Tag Archives: doodling

Zentangle, Graph Paper, and Pancake Art

My recent doodling.

Some recent doodling, by me.

Welcome to this week’s Math Munch!

As you start a new school year, you might be looking for some new mathy doodle games to play in the margins of your notebooks. Doodling helps me to listen sometimes, and I love making neat patterns. I especially like seeing what new shapes I can make.

This summer I was very happy to run across Zentangle®—”an easy-to-learn, relaxing, and fun way to create beautiful images by drawing structured patterns.” I’ve learned a lot about Zentangle from a blog called Tangle Bucket by Sandy Hunter. She shares how to doodle snircles, snafoozles, and oodles. There’s a whole dictionary of zentangle shapes over at tanglepatterns.com.

My favorite idea in Zentangle is trying to combine two kinds of designs. Sometimes this is described as one pattern “versus” another one. For instance, check out these:


Maybe you’ll pick some tangle patterns to combine with each other. If you try some, maybe you’ll share them in our Readers’ Gallery.

Sandy writes:

It’s so true that the more I tangle, the more I see the potential in patterns all around me. I catch myself mentally deconstructing them (whether I want to or not) to figure out if they can be broken down into simple steps without too much effort. That’s the trademark of a good tangle pattern.

Try some of Sandy’s weekly challenges, or check out Tiffany Lovering’s time-lapse videos—here’s one with music and one with an interview. Can you learn the names of any of the shapes she creates? I spy a Rick’s Paradox. There are lots of ways to begin zentangling—I hope you enjoy giving it a try.

Squares and dots and crosses, oh my!

Squares & dots & crosses, oh my!

If zentangling is too freeform for your doodling tastes, then let me share with you one of my longtime favorite websites. I’ve used it for years to help me to do math and to teach math, and it’s great for math doodling, too. I might even call it a trusty friend, one that I met one day through the simple online search: “free online graph paper”.

That’s right, it’s Free Online Graph Paper.

Something I love about the site is that it lets you design different aspects of your graph paper. Then you can print it out. First you get to decide what kind of grid you would like: square? triangular? circular? Then you get to tinker with lots of variables, like how big the grid cells are, how dark the lines are, and what color they are. And more!

Free Online Graph Paper was created by Kevin MacLeod, who composes music and shares it for free. That way other people can use it for creative projects. That’s really awesome! I enjoyed listening to Kevin’s “Winner Winner“. It’s always good to be reminded that everything you use or enjoy was almost certainly made by a person—including custom graph paper websites!

A 7/3 star spirocake.

A 7/3 star spirocake.

Last up this week is some doodly math that you can really munch on. Everyone knows that breakfast is the most important meal of the day and that the most important food group is roulette curves.

To get your daily recommended allowance of groovy math, look no further than the edible doodles of Nathan Shields and his family over at Saipancakes.

I can wait until the Shields family tackles the cissoid of Diocles.

Bon appetit!

Origami Stars, Tessellation Stars, and Chaotic Stars

Welcome to this week’s star-studded Math Munch!

downloadModular origami stars have taken the school I teach in by storm in recent months! We love making them so much that I thought I’d share some instructional videos with you. My personal favorite is this transforming eight-pointed star. It slides between a disk with a hole the middle (great for throwing) and a gorgeous, pinwheel-like eight-pointed star. Here’s how you make one:

Another favorite is this lovely sixteen-pointed star. You can make it larger or smaller by adding or removing pieces. It’s quite impressive when completed and not that hard to make. Give it a try:

type6thContinuing on our theme of stars, check out these beautiful star tessellations. They come from a site made by Jim McNeil featuring oh-so-many things you can do with polygons and polyhedra. On this page, Jim tells you all about tessellations, focusing on a category of tessellations called star and retrograde tessellations.

type3b400px-Tiling_Semiregular_3-12-12_Truncated_Hexagonal.svgTake, for example, this beautiful star tessellation that he calls the Type 3. Jim describes how one way to make this tessellation is to replace the dodecagons in a tessellation called the 12.12.3 tessellation (shown to the left) with twelve-pointed stars. He uses the 12/5 star, which is made by connecting every fifth dot in a ring of twelve dots. Another way to make this tessellation is in the way shown above. In this tessellation, four polygons are arranged around a single point– a 12/5 star, followed by a dodecagon, followed by a 12/7 star (how is this different from a 12/5 star?), and, finally, a 12/11-gon– which is exactly the same as a dodecagon, just drawn in a different way.

I think it’s interesting that the same pattern can be constructed in different ways, and that allowing for cool shapes like stars and different ways of attaching them can open up crazy new worlds of tessellations! Maybe you’ll want to try drawing some star tessellations of your own after seeing some of these.

Screenshot 2014-05-12 10.48.46Finally, to finish off our week of everything stars, check out the star I made with this double pendulum simulator.  What’s so cool about the double pendulum? It’s a pendulum– a weight attached to a string suspended from a point– with a second weight hung off the bottom of the first. Sounds simple, right? Well, the double pendulum actually traces a chaotic path for most sizes of the weights, lengths of the strings, and angles at which you drop them. This means that very small changes in the initial conditions cause enormous changes in the path of the pendulum, and that the path of the pendulum is not a predictable pattern.

Using the simulator, you can set the values of the weights, lengths, and angles and watch the path traced on the screen. If you select “star” under the geometric settings, the simulator will set the parameters so that the pendulum traces this beautiful star pattern. Watch what happens if you wiggle the settings just a little bit from the star parameters– you’ll hardly recognize the path. Chaos at work!

Happy star-gazing, and bon appetit!

Isomorphisms in Five, Parquet Deformations, and POW!

Welcome to this week’s Math Munch!

Here’s a catchy little video. It’s called “Isomorphisms in Five.” Can you figure out why? The note posted below the video says:

An isomorphism is an underlying structure that unites outwardly different mathematical expressions. What underlying structure do these figures share? What other isomorphisms of this structure will you discover?

One of the reasons I LOVE this video is because I really like how the shapes change with the music– which is played in a very interesting time signature. I also love how you can learn a lot about the different growing shape patterns by comparing them. Watch how they grow as the video flips from pattern to pattern. What do you notice? What does the music tell you about their growth?

This video is by a math educator from North Carolina named Stuart Jeckel. The only thing written about him on his “About” page is, “The Art of Math”– so he’s a bit of a mystery! He has three more beautiful videos, all of which present little puzzles for you to solve. Check them out!

(Five-four isn’t a common time-signature for music, but it makes some great pieces. Check out this particularly awesome one. Anyone want to try making a growing shape pattern video to this tune?)


Here is an example of one of my favorite types of geometric patterns– the parquet deformation. To make one, you start with a tessellation. Then you change it- very gradually- until you’ve made a completely different tessellation that’s connected by many tiny steps to the original one.

I love to draw them. It’s challenging, but full of surprises. I never know what it’s going to look like in the end.

2012_10_31-par5Want to try making your own? Check out this site by the professors/architects Tuğrul Yazar and Serkan Uysal. They had one of their classes map out how some different parquet deformations are made. They mostly used computers, but you could follow their instructions by hand, if you like. The image above is a map for the first deformation I showed.

Click on this link to see some awesome deformations made out of tiles. Aren’t they beautiful? And here’s one made by mathematical artist Craig Kaplan. It has a great fractal quality to it:


Finally, here’s something I’ve been meaning to share with you for ages! Do you ever crave a good puzzle and aren’t sure where to find one? Look no farther than the Saint Ann’s School Problem of the Week! Each week, math teacher Richard Mann writes a new awesome problem and posts it on this website. Here’s this week’s problem:

For November 26, 2013– In the picture below, find the shaded right triangle marked A, the equilateral triangle marked B and the striped regular hexagon marked C. Six students make the following statements about the picture below: Anne says “I can find an equilateral triangle three times the area of B.”  Ben says I can find an equilateral triangle four times the area of B.” Carol says, “I can find a find a right triangle triple the area of A.” Doug says, “I can find a right triangle five times the area of A.” Eloise says, “I can find a regular hexagon double the area of C.” Frank says, “I can find a regular hexagon three times the area of C.” Which students are undoubtedly mistaken?

30- 60-90

If you solve this week’s problem, send us a solution!

Bon appetit!

Circling, Squaring, and Triangulating

Welcome to this week’s Math Munch!

How good are you at drawing circles? To find out, try this circle drawing challenge. There are adorable cat pictures for prizes!

What’s the best score you can get? And hey—what’s the worst score you can get? And how is your score determined? Well, no matter how long the path you draw is, using that length to make a circle would surround the most area. How close your shape gets to that maximum area determines your score.

Do you think this is a good way to measure how circular a shape is? Can you think of a different way?

Dido, Founder and Queen of Carthage.

Dido, Founder and Queen of Carthage.

This idea that a circle is the shape that has the biggest area for a fixed perimeter reminds me of the story of Dido and her famous problem. You can find a retelling of it at Mathematica Ludibunda, a charming website that’s home to all sorts of mathematical stories and puzzles. The whole site is written in the voice of Rapunzel, but there’s a team of authors behind it all. Dido’s story in particular was written by a girl named Christa.

If you have any trouble drawing circles in the applet, you might try using pencil and paper or a chalkboard. I bet if you practice your circling and get good at it, you might even be able to challenge this fellow:

The simple perfect squared square of smallest order.

The simple perfect squared square
of smallest order.

Next up is squaring and the incredible Squaring.Net. The site is run by Stuart Anderson, who works at the Reserve Bank of Australia and lives in Sydney.

The site gathers together all of the research that’s been done about breaking up squares and rectangles into squares. It’s both a gallery and an encyclopedia. I love getting to look at the timelines of discovery—to see the progress that’s been made over time and how new things have been discovered even this year! Just within the last month or so, Stuart and Lorenz Milla used computers to show that there are 20566 simple perfect squared squares of order 30. Squaring.Net also has a wonderful links page that can connect you to more information about the history of squaring, as well as some of the delightful mathematical art that the subject has inspired.

trinity-glass2-small sqBox8 wp4f6b3871_0f

Delaunay triangulationLast up this week is triangulating. There are lots of ways to chop up a shape into triangles, and so I’ll focus on one particular way known as a Delaunay triangulation. To make one, scatter some points on the plane. Then connect them up into triangles so that each triangle fits snugly into a circle that contains none of the scattered points.

Fun Fact #1: Delaunay triangulations are named for the Soviet mathematician Boris Delaunay. What else is named for him? A mountain! That’s because Boris was a world-class mountain climber.

Fun Fact #2: The idea of Delaunay triangulations has been rediscovered many times and is useful in fields as diverse as computer animation and engineering.

Here are two uses of Delaunay triangulations I’d like to share with you. The first comes from the work of Zachary Forest Johnson, a cartographer who shares his work at indiemaps.com. You can check out a Delaunay triangulation applet that he made and read some background about this Delaunay idea here. To see how Zach uses these triangulations in his map-making, you’ve gotta check out the sequence of images on this page. It’s incredible how just a scattering of local temperature measurements can be extended to one of those full-color national temperature maps. So cool!


Zachary Forest Johnson

A Delaunay triangulation used to help create a weather map.

A Delaunay triangulation used to help create a weather map.

Finally, take a look at these images that Jonathan Puckey created. Jonathan is a graphic artist who lives in Amsterdam and shares his work on his website. In 2008 he invented a graphical process that uses Delaunay triangulations and color averaging to create abstractions of images. You can see more of Jonathan’s Delaunay images here.

 armandmevis-1  fox

I hope you find something to enjoy in these circles, squares, and triangles. Bon appetit!

A Closet Full of Puzzles, Sphereland, and Math Doodles

Welcome to this week’s Math Munch!

After a few weeks off, we’re back with some exciting things to share.  First up is Futility Closet, a blog featuring “an idler’s miscellany of compendious amusements.”  The blog is full of big-worded phrases like that, but I most love the puzzles they often post – everything from chess to numbers, codes, and devilish word play.  I also love that the name of the person who wrote each puzzle accompanies it.  Take a look at the few I’ve posted below and click here for the full list of puzzles.

Here’s a puzzle called Swine Wave, by Lewis Carroll. The puzzle: Lace 24 pigs in these sties so that, no matter how many times one circles the sties, he always find that the number in each sty is closer to 10 than the number in the previous one. Want to know the solution? Click on the image above to visit Futility Closet.
This puzzle is called Project Management, by Paul Vaderlind. The question: If a blacksmith requires five minutes to put on a horseshoe, can eight blacksmiths shoe 10 horses in less than half an hour? The catch: A horse can stand on three legs, but not on two. Click on the image to visit Futility Closet for the solution!

Next, have you ever wondered what it would be like to visit another dimension?   In 1884, Edwin A. Abbott wrote about life in the second dimension, in a nice little book called Flatland: A Romance of Many Dimesnions.  (Fun fact: the “A” in Edwin’s name stands for Abbott.  So his name is Edwin Abbott Abbott.)  Click on that link and you can read the whole book, if you like.  The book is about a world of flat beings who have no idea that the third dimension exists.  In the book, the main character, A Square, is visited by a sphere from the unknown world “above” him.  Kind of makes me wonder whether we’re just like the characters in Flatland, three-dimensional creatures ignorant of the fourth dimension that exists “above” us…

spherelandWell, the recently released movie Flatland 2: Sphereland deals with precisely that issue.  The Math Munch team had the opportunity to preview this movie, and we loved it.  In Sphereland, the granddaughter of the Square from Flatland, Hex, and her friend Puncto try to understand some mysterious triangles that Puncto thinks will cause the disastrous end of a space exploration mission and go on an adventure to help their three-dimensional friend Spherius with a problem he brought back from the fourth dimension.

portfolio-TorusHigher dimensions can be very difficult to wrap your head around.  This movie does a great job of helping the movie-watcher to understand how higher and lower dimensions relate to each other through the plot twists and challenges that the characters face.  You can really learn a lot about dimensions and the shape of space by watching this movie.  Plus, the characters are engaging and the images are fun.  Sphereland features the voices of a number of really great actors, including Kristen Bell, Danny Pudi, Michael York, and Danica McKellar.

Want to learn more about Sphereland?  Check out the trailer:

And, here’s an interview with Danny Pudi, the voice of Puncto, and Tony Hale, who does a fantastic job as the King of Pointland:

By the way, the makers of Sphereland also made a movie of Flatland!  The Math Munch team loved that one, too.  Here’s a link to the trailer.

tumblr_mgw2ainZDX1s0payeo1_1280Finally, check out this beautiful blog of mathematical doodles by high school math student and artist Chloé Worthington!  Chloé started mathematically doodling a few years ago in… well, in class.  When she doodles in class, Chloé is better able to focus on what’s going on and makes beautiful art.   (We at Math Munch encourage you to pay attention in class while you doodle.)

Chloé does all of her doodles by hand with ink pens.  She does a lot of work with triangles, as shown here.  One of her signature doodles is this nested puzzle piece doodle:


Doodling mathematically is one of the ways that Chloé does math and shares what she loves about it with the world.  She’s a trigonometry student, too.  How do you share what you love about math – or any other subject?

Bon appetit!

Faces, Blackboards, and Dancing PhDs

Welcome to this week’s Math Munch!

What does a mathematician look like? What does a mathematician do? Here are a couple of things I ran across recently that give a window into what it’s like to be a professional research mathematician—someone who works on figuring out new math as their job.

Gary Davis, who blogs over at Republic of Mathematics, recently posted a short piece that challenges stereotypes about mathematicians. It’s called What does a mathematician look like?

Who here is a mathematician? Click through to find out!

Gary’s point is that you can’t tell who is or isn’t a mathematician just by looking at them. Mathematicians come from every background and heritage. Gary followed up on this idea in another post where he highlighted some notable mathematicians who are black women. Here’s a website called Black Women in Mathematics that shares some biographies and history. And here’s a link to the Infinite Possibilities Conference, a yearly gathering “designed to promote, educate, encourage and support minority women interested in mathematics and statistics.” Suzanne Weekes, one of the five mathematicians pictured above, was a speaker at this conference in 2010.

Richard Tapia, another of the mathematicians above, is featured in the following video. His life story both inspires and delights.

And what does this diversity of mathematicians do all day? Well, one thing they do is talk to each other about math! And though there are many new technologies that help people to do and share and collaborate on mathematics (like blogs!), it’s hard to beat a handy chalkboard as a scribble pad for sharing ideas.

At Blackboard of the Day, Mathieu Rémy and Sylvain Lumbroso share the results of these impromptu math jam sessions. Every day they post a photograph of a blackboard covered in doodles and calculations and sketches of ideas. The website is in French, but the mathematical pictures are a universal language.

Diana Davis, putting the finishing touches on a blackboard masterpiece

Sharing mathematical ideas can take many forms, and sometimes choosing the right medium can make all the difference. Mathematicians use pictures, words, symbols, sculptures, movies, songs—even dances! Let me point you to the “Dance your Ph.D.” Contest. It’s exactly what it sounds like—people sharing the ideas of their dissertations (their first big piece of original work) through dance. Entries come in from physicists, chemists, biologists, and more.  Below you’ll find an entry by Diana Davis, a mathematician who completed her dissertation at Brown University this past spring. Diana often studies regular polgyons and especially ways of “dissecting” them—breaking them up into pieces in interesting ways.

Thanks to The Aperiodical—a great math blog—for sharing Diana’s wonderful video!

Some pages from Diana’s notebooks

All kinds of mathematicians study math and share it in so many ways. It’s like a never-ending math buffet!

Bon appetit!

Squiggles, Spheres, and Taxes

Welcome to this week’s Math Munch!

Check out this cool doodle animation from the blog of Matt Henderson. Matt studied math at Cambridge as an undergrad and now does research on speech and language technology. His idea for a doodle was to start with an equilateral triangle and then encircle it with squiggles until it eventually turned into a square.

Matt Henderson

Matt Henderson

Matt’s triangle-to-square squiggle

Matt has all kinds of beautiful and intricate mathematical images on his blog, many of them animated using computer code. He made a similar squiggle-doodle that evolves a straight line into a profile of his face; an animation of rolling a ball on a merry-go-round; a million dot generator; and many more!

Along the same “lines” as Matt’s squiggle, Ted Theodosopoulos wrote an article in Peer Points reviewing a research paper by Stanford mathematician Ravi Vakil. The title of Ravi’s paper is “The Mathematics of Doodling.”

Ravi’s doodle

Next up, check out this cool visualization of a sphere.

The title of the video is Spherikal and was created by Ion Lucin, a graphic artist in Spain.

Something neat comes out about Ion’s attitude toward learning and sharing in a comment he makes:

“Thanks for appreciating my work. I was thinking the same, not to reveal my secrets, but then, i to learned from the videos and tutorials of others, i have been working with 3D for a year and a half, and all i know about it i learned it by myself, by seeing tutorials, im from fine arts. In a way a feel i must share , like other did and helped me”

What a great attitude!

Another spherical idea comes from a post on one of my favorite websites: MathOverflow, a question-and-answer site for research-level mathematicians…and anyone else! The question I have in mind was posted by Joe O’Rourke, a mathematician at Smith College and one of my favorite posters on MathOverflow. It’s about a certain kind of random walk on a sphere. Check it out!

For this step distance, it looks like a random walk will fill up the whole sphere. What about other step distances?

Again, such a cool picture is created by translating a mathematical scenario into some computer code!

Since this week is when federal income taxes are due, I’ll leave you with a few links about taxes and the federal budget. First, here’s the IRS’s website for kids. (Yes, for real.)

Next, this infographic lets you examine how President Obama’s 2011 budget proposal divvied up funds to all of the different departments and projects of the federal government. Can you find NASA’s budget?


On a more personal scale, this applet called “Where did my tax dollars go?” does just that—when you give it a yearly personal income, it will calculate how much of it will go toward different ends.

Finally, this applet lets you tinker with the existing tax brackets and see the effect on total revenue generated for the federal government. Can you find a flat tax rate that would keep total tax revenue the same?

Whew! That was a lot; I hope you didn’t find it too taxing. Bon appetit!