Tag Archives: fractals

Welcome to this week’s Math Munch!

To set up the punchline: if you haven’t played the card game SET before, do yourself a favor and go try it out now!

(Or if you prefer, here’s a video tutorial.)

Are there any sets to be found here?

(And even if you have played before, go ahead and indulge yourself with a round. You deserve a SET break. đź™‚ )

Now, we’ve shared about SET before, but recently there has been some very big SET-related news. Although things have beenÂ quieter around Georgia Tech sinceÂ summer has started, there has been a buzz both here and around the internet about aÂ big breakthrough byÂ Vsevolod Lev, PĂ©ter PĂˇl Pach, and Georgia Tech professorÂ Ernie Croot. Together they have discovered a new approachÂ to estimate how big a SET-less collection of SET cards can be.

In SET there are a total of 81 cards, since each card expresses one combination of four different characteristics (shape, color, filling, number) for which there are three possibilities each. That makes 3^4=81 combinations of characteristics. Of these 81 cards, what do you think is the most cards we could lay out without a SET appearing? This is not an easy problem, but it turns out the answer is 20. An even harder problem, though, is asking the same question but for bigger decks where there are five or ten or seventy characteristicsâ€”and so 3^5 or 3^10 or 3^70 cards. Finding the exact answer to these larger problems would be very, very hard, and so it would be nice if we could at least estimate how big of a collection ofÂ SET-less cards we could make in each case.Â This is called the cap set problem, and Vsevolod,Â PĂ©ter, and Ernie found a much, much better way to estimate the answers than what was previously known.

To find out more on the background of the cap set problem, check out this “low threshold, high ceiling” article by Michigan grad student Charlotte Chan. And I definitely encourage you to check out this article byÂ Erica Klarreich in Quanta Magazine for more details about the breakthrough and for reactions from the mathematical community. Here’s a choice quote:

Now, however, mathematicians have solved the cap set problem using an entirely different method â€” and in only a few pages of fairly elementary mathematics. â€śOne of the delightful aspects of the whole story to me is that I could just sit down, and in half an hour I had understood the proof,â€ť Gowers said.

(For further wonderful math articles, you’ll want to visit Erica’s website.)

 Â Â Â Â Â Â

These are photos of Vsevolod,Â PĂ©ter, Ernie, Charlotte, Erica, and the creator of SET, geneticist Marsha Jean Falco.

Ready for more? Earlier this week, I ran acrossÂ this animation:

It shows two ways of modeling the motions of the sun and the planets in the sky. On the left is a heliocentric model, which means the sun is at the center. On the rightÂ is a geocentric model, which means the earth is at the center.

Around 250 BC, Aristarchus calculated the size of the sun, and decided it was too big to revolve around the earth!

Now, I’m sure you’ve heard that the sun is at the center of the solar system, and that the earth and the planets revolve around the sun. (After all, we call it a “solar system”, don’t we?) But it took a long time for human beings to decide that this is so.

I have toÂ confess: I have a soft spot for the geocentric model. I ran across the animation in a Facebook group of some graduates of St. John’s College, where I studied as an undergrad. We spent a semester or so reading Ptolemy’s Almagestâ€”literally, the “Great Work”â€”on the geocentric model of the heavens. It is an incredible work of mathematics and of natural science. Ptolemy calculated the most accurate table of chordsâ€”a variation on a table of the sine functionâ€”that existed in his time and also proved intricate facts about circular motion. For example, here’s a video that shows that the eccentric and epicyclic models of solar motion are equivalent. What’s really remarkable is that not only does Ptolemy’s system account for the motions of the heavenly bodies,Â it actually gave better predictions of the locations of the planets than Copernicus’s heliocentric system when the latter first debuted in the 1500s. Not bad for somethingÂ that was “wrong”!

Here are Ptolemy and Copernicus’s ways of explaining how Mars appears to move in the sky:

Maybe you would like to learn more about the history of models of the cosmos? Or maybe you would like tinker with a world-system of your own? You might notice that the circles-on-circles of Ptolemy’s model are just like a spirograph or a roulette. I wonder what would happen if we made the orbit circles in much different proportions?

Malin, tiled hyperbolically.

Now, I was very glad toÂ take this stroll down memory lane back to my college studies, but little did I know that I was taking a second stroll as well: the person who created this great animation, I had run across several other pieces of her work before! Her name is Malin Christersson and she’s a PhD student in math education in Sweden. She is also a computer scientist who previously taught high school and also teaches many people about creating math in GeoGebra. You can try out her many GeoGebra applets here. Malin also has a Tumblr where she posts gifs from the applets she creates.

About a year ago I happenedÂ across an applet that lets you create art in the style of artist (and superellipse creator) Piet Mondrian. But it also inverts your artâ€”reflects it across a circleâ€”so that you can view your own work from a totally different perspective.Â Then just a few months later I delighted in finding anotherÂ applet where you can tile the hyperbolic plane with an image of your choice. (I used one tiling I produced as my Twitter photo for a while.)

Mondrainverted.

Me, tiled hyperbolically.

And now come to find out these were both made by Malin, just like the astronomy animation above! And Malin doesn’t stop there, no, no. You should see her fractal applets depicting Julia sets. And herÂ Rolling Hypocycloids and EpicycloidsÂ are can’t-miss. (Echoes of Ptolemy there, yes?!)

And please don’t miss out on Malin’s porfolioÂ of applets made in the programming language Processing.

It’s a good feeling to finally put the pieces together and to have a new mathematician, artist, and teacher who inspires me!

I hopeÂ you’ll find some inspiration, too. Bon appetit!

Web Applets, Space Fillers, and Sisters

Welcome to this week’s Math Munch!

Recently I’ve been running across tons of neat, slick math applets. I feel like they all go together. What do they have in common? Maybe you’ll be able to tell me.

First up, you can tinker with some planetary gears. Then try out these chorded polygons. And then how about some threaded lines?

Ready for some more? Because with these sorts of visualizations, Dan Anderson has been on fire lately. Dan is a high school math teacher in New York state. He and his students had fifteen minutes of fame last year when they investigated whether or not Double Stuf Oreos really have double the stuf.

Here is Dan’s page on OpenProcessing. (Processing is the computer language in which Dan programs his applets.) And check out the images and gifs on Dan’s Tumblr. Here’s a sampling!

Dan also coordinates Daily Desmos, which we’ve feature previously. Check out the latest periodic and “obfuscation” challenges!

That’s a chunk of math to chew on already, but we’re just getting started! Next up, check out the space-filling artwork of John Shier.

John’s artwork places onto the canvas shapes of smaller and smaller sizes. Notice that the circles below fill in gaps, but they don’t touch each other, they way circles do in an Apollonian gasket.

You can learn more about John’s space-filling shapes on this page and find further details in this paper.

Thanks for making us this sweet banner, John!

Last up this week, head to this site to watch an awesome trailer of a film about Julia Robinson. The short clip focuses on Julia’s work on Hilbert’s tenth problem. It includes interviews with a number of people who knew Julia, including her sister Constance Reid. Constance wrote extensively about mathematics and mathematicians. I’ve read her biography of Hilbert and can highly recommend it. You can read more about Julia and Constance here and here.

 Julia Robinson Julia’s sister, Constance Reid

Julia and Constance as young girls.

You might enjoy visiting the site of the Julia Robinson Mathematics Festival. Check to see if a festival will be hosted in your area sometime soon, or find out how you can run one yourself!

With May wrapped up and June getting started, I hope you have a lot of math to look forward to this summer. Bon appetit!

Continents, Math Explorers’ Club, and “I use math for…”

Welcome to this week’s Math Munch!

Steven Strogatz.

All of our munchesÂ this week come from theÂ recent tweetsÂ of mathematician, author, and friend of the blog Steven Strogatz. Steve works at Cornell University as an applied mathematician, tackling questions like “If people shared taxis with strangers, how much money could be saved?” and “What caused Londonâ€™s Millennium Bridge to wobble on its opening day?”

On top of his research,Â Steve is great at sharing math with others. (This week I learnedÂ one great piece of mathÂ from him, and then another, and suddenly there was a very clear theme to my post!) SteveÂ has written for the New York Times and was recently awarded the Lewis Thomas Prize as someone “whoseÂ voice and vision can tell us about scienceâ€™s aesthetic and philosophical dimensions, providing not merely new information but cause for reflection, even revelation.”

This Saturday, Steve will be presenting at the first-ever National Math Festival. The free and fun main event isÂ at the SmithsonianÂ inÂ Washington, DC, andÂ there are relatedÂ math events all around the country this weekend. Check and see if there’s one near you!

Here are a few pieces of math that Steve liked recently. I liked them as well, and I hope you will, too.

First up, check out this lovely image:

It appeared on Numberplay and was created by Hamid Naderi Yeganeh, a student at University of Qom in Iran. Look at the way the smaller and smaller tiles fit together to make the design. It’s sort of like a rep-tile, or this scaly spiral. And do those shapes look familiar? Hamid was inspired by the shapes of the continents of Africa and South America (if you catch my continental drift). Maybe you can create your own Pangaea-inspired tiling.

If you think that’s cool, you should definitely check out Numberplay, where there’s a new math puzzle to enjoyÂ each week!

Next, up check out the Math Explorers’ Club, a collection of great math activities for people of all ages. The Club is a project of Cornell University’s math department, where Steve teaches.

The first item every sold on the auction site eBay. Click through for the story!

One of the bits of math that jumped out to me was this page aboutÂ auctions. There’s so much strategy and scheming that’s involved in auctions! I remember being blown away when I first learned about Vickrey auctions, where the winner pays not what they bid but what the second-highest bidder did!

If auctions aren’t your thing, there’s lots more great math to browse at the Math Explorer’s Clubâ€”everything from chaos and fractals to error correcting codes. EvenÂ Ehrenfeucht-FraĂŻssĂ© games, which are brand-new to me!

And finally this week: have you ever wondered “What will I ever use math for?” Well, SIAMâ€”theÂ Society for Industrial and Applied Mathematicsâ€”has just the video for you.Â They asked people attending one of their meetings to finish the sentence, “I use math for…”. Here are 32 of their answers in just 60 seconds.

Thanks for sharing all this great math, Steve! And bon appetit, everyone!

SquareRoots, Concave States, and Sea Ice

Welcome to this week’s Math Munch!

The most epic Pi Day of the century will happen in just a few weeks: 3/14/15! I hope you’re getting ready. To help you get into the spirit, check out these quilts.

 American Pi. African American Pi.

There’s an old joke that “pi is round, not square”â€”a punchline to the formula for the area of a circle. But in these quilts, we can see that pi really can be square! Each quilt shows the digits ofÂ pi in base 3.Â The quilts are a part of a projectÂ called SquareRoots by artist and mathematician John Sims.

John Sims.

There’s lots more to explore and enjoy on John’sÂ website, including a musical interpretation of pi and some fractal trees that he has designed.Â John studied mathematics as an undergrad at Antioch College and has pursued graduate workÂ at Wesleyan University. He even created a visual math course for artists when he taught at the Ringling College of Art and Design in Florida.

I enjoyed reading several articlesÂ (1, 2, 3, 4) about John and his quilts, as well as thisÂ interviewÂ with John. Here’s one of my favorite quotes from it, in response to “How do you begin a project?”

It can happen in two ways. I usually start with an object, which motivates an idea. That idea connects to other objects and so on, and, at some point, there is a convergence where idea meets form. Or sometimes I am fascinated by an object. Then I will seek to abstract the object into different spatial dimensions.

Cellular Forest and Square Root of a Tree, by John Sims.

You can find more ofÂ John’s work on hisÂ YouTube channel.Â Check outÂ this video, which features some of John’s music and an artÂ exhibitÂ heÂ curated called Rhythm of Structure.

Next up: Some of our US states are nice and boxyâ€”like Colorado. (Or isÂ it?) Other states have very complicated, very dent-y shapesâ€”way more complicated than the shapesÂ we’re used to seeing in math class.

Which state is the most dent-y? How would you decide?

West Virginia is pretty dent-y. By driving “across” it, you can pass through many other states along the way.

The mathematical term for dent-y is “concave”. One way you might try toÂ measureÂ the concavity of a state is to see how far outside of the state you can get by moving in a straight line from one point in it to another. For example, you can drive straight from one place in West Virginia to another,Â and along the way pass through four other states. That’s pretty crazy.

But is it craziest? Is another state even more concave? That’s what this study set out to investigate. Click through to find out their results. And remember that thisÂ is just one way to measure how concave a state is. A different way of measuring might give a different answer.

Awesome animal kingdom gerrymandering video!

This puzzle about the concavity of statesÂ isÂ silly and fun, but there’s more here, too. Thinking about theÂ denty-ness of geographic regions is very important to our democracy. After all, someone has to decide where to draw the lines.Â When regions and districts are carved out inÂ a way that’s unfair to the voters and their interests, that’s called gerrymandering.

Karen Saxe.

To find out more about the process of creating congressional districts, you can listen to a talk byÂ Karen Saxe, aÂ math professor at Macalester College. Karen was a part of a committeeÂ that worked to draw new congressional districts in Minnesota after the 2010 US Census. (Karen speaks about compactness measures startingÂ here.)

Recently I ran across an announcement for a conferenceâ€”a conference thatÂ was all about the math of sea ice! I never grow tired of learning new and exciting ways that math connects with the world.Â Check out this video featuringÂ Kenneth Golden, a leading mathematician in the study of sea ice who works at the University of Utah. I love the line from the video: “People donâ€™t usually think aboutÂ mathematicsÂ as a daring occupation.” Ken and his team show that math can take you anywhere that you can imagine.

Bon appetit!

Reflection sheet – SquareRoots, Concave States, and Sea Ice

Nice Neighbors, Spinning GIFs, and Breakfast

A minimenger.

Welcome to this week’s Math Munch!

Math projects are excitingâ€”especially whenÂ a whole bunch of people work together. One example of big-time collaboration isÂ the GIMPS project, where anyone can use their computer to help find the next large prime number. Another is the recent MegaMenger project, where people from all over the world helped to build a giant 3D fractal.

But what if I told you that you canÂ join up with others on the internetÂ to discover some brand-new math by playing a webgame?

Chris Staecker is a math professor at Fairfield University. This past summer he led a small group of students in a research project. Research Experiences for Undergraduatesâ€”or REUs, as they’re calledâ€”areÂ summer opportunities for college students to be mentoredÂ by professors. Together they work to figure out some brand-new math.

The crew from last summer’s REU at Fairfield. Chris is furthest in the back.

The irreducible digital images containing 1, 5, 6, and 7 “chunks”.

Chris and his students Jason Haarmann, Meg Murphy, and Casey Peters worked on a topic in graph theory called “digital images”. ComputerÂ images areÂ made of discrete chunks, but we often want to make them smallerâ€”like with pixel art. So how can we make sure that we can make them smaller without losing too much information? That’s an important problem.

Now, the pixelsÂ on a computer screen are in a nice grid, but we could also wonder about the same question on an arbitrary connected networkâ€”and that’s what Chris, Jason, Meg, and Casey did. Some networks can be made smaller through one-step “neighbor” moves while still preserving the correct connection properties. Others can’t. By the end of the summer, the team had come up with enough results about digital images with up to eight chunks to write about them in a paper.

To help push their research further, Chris has made a webgame that takes larger networks and offers them as puzzles to solve.Â Here’s how I solved one of them:

See how the graph “retracts” onto itself, just by moving some of the nodes on top of their neighbors? That’s the goal. And there are lots of puzzles to work on. For many of them, if you solve them, you’ll be the first person ever to do so! Mathematical breakthrough! Your result will be saved, the number at the bottom of the screen will go up by one, and Chris and his students will be one step closer to classifying unshrinkable digital images.

Starting with the tutorial forÂ Nice NeighborsÂ is a good idea. Then you can try out the unsolved experimental puzzles.Â If you find success, please let us know about in the comments!

Do you have a question for Chris and his students? Then send it to us and we’ll try to include it in our upcoming Q&A with them.

Next up: you probably know by now thatÂ at Math Munch, weÂ just can’t get enough of great mathy gifs. Well,Â Sumit Sijher has us covered this week, with his Tumblr called archery.

Here are four of Sumit’s gifs. There are plenty more where these came from. This is a nice foursome, though, because theyÂ all spin.Â Click to see the images full-sized!

 How many different kinds of cubes can you spot? This one reminds me of the Whitney Music Box. Whoa. Clockwise or counterclockwise?

I really appreciate how Sumit also sharesÂ the computer code that he uses to make each image. It gives a whole new meaning to “show your work”!

Through Sumit’s work I discovered that WolframAlphaâ€”an online calculator that is way more than a calculatorâ€”has a Tumblr, too. By browsing it you can find some groovy curves and crazy estimations. Sumit won an honorable mention in Wolfram’sÂ One-Liner Competition back in 2012. You can see his entry in this video.

And now for the most important meal of the day: breakfast. Mathematicians eat breakfast, just like everyone else. What do mathematicians eat for breakfast? Just about any kind of breakfast you might name. For some audio-visual evidence, here’s a collection of sound checks by Numberphile.

Sconic sections. Yum!

If that has you hungry for a mathematical breakfast, you might enjoy munching on some sconic sections,Â a linked-to-itself bagel, or some spirograph pancakes.

Bon appetit!

Girls’ Angle, Spiral Tilings, and Coins

Welcome to this week’s Math Munch!

Girls’ Angle is a math club for girls. Since 2007 it has helped girls to grow their love of math through classes, events, mentorship, and a vibrant mathematical community. Girls’ Angle isÂ based in Cambridge, Massachusetts, but its ideas and resources reach around the world through the amazing power of the internet. (And don’t you worry, gentlemen—there’s plenty for you to enjoy on the site as well.)

Amazingly, the site contains an archive of everyÂ issue of Girls’ Angle Bulletin, a wonderful bimonthly journal to “foster and nurture girlsâ€™ interest in mathematics.”Â In their most recent issue, you’ll find an interview with mathematician Karen E. Smith, along with several articles and puzzles about balance points of shapes.

There’s so much to dig into at Girls’ Angle! In addition to the Bulletins,Â there are two pages of mathematicalÂ videos.Â The first page shares a host of videos of women in mathematics sharing a piece of math that excited them when they were young. The most recent one is by Bridget Tenner, who shares about Pick’s Theorem.Â The second pageÂ includes several videos produced by Girls’ Angle, including this one called “Summer Vacation”.

Girls’ Angle can even help you buy a math book that you’d like, if you can’t afford it. For so many reasons, I hope you’ll find some time to explore the Girls’ Angle site over your summer break. (And while you’ve got your explorer’s hat on, maybe you’ll tour around Math Munch, too!)

I did a Google search recently for “regular tilings.” I needed a few quick pictures of the usual triangle, square, and hexagon tilings for a presentation I was making. As I scrolled along, this image jumped out at me:

What is that?! It certainly is a tiling, and all the tiles are the “same”—even if they are different sizes. Neat!

Clicking on the image, I found myself transported to a page all about spiral tilings at the Geometry Junkyard. The site is aÂ whole heap of geometrical odds and ends—and a placeÂ thatÂ I’ve stumbled across many times over the years. Here areÂ a few places to get started. I’m sure you’ll enjoy poking around the site to find some favorite “junk”Â of your own.

 Spirals Circles &Â spheres Coloring

Last up this week, you may have seen this coin puzzle before. Can you make the triangle point downwards by moving just three pennies?

There are lots of variants ofÂ this puzzle. You can find some in an online puzzle game called Coins. In the gameÂ you have to make arrangements of coins, but the twist is that you can only move a coin to a spot where would it touch at least two other coins. I’m enjoying playing Coins—give it a try!

I solved this Coins puzzle in four moves. Can you? Can you do better?

That’s it for this week’s Math Munch. Bon appetit!

Fullerenes, Fibonacci Walks, and a Fourier Toy

Welcome to this week’s Math Munch!

Stan and James

Earlier this month, neuroscientists Stan Schein and James Gayed announced the discovery of a new class of polyhedra. We’ve often posted about Platonic solids here on Math Munch. The shapes that Stan and James found have the same symmetries as the icosahedron and dodecahedron, and they also have all equal edge lengths.

One of Stan and James’s shapes, made of equilateral pentagons and hexagons.

These new shapes are examples of fullerenes, a kind of shape named after the geometer, architect, and thinker Buckminster Fuller. In the 1980s, chemists discovered that molecules made of carbon can occur in polyhedral shapes, both in the lab and in nature. Stan and James’s new fullerenes are modifications of some existing shapes first described in 1937 by Michael Goldberg. The faces of Goldberg’s shapes were warped, not flat, and Stan and James showed that flattening can be achieved—thus turning Goldberg’s shapes into true polyhedra—while also having all equal edge lengths.Â There’s great coverage of Stan and James’s discovery in this article at Science News and a fascinating survey of the media’s coverage of the discovery by Adam Lore on his blog. Adam’s post includes an interview with Stan!

Next up—how much fun is it to find a fractal that’s new to you? That happened to me recently when I ran across the Fibonacci word fractal.

A portion of a Fibonacci word curve.

Fibonacci “words”—really just strings of 0’s and 1’s—are constructed kind of like the numbers in the Fibonacci sequence. Instead of adding numbers previous numbers to get new ones, we link up—or “concatenate”—previous words. The first few Fibonacci words are 1, 0, 01, 010, 01001, and 01001010. Do you see how new words are made out of the two previous ones?

Here’s a variety of images of Fibonacci word fractals, and you can find more details about the fractal in this article. The infinite Fibonacci word has anÂ entryÂ at the OEIS, and you can findÂ a Fibonacci word necklace on Etsy.Â Dale Gerdemann, a linguist at the University of TĂĽbingen,Â has a whole series of videos that show off patterns created out of Fibonacci words. Here is one of my favorites:

Last but not least this week, check out this groovy applet!

Lucas’s applet showing the relationship between epicycles and Fourier series

A basic layout of Ptolemy’s model, including epicycles.

Sometime around the year 200 AD, the astronomer Ptolemy proposed a way to describe the motion of the sun, moon, and planets. Here’s a video about his ideas. Ptolemy relied on many years of observations, a new geometrical tool we call “trigonometry”, and a lot of ingenuity. He said that the sun, moon, and planets move around the earth in circles that moved around on other circles—not just cycles, but epicycles. Ptolemy’s model of the universe was incredibly accurate and was state-of-the-art for centuries.

Joseph Fourier

In 1807, Joseph Fourier turned the mathematical world on its head. He showed that periodic functions—curves with a repeated pattern—can be built by adding together a very simple class of curves. Not only this, but he showed that curves created in this way could have breaks and gaps even though they are built out of continuous curves called “sine” and “cosine”. (Sine and cosine are a part of the same trigonometry that Ptolemy helped to found.) Fourier series soon became a powerful tool in mathematics and physics.

A Fourier series that converges to a discontinuous function.

And then in the early 21st century Lucas Vieira created an applet that combines and sets side-by-side the ideas of Ptolemy and Fourier. And it’s a toy, so you can play with it! What cool designs can you create? We’ve featured some ofÂ Lucas’s work in the past. Here is Lucas’s short post about his Fourier toy, including some details about how to use it.

Bon appetit!