Author Archives: Paul Salomon

Scary-o-graphic Projection, Thinky the Dragon, and Martin Gardner

Welcome to this week’s Math Munch!

Halloween is quickly approaching, which is why last week, Anna shared some pumpkin polyhedra. It just so happens that Justin did some pumpkin-y math of his own last year. He created a must-watch video called “Scary-o’-graphic Projection,” which was shown in the 2014 Bridges Short Film Festival. Enjoy, but don’t get too scared.

A stereographic projection sculpture by Henry Segerman.

A stereographic projection sculpture by Henry Segerman.

To learn some more about stereographic projection, watch one of Henry Segerman’s videos. You’ll also get to see some of his 3D printed sculptures.  (1 2)

In other news, Oct. 21 marked the 100th anniversary of the birthday of Martin Gardner!! (previously featured here, here, and here, among others) Around this time every year people get together to do math in his honor as part of Celebration of Mind.

This year we’re featuring one of Gardner’s optical illusions. Let’s begin with a video. Meet Thinky the Dragon.

You can find printable make-your-own templates here. (There are other colors as well.) Thinky is an example of a “hollow face” illusion, many more of which can be found on mathaware.org. There you can also find this video explaining the geometry behind this illusion.

Can you fold this strip of 7 squares into a cube?

Can you fold this strip of 7 squares into a cube?

Whats special about this square?  More than you think!

Whats special about this square? More than you think!

Thank you to Colm Mulcahy for his recent post on the BBC website, where Colm put together a list of 10 really wonderful problems from the hundreds that Gardner wrote about and popularized during his career. Gardner helped show the world that thinking about problems and mathematics was a really fun way to spend time. Watch the video below to learn more about Celebration of Mind events, and click here to see if there’s an event near you.  Note: You can even host an event of your own.

BONUS: I just have to mention MoSAIC for any math art enthusiasts in our audience. Around the country, small mathematical art conferences and exhibitions will go on this year. Click to learn more or find an event near you.

Munch in honor of Martin Gardner. Bon appetit!

Picture from the MoSAIC website.

Picture from the MoSAIC website.

Marc Chamberland, Math Fonts, and Congruent Triangles

Welcome to this week’s Math Munch!

Marc Chamberland

Marc Chamberland

Up first, a follow up to our post about the World Cup a while back. We received an email from Marc Chamberland linking us to a nice little video (below) about World Cup Balls and their various properties. You may remember seeing Marc’s mathematical art in this post. Below you can see another nice piece that was included in the mathematical art exhibit at the 2013 Joint Mathematics Meetings. Click for a nice description of the math puzzle it solves.  (in short: What’s the area of the red square?)

"Inner Square" by Marc Chamberland

“Inner Square” by Marc Chamberland

Marc is a math professor at Grinnell College. In March of 2014 (3-14?) he began working on Tipping Point Math, a youtube channel full of videos showing “math as you never imagined.” I encourage you to find something nice there. For now, here’s that video about World Cup Balls I promised you.

A font based on glass bending

A font based on glass bending

Up next are some nifty, fun fonts based on mathematics. Erik Demaine is no stranger to Math Munch readers, and it’s no wonder why. His stuff is clever and downright intriguing. He and his father Martin published a very interesting paper last April about a series of mathematical typefaces they’ve created over the course of their last decade of research and play.

A Conveyor Belt Font by Erik and Martin Demaine

A conveyor belt typeface by Erik and Martin Demaine

Their paper was published to the arXiv (pronounced “archive”) where it is publicly available.  You can read it here. Or, if you like something slightly more plain-language, here’s a nice review over on medium.com.

Screen Shot 2014-09-15 at 9.05.54 PM Screen Shot 2014-09-15 at 9.07.28 PM
The 4051 Tektronix Graphics Terminal

The 4051 Tektronix Graphics Terminal

Finally, I want to share a sleepy little video called “Congruent Triangles.”  I like to think of it as a slice of mathematical cultural history. This film was made in 1977 on an early computer called a Tektronix 4051 Graphics Terminal. It was made by Bruce and Katherine Cornwell as part of a series of mathematical videos. The way the shapes move and deform to present the ideas and connect the pieces together is so very cool. I also love the choice of music. It tells you something about what math was like for people then. I’d say sort of “groovy.”

Screen Shot 2014-09-15 at 9.22.09 PMThere’s more to the story and many more cool videos to enjoy.  You can look forward to seeing more from the Cornwells, but for now, enjoy this one video and do some hunting on your own if you’re interested.  That’s called “research.”

Bon appetit!

The Art of Merete Rasmussen, a Game About Squares, and VAX!

Welcome to this week’s Math Munch! We’ve got a pair of new games for you to play later, but first I want to share something beautiful and impressive.

Hyperseeing Summer '14

Ready for some mathematical art? The new issue of Hyperseeing begins with a review of Merete Rasmussen’s ceramic sculpture. Merete is a Danish artist who lives in London, and her recent work features complex and beautiful, smooth two-dimensional surfaces.

Editor Nat Friedman’s writeup begins with this wonderful quote by Rasmussen:

 

“I want to create a form that you can’t understand until you see the other side. You have to look at it for a while to realize how it is connected.”

Merete Rasmussen at work

Merete Rasmussen at work

A lot of mathematical work is done just trying to describe and understand the ideas or pictures in our head. Merete’s sculpture get us to do math as we try to understand the nature of her sculptural surfaces. How many sides do they have? How many edges? How many holes? I just love that.

Blue Gray

The article is very enjoyable, and I encourage you to read the entire text, but what got me hooked, what completely mesmerized and inspired me, was a video about Merete’s work and process that I found referenced at the end of the article. The video is presented in dual screen, which is really fantastic, because just like Merete’s sculptures, you may need to view it a couple of times to catch all that’s going on.

I recommend the full video. I recommend full screen.

You can learn more about Merete Rasmussen and view more of her work at her website, mereterasmussen.com.

* * *

OK, now on to a couple of new games.

Game About Squares

Game About Squares“Right from the start I was thinking about creating a simple game, with simple graphics and simple game design.” That’s what 26-yr old Andrey Shevchuk said about his recent creation, “Game About Squares.” You’ll find as you play, however, that these little puzzles can get oh so complicated, despite their simple presentation.

I love imagining how Andrey must have had to think creatively to keep developing his simple idea in new ways, and I love the way that the puzzles get us to think in new ways. All in all, this is just a wonderful game.

Oh, and thinking about the very viral 2048, Andrey had this to say,

“Squares are trendy.  Hexagons aren’t even close, let alone triangles.”

VAX!

Screen Shot 2014-08-08 at 9.29.36 PMThat’s short for vaccine, in case you don’t know.  The Salathé Group recently released a game about vaccinations and fighting the spread of epidemics (previously). The game is called VAX!, and it’s based on a graph theory representation for the spread of disease. Take the tour and you’ll learn everything you need to play.

There’s also a module that explains herd immunity. That’s where random vaccines are used to isolate the potentially infected from potential carriers of the disease.

Bon appetit.  Dig in!

The World Cup Group Stage, Math at First Sight, and Geokone

Welcome to this week’s Math Munch! We’ve got some World Cup math from a tremendous recreational mathematics blog and a new mathematical art tool. Get ready to dig in!

Brazuca: The 2014 World Cup Ball

Brazuca: The 2014 World Cup Ball

I’ve been meaning to share the really fantastic Puzzle Zapper Blog, because it’s so full of cool ideas, but the timing is perfect, because IT’S WORLD CUP TIME!!! and the most recent post is about the math of the world cup group stage! It’s called “World Cup Group Scores, and “Birthday Paradox” Paradoxes,” and I hope you’ll give it a read. (For some background on the Birthday Paradox, watch this Numberphile video called 23 and Football Birthdays.)

The thing that got me interested in the article was actually just this chart. I think it’s really cool, probably because I always find myself two games through the group stage, thinking of all the possible outcomes. If you do nothing else with this article, come to understand this chart. I was kind of surprised how many possible outcomes there are.

All Possible World Cup Group Stage Results

All Possible World Cup Group Stage Results

Long story short (though you should read the long story), there’s about a 40% chance that all 8 world cup groups will finish with different scores.

Alexandre Owen Muñiz, Author of Puzzle Zapper.  (click for an interview video about Alexandre's interactive fiction)

Alexandre Owen Muñiz, Author of Puzzle Zapper.  (click for an interview video about Alexandre’s interactive fiction)

Puzzle Zapper is the recreational mathematics blog of Alexandre Owen Muñiz. You can also find much of his work on his Math at First Sight site. He has a lot of great stuff with polyominoes and other polyforms (see the nifty pics below). Alexandre is also a writer of interactive fiction, which is basically a sort of text-based video game. Click on Alexandre’s picture to learn more.

The Complete Set of "Hinged Tetriamonds"

The complete set of “hinged tetrominoes”

A lovely family portrait of the hinged tetriamonds.

A lovely, symmetric family portrait of the “hinged tetriamonds”

I hope you’ll poke around Alexandre’s site and find something interesting to learn about.

For our last item this week, I’ve decided to share a new mathematical art tool called Geokone. This app is a recursive, parametric drawing tool. It’s recursive, because it is based on a repeating structure, similar to those exhibited by fractals, and it’s parametric, because the tool bar on the right has a number of parameters that you can change to alter the image. The artistic creation is in playing with the parameter values and deciding what is pleasing. Below are some examples I created and exported.

geokone2 geokone1

geokone3

I have to say, Geokone is not the easiest thing in the world to use, but if you spend some time playing AND thinking, you can almost certainly figure some things out! As always, if you make something cool, please email it to us!

Now go create something!  Click to go to Geokone.net.

I hope you find something tasty this week. Bon appetit!

Halving Fun, Self-Tiling Tile Sets, and Doodal

Welcome to this week’s Math Munch!

Print out two copies of this pattern, cut them out, and fold each along the dotted lines, making two identical solids. Then fit these two pieces together to make a regular tetrahedron.

Print out two copies of this pattern, cut them out, and fold each along the dotted lines, making two identical solids. Then fit these two pieces together to make a regular tetrahedron.

Our first bit of fun comes from a blog called Futility Closet (previously featured). It’s a neat little cut-and-fold puzzle. The shape to the right can be folded up to make a solid with 5 sides. Two of them can be combined to make a solid with only 4 sides, the regular tetrahedron. If you’d like, you can use our printable version, which has two copies on one sheet.

What do you know, I also found our second item on Futility Closet! Check out the cool family of tiles below. What do you notice?

A family of self-tiling tiles

A family of self-tiling tiles

Did you notice that the four shapes in the middle are the same as the four larger shapes on the outside? The four tiles in the middle can combine to create larger versions of themselves! They can make any and all of the original four!!

Lee Sallows

Recreational Mathematician, Lee Sallows

Naturally, I was reminded of the geomagic squares we featured a while back (more at geomagicsquares.com), and then I came to realize they were designed by the same person, the incredible Lee Sallows! (For another amazing one of Lee Sallows creations, give this incredible sentence a read.) You can also visit his website, leesallows.com.

reptile3

A family of 6 self-tiling tiles

For more self-tiling tiles (and there are many more amazing sets) click here. I have to point out one more in particular. It’s like a geomagic square, but not quite. It’s just wonderful. Maybe it ought to be called a “self-tiling latin square.”

And for a final item this week, we have a powerful drawing tool. It’s a website that reminds me a lot of recursive drawing, but it’s got a different feel and some excellent features. It’s called Doodal. Basically, whatever you draw inside of the big orange frame will be copied into the blue frames.  So if there’s a blue frame inside of an orange frame, that blue frame gets copied inside of itself… and then that copy gets copied… and then that copy…!!!

To start, why don’t you check out this amazing video showing off some examples of what you can create. They go fast, so it’s not really a tutorial, but it made me want to figure more things out about the program.

I like to use the “delete frame” button to start off with just one frame. It’s easier for me to understand if its simpler. You can also find instructions on the bottom. Oh, and try using the shift key when you move the blue frames. If you make something you like, save it, email it to us, and we’ll add it to our readers’ gallery.

Start doodaling!

Make something you love. Bon appetit!

A fractal Math Munch Doodal

A fractal Math Munch Doodal

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!