Tag Archives: sculpture

Squricangle, Magic Angle Sculpture, and …

Welcome to this week’s Math Munch!

There’s a neat old problem/puzzle that goes like this: make a 3-D shape that could fit snugly through each of three holes—one a square, one a circle, and one a triangle. To make a shape that works for just two holes isn’t so tricky. For example, a cylinder that is just as tall as it is across would fit snugly through a circle hole and a square hole. Can you think of what would work for each of the other two shape combos? What about all three?


Three holes, three shapes…and what’s that over in the corner??

If you’re curious about the answer, you might enjoy this post by Kit Wallace or this page by George Hart or—believe it or not—roundsquaretriangle.com. I don’t know the origin of this puzzle and would love to. I haven’t found any info about it after to poking around the internet for a while. So if you locate any information about the backstory of the squircangle—which is not its real name, just one that I made up—please let us know!

Even though I knew about the square-circle-triangle problem, I was not at all prepared to encounter the solution to the jet-butterfly-dragon problem!


Dragon Butterfly Jet is just one of several “magic angle sculptures” created by artist, chemist, and PhD, and high school dropout John V. Muntean. John writes the following in his Artist Statement:

As a scientist and artist, I am interested in the how perception influences our theory of the universe. … Every 120º of rotation, the amorphous shadows evolve into independent forms. Our scientific interpretation of nature often depends upon our point of view. Perspective matters.

There’s much more to see on John’s website. And you can check out Dragon Butterfly Jet in action in the video below, along with Knight Mermaid Pirate-Ship. I also recommend this video made by John where he demonstrates how his sculpture works himself. It also includes a stop-frame animation of the sculpture being built! So cool.


No, not ellipses…

And finally, what you’ve all been waiting for…


That’s right! My final share of the week is that most outspoken of punctuation marks, the ellipsis. Because often what you don’t say says a whole lot! That’s true when writing a story or some dialogue, and it’s also true in mathematics. Watch: 1+2+3+…+100. See? Pretty neat! Those three dots sure say a mouthful…

The ellipsis is probably my second favorite punctuation mark—after the em dash, of course. But don’t take my word for it. Instead, check out this article about the history and uses—mathematical and otherwise—of the humble ellipsis. Author Cameron Hunt McNabb writes:

Thus the ellipsis has been used to indicate anything from the erroneous to the irrational, and its intrigue lies in resistance to meaning. As long as we have things to say, we will have things to omit.


The very first equals sign, in 1557.

I could go on and on about the ellipsis, just like pi does: 3.1415… But anyway, while we’re on the subject of punctuation, let me point you to one of my favorite sites on the mathematical internet: the Earliest Uses of Various Mathematical Symbols page, maintained by Jeff Miller. Jeff teaches high school math in Florida and also has some other great pages, too, including this one about mathematicians featured on stamps.




A nice visualization of the squircangle by Matt Henderson


Lucea, Fiber Bundles, and Hamilton

Welcome to this week’s Math Munch!

The Summer Olympics are underway in Brazil. I have loved the Olympics since I was a kid. The opening ceremony is one of my favorite parts—the celebration of the host country’s history and culture, the athletes proudly marching in and representing their homeland. And the big moment when the Olympic cauldron is lit! This year I was just so delighted by the sculpture that acted as the cauldron’s backdrop.

Isn’t that amazing! The title of this enormous metal sculpture is Lucea, and it was created by American sculptor Anthony Howe. You can read about Anthony and how he came to make Lucea for the Olympics in this article. Here’s one quote from Anthony:

“I hope what people take away from the cauldron, the Opening Ceremonies, and the Rio Games themselves is that there are no limits to what a human being can accomplish.”

Here’s another view of Lucea from Anthony’s website:

Lucea is certainly hypnotizing in its own right, but I think it jumped out at me in part because I’ve been thinking a lot about fiber bundles recently. A fiber bundle is a “twist” on a simpler kind of object called a product space. You are familiar with some examples of products spaces. A square is a line “times” a line. A cylinder is a line “times” a circle. And a torus is a circle “times” a circle.


Square, cylinder, and torus.

So, what does it mean to introduce a “twist” to a product space? Well, it means that while every little patch of your object will look like a product, the whole thing gets glued up in some fancy way. So, instead of a cylinder that goes around all normal, we can let the line factor do a flip as it goes around the circle and voila—a Mobius strip!


Now, check out this image:


It’s two Mobius strips stuck together! Does this remind you of Lucea?! Instead of a line “times” a circle that’s been twisted, we have an X shape “times” a circle.

Do you think you could fill up all of space with an infinity of circles? You might try your hand at it. One answer to this puzzle is a wonderful example of a fiber bundle called the Hopf fibration. Just as you can think about a circle as a line plus one extra point to close it up, and a sphere as a plane with one extra point to close it up, the three-sphere is usual three-dimenional space plus one extra point. The Hopf fibration shows that the three-sphere is a twisted product of a sphere “times” a circle. For a really lovely visualization of this fact, check out this video:

That is some tough but also gorgeous mathematics. Since you’ve made it this far in the post, I definitely think you deserve to indulge and maybe rock out a little. And what’s the hottest ticket on Broadway this summer? I hope you’ll enjoy this superb music video about Hamilton!

William Rowan Hamilton, that is. The inventor of quaternions, explorer of Hamiltonian circuits, and reformulator of physics. Brilliant.

citymapHere are a couple of pages of Hamiltonian circuit puzzles. The goal is to visit every dot exactly once as you draw one continuous path. Try them out! Rio, where the Olympics is happening, pops up as a dot in the first one. You might even try your hand at making some Hamiltonian puzzles of your own.

Happy puzzling, and bon appetit!

MoMA, Pop-Up Books, and A Game of Numbers

Welcome to this week’s Math Munch!

Thank you so much to everyone who participated in our Math Munch “share campaign” over the past two weeks. Over 200 shares were reported and we know that even more sharing happened “under the radar”. Thanks for being our partners in sharing great math experiences and curating the mathematical internet.

Of course, we know that the sharing will continue, even without a “campaign”. Thanks for that, too.

All right, time to share some math. On to the post!

N_JoshiTo kick things off, you might like to check out our brand-new Q&A with Nalini Joshi. A choice quote from Nalini:

In contrast, doing math was entirely different. After trying it for a while, I realized that I could take my time, try alternative beginnings, do one step after another, and get to glimpse all kinds of possibilities along the way.

By Philippe Decrauzat.

By Philippe Decrauzat.

I hope the math munches I share with you this week will help you to “glimpse all kinds of possibilities,” too!

Recently I went to the Museum of Modern Art (MoMA) in New York City. (Warning: don’t confuse MoMA with MoMath!) On display was an exhibit called Abstract Generation. You can view the pieces of art in the exhibit online.

As I browsed the galley, the sculptures by Tauba Auerbach particularly caught my eye. Here are two of the sculptures she had on display at MoMA:

CRI_244599 CRI_244605

Just looking at them, these sculptures are definitely cool. However, they become even cooler when you realize that they are pop-up sculptures! Can you see how the platforms that the sculptures sit on are actually the covers of a book? Neat!

Here’s a video that showcases all of Tauba’s pop-ups in their unfolding glory. Why do you think this series of sculptures is called [2,3]?

This idea of pop-up book math intrigued me, so I started searching around for some more examples. Below you’ll find a video that shows off some incredible geometric pop-ups in action. To see how you can make a pop-up sculpture of your own, check out this how-to video. Both of these videos were created by paper engineer Peter Dahmen.

Taura Auerbach.

Tauba Auerbach.

Tauba got me thinking about math and pop-up books, but there’s even more to see and enjoy on her website! Tauba’s art gives me new ways to connect with and reimagine familiar structures. Remember our post about the six dimensions of color? Tauba created a book that’s a color space atlas! The way that Tauba plays with words in these pieces reminds me both of the word art of Scott Kim and the word puzzles of Douglas Hofstadter. Some of Tauba’s ink-on-paper designs remind me of the work of Chloé Worthington. And Tauba’s piece Componants, Numbers gives me some new insight into Brandon Todd Wilson’s numbers project.

0108 MM MM-Tauba-Auerbach-large

This piece by Tauba is a Math Munch fave!

For me, both math and art are all about playing with patterns, images, structures, and ideas. Maybe that’s why math and art make such a great combo—because they “play” well together!

Speaking of playing, I’d like to wrap up this week’s post by sharing a game about numbers I ran across recently. It’s called . . . A Game of Numbers! I really like how it combines the structure of arithmetic operations with the strategy of an escape game. A Game of Numbers was designed by a software developer named Joseph Michels for a “rapid” game competition called Ludum Dare. Here’s a Q&A Joseph did about the game.

A Game of Numbers.

A Game of Numbers.

If you enjoy A Game of Numbers, maybe you’ll leave Joseph a comment on his post about the game’s release or drop him an email. And if you enjoy A Game of Numbers, then you’d probably enjoy checking out some of the other games on our games page.

Bon appetit!

PS Tauba also created a musical instrument called an auerglass that requires two people to play. Whooooooa!

Reflection Sheet – MoMA, Pop-Up Books, and A Game of Numbers