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


Squaring, Water Calculator, and Snap the Turtle

Welcome to this week’s Math Munch!

I’ve been really into squares lately. Maybe it’s because I recently ran across a new puzzle involving squares– something called Mrs. Perkin’s quilt.

Mrs. Perkin's quilt 1

69 by 69 Mrs. Perkin’s quilt.

The original version of the puzzle was published way back in 1907, and it went like this: “For Christmas, Mrs. Potipher Perkins received a very pretty patchwork quilt constructed of 169 square pieces of silk material. The puzzle is to find the smallest number of square portions of which the quilt could be composed and show how they might be joined together. Or, to put it the reverse way, divide the quilt into as few square portions as possible by merely cutting the stitches.”

Mrs. Perkin's quilt 18

18 by 18 Mrs. Perkin’s quilt

Said in another way, if you have a 13 by 13 square, how can you divide it up into the smallest number of smaller squares? Don’t worry, you get to solve it yourself– I’m not including a picture of the solution to that version of the puzzle because there are so many beautiful pictures of solutions to the puzzle when you start with larger and smaller squares. Some are definitely more interesting than others. If you want to start simple, try the 4 by 4 version. I particularly like the look of the solution to the 18 by 18 version.

Mrs. Perkin's quilt 152

152 by 152 Mrs. Perkin’s quilt

Maybe you’re wondering where I got all these great pictures of Mrs. Perkin’s quits. And– wait a second– is that the solution to the 152 by 152 version? It sure is– and I got it from one of my favorite math websites, the Wolfram Demonstrations Project. The site is full of awesome visualizations of all kinds of things, from math problems to scans of the human brain. The Mrs. Perkin’s quilts demonstration solves the puzzle for up to a 1,098 by 1,098 square!

Next up, we here at Math Munch are big fans of unusual calculators. Marble calculators, domino calculators… what will we turn up next? Well, here for your strange calculator enjoyment is a water calculator! Check out this video to see how it works:

I might not want to rely on this calculator to do my homework, but it certainly is interesting!

Snap the TurtleFinally, meet Snap the Turtle! This cute little guy is here to teach you how to make beautiful math art stars using computer programming.

On the website Tynker, Snap can show you how to design a program to make intricate line drawings– and learn something about computer programming at the same time. Tynker’s goal is to teach kids to be programming “literate.” Combine computer programming with a little math and art (and a turtle)– what could be better?

I hope something grabbed your interest this week! Bon appetit!

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!

Maria Chudnovsky, Puzzlebomb, and Some Futility

Welcome to this week’s Math Munch!

This week we meet an incredible mathematician, take on a tough number puzzle, check out a wonderful mathematical card trick, and much more.

Maria Chudnovsky

A while ago we shared an interview with mathematician Fan Chung Graham.  The interview was posted by Anthony Bonato, The Intrepid Mathematician. Well, this week we share another of his interviews, this time with Maria Chudnovsky, graph theorist and star of not one, but two television commercials. (A rare feat for a mathematician.) Maria is also a winner of the extraordinary MacArthur “Genius” Grant. You can check out the video below or click here for the full interview.

Up next, our friends over at The Aperiodical do a lot of great things for the math world. One contribution is the monthly Puzzlebomb put on by Katie Steckles.

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This month’s puzzle is MODOKU, a sort of sudoku style puzzle where columns and rows span the possible remainders mod 7 and mod 5. Check it out! Thanks to Katie for such a lovely puzzle! You can click below for an interactive version with complete instructions.

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Finally this week, it’s time again to look at a Futility Closet, a phenomenal blog containing the odd mathematical tidbit. We’ll take a look at three of them.

Screen Shot 2016-07-21 at 12.51.33 AMHere’s a weird arithmetic fact I found there. Do you see what’s going on there? I have absolutely no idea how often this kind of thing is true, if ever again, but it gets me thinking.

2016-07-14-a-square-triangleHere’s another incredible one. We’ve posted about Pascal’s (Yang-Hui’s) Triangle lots of times (1 2), and I’ve come across a lot of fascinating stuff about it, but this is new to me. Apparently, “the product of the six numbers surrounding any interior number in Pascal’s triangle is a perfect square.” Can you prove it?

Now on to the biggie…  This is such a cool card trick! Here’s the trick as explained by Futility Closet:

“I hand you an ordinary deck of 52 cards. You inspect and shuffle it, then choose five cards from the deck and hand them to my assistant. She looks at them and passes four of them to me. I name the fifth card.”         !!!!!!!!!!


The key to the magic is this chart:

{low, middle, high} = 1
{low, high, middle} = 2
{middle, low, high} = 3
{middle, high, low} = 4
{high, low, middle} = 5
{high, middle, low} = 6

Can you figure out how it works from the chart alone? You’ll need a good assistant to get on board, and it wouldn’t hurt to practice a bit. Then get ready to impress. Oh, and if you can’t figure out the trick from the chart alone, then just head over to Futility Closet and read the full explanation.

Well that’s it for this week. Hope you found something delicious. Bon appetit!

Fractions, Sam Loyd, and a MArTH Journal

This week we’re rewinding to July 2012 for some fun with the fabulous Farey Fractions—which have been on my mind recently—plus lots more! Bon appetit!

Math Munch

Welcome to this week’s Math Munch!

Check out this awesome graph:

What is it?  It’s a graph of the Farey Fractions, with the denominator of the (simplified) fraction on the vertical axis and the value of the fraction on the horizontal axis, made by mathematician and professor at Wheelock College Debra K. Borkovitz (previously).  Now, I’d never heard of Farey Fractions before I saw this image – but the graph was so cool that I wanted to learn all about them!

So, what are Farey Fractions, you ask?  Debra writes all about them and the cool visual patterns they make in this post.  To make a list of Farey Fractions you first pick a number – say, 5.  Then, you list all of the fractions between 0 and 1 whose denominators are less than or equal to the number you picked.  So, as Debra writes in…

View original post 560 more words

Wild Maths, Ambiguous Cylinders, and 228 Women

Welcome to this week’s Math Munch!

You should definitely take some time to explore Wild Maths, a site dedicated to the creative aspects of mathematics. Wild Maths is produced by the Millennium Mathematics Project, which also makes NRICH and Plus.


I won!

One fun things you’ll find on Wild Maths is a game called Square It! You can play it with a friend or against the computer. The goal is to color dots on a square grid so that you are the first to make a square in your color. It is quite challenging! To the left you’ll find my first victory against the computer after losing the first several matches.

You’ll find lots more on Wild Maths, including an equal averages challenge, a number grid journey, and some video interviews with mathematicians Katie Steckles and Nira Chamberlain. Wild Maths also has a Showcase of work that has been submitted by their readers, much like our own Readers’ Gallery. (We love hearing from you and seeing your creations!)

Next up is a video of an amazing illusion:

Now, I am as big of a fan of squircles as anyone, but this video really threw me for a loop. The illusion just gets crazier and crazier! The illusion was designed by Kokichi Sugihara of Meiji University in Japan. It recently won second place in the Best Illusion of the Year Contest.

We are fortunate that Dave Richeson has hit it out of the park again, this time sharing both an explanation of the mathematics behind the illusion and a paper template you can use to make your own ambiguous cylinder!

PWinmathFinally this week, I’d like to share a fascinating document with you. It is a supplement to a book called Pioneering Women in American Mathematics: The Pre-1940s PhD’s by Judy Green and Jeanne LaDuke.

The supplement gives biographies of all 228 American women who earned their PhD’s in mathematics during the first four decades of the 20th century. You might enjoy checking out this page from the National Museum of American History, which describes some about the origin of the book project.


Judy Green, Jeanne LaDuke, and fifteen women who received their PhD’s in math before 1940.

I hope you will find both pleasure and inspiration in reading the stories of these pioneers in American mathematics. I have found them to be a lot of fun to read.

Bon appetit!

SliceForm, Rinus Roelofs, and krazydad

Welcome to this week’s Math Munch!

For the 5th and final Thursday of June we will once again take a look at some of the goodness over on our facebook page, and oh my goodness what a huge load of goodness we have indeed! For an appetizer, how about this little visual problem posted by ThinkFun Games? (If you remind me in the comments, I’ll tell you the neat way I thought about solving it.)

Circle areasThe shape consists of overlapped color circles.  Which two colors have their total visible areas equal? (click to enlarge)

Now onto the main course. I have to show you this incredible new math art tool called SLICEFORM STUDIO. Click over and check out their gallery to begin with. Just gorgeous.

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My first creation on SliceForm.com

There’s a tutorial page as well, but the best thing to do is probably just to start playing with the app itself.  DIG IN! The site is sort of made for people who can use laser cutters to do the paper and stuff, but you can also just click “trace and export strips” and then color it in and export the image. On the right, you can see my first creation. Email yours to mathmunchteam@gmail.com and we’ll stick it in our readers’ gallery.

Alright, up next is an amazing mathematical artist by the name of Rinus Roelofs. (You might remember the paper project of his that we shared at new year.) Well, Rinus is just an unblievable and prolific maker of incredible and beautiful things. Check out his website. (He has two, I think)

I follow Rinus on facebook, and he’s always posting pictures of his works in progress, and they are stunning. First, check out this gallery of Interwoven Ring Patterns he recently posted. Then take a look at his timeline photos. Lots of overlapping patterns and Möbius shapes.

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A completed galaxy puzzle.  Each colored area has rotational symmetry

Finally, have you ever heard of Galaxy Puzzles?  I hadn’t either, but you can find lots of them over on the wonderful puzzling site, krazydad. The puzzle begins with lots of dots, and your goal is to separate the dots by making enclosures that have 180 degree rotational symmetry. Print and play galaxy puzzles are available as well as an interactive online version. There are lots of other puzzles available as well, but I think Battleships is a pretty cool. You might give that a try too.


But wait, there’s more. With 5 Thursdays in a month, there’s just lots to share, so you also get some bonus stuff!

That’s it for June. See you next time. Bon appetit!