometry galore, starting with a series of historical math diagrams and a color update to Euclid’s Elements. Then it’s onto modern day paper artist Helen Friel, and finally a nifty new app that makes music from polygons. Let’s get into it.

Euclid’s “Elements” was written around 300BC. It was the first great compilation of geometric knowledge, broken into 13 books, and it is one of the most influential books of all time. Euclid’s proof of the Pythagorean Theorem may be his most famous proof from the book (and all of mathematics for that matter), and in the pictures below you can see three diagrams of the proof, spanning seven centuries.

Nasir al-Din al-Tusi's 13th century arabic translation of Euclid's proof.

Persian mathematician Nasir al-Din al-Tusi‘s 13th century arabic translation of Euclid’s proof.

Late 14th century English manuscript

A late 14th century English manuscript of Euclid’s “Elements.”

The idea in each picture is that the area of the top two squares adds up exactly to the area of the bottom square. In the picture below, we see the big square broken up into blue and yellow pieces, whose areas are the same as the squares above them.

Oliver Byrne's 1871 color edition

Oliver Byrne’s 1847 color edition.  Click the image for the full proof of the Pythagorean Theorem as presented by Oliver Byrne in 1847.

This color version comes from Oliver Byrne’s 1847 edition, “The First Six Books of the Elements of Euclid, with Coloured Diagrams and Symbols.” (completely available online). I find the diagrams really beautiful and charming. There’s something extremely modern about them (see De Stijl) though they’re more than 150 years old now. See if you can follow his Oliver Byrne’s version of Euclid’s proof. It’s quite short.

Paper Engineer Helen Friel
Paper Engineer Helen Friel

 

“They’re an absolutely beautiful piece of work and far ahead of their time,” said paper engineer Helen Friel. Helen lives in London, and and as part of a charity project, she designed paper sculptures of Oliver Byrne’s diagrams.

Euclid 2 Euclid 4 Euclid 3 Euclid 1

In an interview, she explained, “It’s a more visual and intriguing way to describe the geometry. I love anything that simplifies. I find it very appealing!” In the interview, Helen also talks a little about her attraction to math. “There’s order in straight lines and geometry. Although my job is creative, I use as much logical progression as possible in my work.”

It’s also cool to see Helen’s work side by side with Oliver Byrne‘s, so click for that.

Screen Shot 2014-02-05 at 11.46.00 PM

Click to send us a pic.  Yes, that is a paper camera Helen made.

Downloadable model

Downloadable model

Perhaps the best part in all of this, though, is that you can download Helen’s Pythagaorean Theorem model and make your own! There are plain white version as well as color. If you end up making one, definitely email us a picture, and we’ll show it off here on Math Munch.

Oh, and here’s a quick video documenting the many versions Helen decided not to use.  So cool.

Now, on to our final bite.
PolygonJazz Recently, John Miller sent me an email showing off his new iPad app called PolygonJazz. In the app, you control the starting direction for a ball inside a polygon. Once you start it moving, the ball bounces off the walls, making a sound every time it hits a side. Check out the video below. I noticed something about the speed of the ball. Can you spot it? (PolygonJazz is available for $0.99 on the iTunes store.)

Speaking of bouncing around, here‘s a previous Math Munch featuring some billiards, and here‘s another bouncy post that features one of my favorite juggling routines. Michael Moschen built a gigantic equilateral triangle and juggles silicon balls inside and off of it. As with the app, Michael is utilizing the sound and geometry of the collisions to make something beautiful. It’s quite mesmerizing.

Have a bouncy week, and bon appetit!