Tag Archives: origami

Mathematical Impressions, Modular Origami, and the Tenth Dimension

Welcome to this week’s Math Munch!

First up, check out the latest video in George Hart‘s series called “Mathematical Impressions.”  George has been making videos for “Mathematical Impressions,” which is sponsored by the Simons Foundation, since summer, when he made his video debut – so there are many videos to watch!  Here’s his newest video, called, “Attesting to Atoms,” about how the geometric structure of crystals gives clues to the existence of atoms.  (Click on the picture below to watch the video.)

Atoms video

I love how this video shows a real way in which knowledge of mathematics – which can seem very abstract at times – can help us to understand the structure of the world, which is very concrete.  In this second video, one of my favorites, George talks about the reverse of that – allowing our knowledge of something concrete to help us understand abstract mathematics.  This video is called, “Knot Possible.”  (Again, click on the picture to watch the video!)

Knot video

I could have used these words of wisdom from George when I was thinking about the problem he poses in this video: “Don’t let your knowledge of mathematics artificially limit what you think is physically possible.  Quite to the contrary!  Mathematics is a tool which can empower us to do amazing things that no one has ever done before.”  Well said, George!

sierpinski-tetrahedron-tri-2Speaking of using mathematics to do and make amazing things, check out this website of modular origami models and patterns!

This site was put together by Michal Kosmulski, who lives in Poland and works in information technology.  In addition, however, he folds these amazing modular origami polyhedra, fractals, and other awesome mathematical objects!  Michal’s site is full of pictures of his modular origami creations and links to patterns for how to make them yourself as well as information about the mathematics behind the objects.  He has also included some useful tips on how to make the more challenging shapes.

fit-five-intersecting-tetrahedra-60deg-2One of my favorites is the object to the left, “Five Intersecting Tetrahedra.”  I think that this structure is both beautiful and very interesting.  It can be made by intersecting five tetrahedra, or triangular-based pyramids, as shown, or by making a stellation of an icosahedron.  What does that mean?  Well, an icosahedron is a polyhedron with twenty equilateral triangular faces.  To stellate a polyhedron, you extend some element of the polyhedron – such as the faces or edges – in a symmetric way until they meet to form a new polyhedron.  There are 59 possible stellations of the icosahedron!  Michal has models of several of them, including the Five Intersecting Tetrahedra and the great stellated icosahedron shown below on the left.  The figure on the right is called “Cube.”


Finally, all the talking about dimensions that we’ve been doing for the past few weeks reminded me of my favorite video about higher dimensions.  It’s called, “Imagining the Tenth Dimension,” and it shows a way of thinking about dimensions, from the zero dimension all the way up to the tenth.  I can watch this video again and again and still find it mind-blowing and fascinating.

Bon appetit!

Newroz, a Math Factory, and Flexagons

Welcome to this week’s Math Munch!

You’ve probably seen Venn diagrams before. They’re a great way of picturing the relationships among different sets of objects.

But I bet you’ve never seen a Venn diagram like this one!

Frank Ruskey

That’s because its discovery was announced only a few weeks ago by Frank Ruskey and Khalegh Mamakani of the University of Victoria in Canada. The Venn diagrams at the top of the post are each made of two circles that carve out three regions—four if you include the outside. Frank and Khalegh’s new diagram is made of eleven curves, all identical and symmetrically arranged. In addition—and this is the new wrinkle—the curves only cross in pairs, not three or more at a time. All together their diagram contains 2047 individual regions—or 2048 (that’s 2^11) if you count the outside.

Frank and Khalegh named this Venn diagram “Newroz”, from the Kurdish word for “new day” or “new sun”. Khalegh was born in Iran and taught at the University of Kurdistan before moving to Canada to pursue his Ph.D. under Frank’s direction.

Khalegh Mamakani

“Newroz” to those who speak English sounds like “new rose”, and the diagram does have a nice floral look, don’t you think?

When I asked Frank what it was like to discover Newroz, he said, “It was quite exciting when Khalegh told me that he had found Newroz. Other researchers, some of my grad students and I had previously looked for it, and I had even spent some time trying to prove that it didn’t exist!”

Khalegh concurred. “It was quite exciting. When I first ran the program and got the first result in less than a second I didn’t believe it. I checked it many times to make sure that there was no mistake.”

You can click these links to read more of my interviews with Frank and Khalegh.

I enjoyed reading about the discovery of Newroz in these articles at New Scientist and Physics Central. And check out this gallery of images that build up to Newroz’s discovery. Finally, Frank and Khalegh’s original paper—with its wonderful diagrams and descriptions—can be found here.

A single closed curve—or “petal”— of Newroz. Eleven of these make up the complete diagram.

A Venn diagram made of four identical ellipses. It was discovered by John Venn himself!

For even more wonderful images and facts about Venn diagrams, a whole world awaits you at Frank’s Survey of Venn Diagrams.

On Frank’s website you can also find his Amazing Mathematical Object Factory! Frank has created applets that will build combinatorial objects to your specifications. “Combinatorial” here means that there are some discrete pieces that are combined in interesting ways. Want an example of a 5×5 magic square? Done! Want to pose your own pentomino puzzle and see a solution to it? No problem! Check out the rubber ducky it helped me to make!

A pentomino rubber ducky!

Finally, Frank mentioned that one of his early mathematical experiences was building hexaflexagons with his father. This led me to browse around for information about these fun objects, and to re-discover the work of Linda van Breemen. Here’s a flexagon video that she made.

And here’s Linda’s page with instructions for how to make one. Online, Linda calls herself dutchpapergirl and has both a website and a YouTube channel. Both are chock-full of intricate and fabulous creations made of paper. Some are origami, while others use scissors and glue.

I can’t wait to try making some of these paper miracles myself!

Bon appetit!

Music Box, FatFonts, and the Yoshimoto Cube

Welcome to this week’s Math Munch!

The Whitney Music Box

Jim Bumgardner

Solar Beat

With the transit of Venus just behind us and the summer solstice just ahead, I’ve got the planets and orbits on my mind. I can’t believe I haven’t yet shared with you all the Whitney Music Box. It’s the brainchild of Jim Bumgardner, a man of many talents and a “senior nerd” at Disney Interactive Labs. His music box is one of my favorite things ever–so simple, yet so mesmerizing.

It’s actually a bunch of different music boxes–variations on a theme. Colored dots orbit in circles, each with a different frequency, and play a tone when they come back to their starting points. In Variation 0, for instance, within the time it takes for the largest dot to orbit the center once, the smallest dot orbits 48 times. There are so many patterns to see–and hear! There are 21 variations in all. Go nuts! In this one, only prime dots are shown. What do you notice?

You can find a more astronomical version of this idea at SolarBeat.

Above you’ll find a list of the numerals from 1 to 9. Or is it 0 to 9?

Where’s the 0 you ask? Well, the idea behind FatFonts is that the visual weight of a number is proportional to its numerical size. That would mean that 0 should be completely white!

FatFonts can also be nested. The first number below is 64. Can you figure out the second?

This is 64 in FatFonts.

What number is this?
Click to zoom!

FatFonts was developed by the team of Miguel NacentaUta Hinrichs, and Sheelagh Carpendale. You can see some uses that FatFonts has been put to on their Gallery page, and even download FatFonts to use in your word processor. Move over, Times New Roman!

This past week, Paul pointed me to this cool video by George Hart about interlocking complementary polyhedra that together form a cube. It reminded me of something I saw for the first time a few years ago that just blew me away. You have to see the Yoshimoto Cube to believe it:

In addition to its more obvious charms, something that delights me about the Yoshimoto Cube is how it was found so recently–only in 1971, by Naoki Yoshimoto.  (That other famous cube was invented in 1974 by Ernő Rubik.) How can it be that simple shapes can be so inexhaustible? If you’re feeling inspired, Make Magazine did a short post on the Yoshimoto Cube a couple of years that includes a template for making a Yoshimoto Cube out of paper. Edit: These template and instructions aren’t great. See below for better ones!

Since it’s always helpful to share your goals to help you stick to them, I’ll say that this week I’m going to make a Yoshimoto Cube of my own. Begone, back burner! Later in the week I’ll post some pictures below. If you decide to make one, share it in the comments or email us at


We’d love to hear from you.

Bon appetit!


Here are the two stellated rhombic dodecahedra that make the Yoshimoto Cube that Paul and I made! Templates, instructions, and video to follow!

Here are two different templates for the Yoshimoto cubelet. You’ll need eight cubelets to make one star.

And here’s how you tape them together:

Polyominoes, Rubix, and Emmy Noether

Welcome to this week’s Math Munch!

Check out the Pentomino Project, a website devoted to all things about polyominoes by students and teachers from the K. S. O. Glorieux Ronse school in Belgium.

Their site is full of lots of useful information about polyominoes, such as what the different polyominoes look like and how they are formed.

In this puzzle, place the twelve pentominoes as "islands in a sea" so that the area of the sea is a small as possible. The pentominoes can't touch, even at corners. Here's a possible solution.

Even more awesome, though, is their collection of polyomino puzzles – about dissections, congruent pieces, tilings, and more!  They have a contest every year  – and people from around the world are encouraged to participate!  If you solve a puzzle, you can send them your solution and they might post it on their site.

Next, have you ever thought to yourself, “Gee, I wonder if I can make my own Rubix Cube?”  Well, sixth grader August did just that.  And, after several days of searching for patterns and working hard with paper, scissors, string, and tape, August succeeded!  His 2-by-2 Rubix Cube works just like any other, is fun to play with, and – even better – was fun to make.

Try it yourself:

Finally, ever heard of Emmy Noether?  It’s not surprising if you haven’t, because, according a New York Times article about her, “few can match in the depths of her perverse and unmerited obscurity….”  But, she was one of the most influential mathematicians and scientists of the 20th century – and was named by Albert Einstein the most “significant” and “creative” woman mathematician of all time.  You can read about Emmy’s influential theorem, and her struggles to become accepted in the mathematical community as a Jewish woman, in this article.

Want to learn more about women mathematicians throughout history?  Check out this site of biographies from Agnes Scott College.

Bon appetit!

Origami, Games, and the Huang Twins

Welcome to this week’s Math Munch!

Origami Whale

We’ve had a few posts (like this, this, and this) that included paper folding, but this week we really focus on doing it yourself.  Check out Origamiplayer.com, a terrific website that doesn’t just show you origami models.  It has an animator that folds them in front of you and waits for you to fold along with it.  I really like this origami pentagon, but there’s lots of designs and you can even sort them by type or difficulty.  You can change the speed or click around to different steps, so find a model you like and get folding!

Up next, meet the Huang Twins, 14-year old brothers from California.  Mike and Cary have been working as a team to design and program all kinds of great web stuff.  They actually have their own orgami animator to fold polyhedra.  But my favorite thing of theirs is The Scale of the Universe 2, an incredible applet that let’s you compare the sizes of all kinds of things big and small.  It uses scientific notation to describe the sizes, so if you’ve never seen that before, you might want to read up.  It’s genius.

They’ve also written several excellent games, which we’ve added to our Math Games page.  Cube Roll has a familiar format with a twist; The cube has to land on the correct side.  I really like that one.  No Walking, No Problem is another neat little puzzler.  Use the objects to move side to side, because you can’t walk!  Lastly, (though the Huangs didn’t write it) we’ve added Morpion Solitaire, a tough little game you can play online or on paper.

Bon appetit!

Cube Roll

No Walking, No Problem

Morpion Solitaire

Cubes, Curves, and Geometric Romance

Welcome to this week’s Math Munch!

If you like Rubik’s Cubes, then check out Oskar van Deventer’s original Rubik’s cube-type puzzles!  Oskar is a Dutch scientist who has been designing puzzles since he was 12 years old.  He makes many of his puzzles using a 3D printer, with a company called Shapeways.

Oskar has posted a number of videos of himself explaining his creations.  Here’s him demonstrating the Oh Cube:

Next, take a look at these beautiful curved-crease sculptures made by MIT mathematician and origami artist Erik Demaine and his father, Martin Demaine.  Erik and Martin make these hyperbolic paraboloid structures by folding rings of creases in a circular piece of paper.  They have exhibits of their artwork in various museums and galleries, including in the MoMA permanent collection and the Guided By Invoices gallery in Chelsea, NYC.  So, if you live in NYC, then you could go see these!

Want to learn how to fold your own hyperbolic paraboloid?  Erik has these instructions for making one out of a square piece of paper with straight folds.

Finally, here is a wonderful video made of Norton Juster’s picture book, The Dot and the Line.  Enjoy!

Bon appetit!

Math Craft, Philippa Fawcett, and Mandelbrot

Welcome to this week’s Math Munch!

Math Craft is a supersweet website where members submit their mathematically inspired art and instructions about how to make your own.  I love the polyhedra made out of pennies in the masthead, these curve stitches, and these polyhedral pumpkins!  Here is a link to Math Craft’s welcome page, authored by admin Cory Poole.  Cory is a math and physics teacher at University Preparatory School in California. The welcome page includes some instructions for creating some great paper polyhedra. Math Craft is just starting up; I’m sure there will be many more great project to be found there in the future!

Philippa Fawcett, who broke the glass ceiling of Cambridge mathematics

An article recently appeared on the Past Imperfect blog on Smithsonian.com about the compelling story of Philippa Fawcett. Fawcett was the first and only woman to make the highest score on the Cambridge tripos mathematical exam.  She did so during an age when the predominant opinion was that women were incapable and weak and certainly couldn’t excel at mathematics.  Fawcett’s performance on this exam did much to dispel this prejudice.  The article not only relates an interesting chapter from history, but also give an inspiring account of a person’s drive to success despite enormous obstacles.

Finally, by request, a journey through the Mandelbrot set:

[youtube http://www.youtube.com/watch?v=F_nfHY61T-U&feature=related]

Benoit Mandelbrot, the father of fractal geometry, passed away about a year ago. You can listen to his outstanding TED talk about his life’s work here. I love his enthusiasm and curiosity, as well as how he can find marvels in the seemingly ordinary.  Also, how much fun is the way he pronounces “cauliflower”?!  You can find a memorial to Benoit Mandelbrot in last November’s edition of Peer Points.

Bon appetit!