Tag Archives: programming

SET, Ptolemy, and Malin Christersson

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.)

ThereAreNoSetsHere

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. ūüôā )

Now, we’ve shared about SET before, but recently there has been some very big SET-related news. Although things have been¬†quieter around Georgia Tech since¬†summer has started, there has been a buzz both here and around the internet about a¬†big breakthrough by¬†Vsevolod Lev, P√©ter P√°l Pach, and Georgia Tech professor¬†Ernie Croot. Together they have discovered a new approach¬†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 characteristics‚ÄĒand 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 of¬†SET-less cards we could make in each case.¬†This is called the cap set problem, and Vsevolod,¬†P√©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 by¬†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 ‚ÄĒ and in only a few pages of fairly elementary mathematics. ‚ÄúOne 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,‚ÄĚ Gowers said.

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

 Vsevolod  Peter  Ernie
 Charlotte  Erica  Marsha

These are photos of Vsevolod, Péter, Ernie, Charlotte, Erica, and the creator of SET, geneticist Marsha Jean Falco.

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

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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 right is a geocentric model, which means the earth is at the center.

suntriangle

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 to¬†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 Almagest‚ÄĒliterally, the “Great Work”‚ÄĒon the geocentric model of the heavens. It is an incredible work of mathematics and of natural science. Ptolemy calculated the most accurate table of chords‚ÄĒa variation on a table of the sine function‚ÄĒthat 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,¬†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 something¬†that was “wrong”!

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

ptolemy Copernicus_Mars

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

Malin, tiled hyperbolically.

Now, I was very glad to¬†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 happened¬†across an applet that lets you create art in the style of artist (and superellipse creator) Piet Mondrian. But it also inverts your art‚ÄĒreflects it across a circle‚ÄĒso that you can view your own work from a totally different perspective.¬†Then just a few months later I delighted in finding another¬†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.)

Mondrian

Mondrainverted.

tiling (4)

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 her¬†Rolling Hypocycloids and Epicycloids¬†are can’t-miss. (Echoes of Ptolemy there, yes?!)

And please don’t miss out on Malin’s porfolio¬†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 hope¬†you’ll find some inspiration, too. Bon appetit!

Braids, Hacktastic, and Rock Climbing

Welcome to this week’s Math Munch!

lym_angel

Math hair braiding art by So Yoon Lym, shown at the 2014 Joint Mathematics Meetings.

First up, a little about one of my favorite things to do (and part of what got me into math in the first place!): hair braiding. If you’ve ever done a complicated braid in someone’s hair before, you might have had an inkling that something mathematical was going on. Well, you’re right! Mathematicians Gloria Ford Gilmer and Ron Eglash¬†have spent much of their careers studying and teaching about the math that goes into hair braiding.

SYL_Diosnedys_new1

See the tessellation?

In their research, Gloria and Ron investigate how math can improve hair braiding, how hair braiding can improve math, and how the overlap between the two can teach us about how different cultures use and understand math. As Gloria shows in her article on math and braids, tessellations are very important to braided designs.

braids

And so are fractals! Ron studies how fractals are used in African and African American designs, including in the layouts of towns, tile patterns, and cornrow braids. (Watch his TED Talk to learn more!) On his beautiful website dedicated to the math of cornrows, Ron shows how braiders use tools essential to making fractals to design their braids.

programmed braid

Just like when making a fractal, braid designers repeat the same shape while shifting, rotating, reflecting, and shrinking it. You can design your own mathematical cornrow braid using Ron’s braid programming app!¬†If you’ve ever used Scratch, this app will look very familiar. I made the spiral braid on the right using the app. Next challenge: try to make your braid on a real head of hair…

trig bracelets Laura Taalman

Next up, a little about something I wish I could do: make awesome 3D-printed art! Here’s a blog that might help me (and you) get started. Mathematician Laura Taalman¬†(who calls herself¬†@mathgrrl¬†on Twitter) writes a blog called Hacktastic all about making math designs, using a 3D-printer and many other tools. She has designs for all kinds of awesome things, from Menger sponges to trigonometric bracelets. One of the best things about Laura’s site is that she tells you the story behind how she came up with her designs, along with all the instructions and code you’ll ever need to make her designs yourself.

Rock climbing Skip

Skip Garibaldi, climbing

Finally, a little about something I’m trying to learn to do better: rock climbing! Mathematician Skip Garibaldi¬†loves both math and rock climbing– so he decided to combine his interests for the better of each. In this video, Skip discusses some of the mathematical ideas important to rock climbing– including some essential to a type of climbing that I find most intimidating, lead climbing. Check it out!

Bon appetit!

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!

 

“Happy Birthday, Euler!”, Project Euler, and Pants

Welcome to this week’s Math Munch!

Did you see the Google doodle on Monday?

Leonhard Euler Google doodleThis medley of Platonic solids, graphs, and imaginary numbers honors the birthday of mathematician and physicist Leonhard Euler. (His last name is pronounced “Oiler.” Confusing because the mathematician Euclid‘s name is not pronounced “Oiclid.”) Many mathematicians would say that Euler was the greatest mathematician of all time – if you look at almost any branch of mathematics, you’ll find a significant contribution made by Euler.

480px-Leonhard_Euler_2Euler was born on April 15, 1707, and he spent much of his life working as a mathematician for one of the most powerful monarchs ever, Frederick the Great of Prussia. In Euler’s time, the kings and queens of Europe had resident mathematicians, philosophers, and scientists to make their countries more prestigious. ¬†The monarchs could be moody, so mathematicians like Euler had to be careful to keep their benefactors happy. (Which, sadly, Euler did not. After almost 20 years, Frederick the Great’s interests changed and he sent Euler away.) But, the academies helped mathematicians to work together and make wonderful discoveries.

Want to read some of Euler’s original papers? Check out the Euler Archive. Here’s a little bit of an essay called, “Discovery of a Most Extraordinary Law of Numbers, Relating to the Sum of Their Divisors,” which you can find under the subject “Number Theory”:

Mathematicians have searched so far in vain to discover some order in¬†the progression of prime numbers, and we have reason to believe that it is a¬†mystery which the human mind will never be able to penetrate…¬†This situation is all the¬†more surprising since arithmetic gives us unfailing rules, by means of which¬†we can continue the progression of these numbers as far as we wish, without¬†however leaving us the slightest trace of any order.

Mathematicians still find this baffling today! If you’re interested in dipping your toes into Euler’s writings, I’d suggest checking out other articles in “Number Theory,” such as “On Amicable Numbers,” or some articles in “Combinatorics and Probability,” like “Investigations on a New Type of Magic Square.”

pe_banner_lightWant to work, like Euler did, on important math problems that will stretch you to make connections and discoveries? Check out Project Euler, an online set of math and computer programming problems. You can join the site and, as you work on the problems, talk to other problem-solvers, contribute your solutions, and track your progress. The problems aren’t easy – the first one on the list is, “Find the sum of all the multiples of 3 and 5 below 1000” – but they build on one another (and are pretty fun).

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Pants made from a crocheted model of the hyperbolic plane, by Daina Taimina.

Finally, if someone asked you what a pair of pants is, you probably wouldn’t say, “a sphere with three open disks removed.” But maybe you also didn’t know that pants are important mathematical objects!

I ran into a math problem involving pants on¬†Math Overflow¬†(previously). Math Overflow is a site on which mathematicians can ask and answer each other’s questions.¬†The question I’m talking about was asked by Tony Huynh. He knew it was possible to turn pants inside-out if your feet are tied together. (Check out the video below to see it done!) Tony was wondering if it’s possible to turn your pants around, so that you’re wearing them backwards, if your feet are tied together.

Is this possible? Another mathematician answered Tony’s question – but maybe you want to try it yourself before reading about the solution. Answering questions like this about transformations of surfaces with holes in them is part of a branch of mathematics called topology – which Euler is partly credited with starting. A more mathematical way of stating this problem is: is it possible to turn a torus (or donut) with a single hole in it inside-out? Here’s another video, by James Tanton, about turning things inside-out mathematically.

Bon appetit!

MMteam-240x240P.S. – The Math Munch team will be speaking next weekend, on April 27th, at TEDxNYED! We’re really excited to get to tell the story of Math Munch on the big stage. Thank you for being such enthusiastic and curious readers and allowing us to share our love of math with you. Maybe we’ll see some of you there!

Harmonious Sum, Continuous Life, and Pumpkins

Welcome to this week’s Math Munch!

We’ve posted a lot about pi on Math Munch – because it’s such a mathematically fascinating little number. ¬†But here’s something remarkable about pi that we haven’t yet talked about. Did you know that pi is equal to four times this? Yup. ¬†If you were to add and subtract fractions like this, for ever and ever, you’d get pi divided by 4. ¬†This remarkable fact was uncovered by the great mathematician Gottfried Wilhelm Leibniz, who is most famous for developing the calculus. ¬†Check out this interactive demonstration from the Wolfram Demonstrations Project to see how adding more and more terms moves the sum closer to pi divided by four. ¬†(We’ve written about Wolfram¬†before.)

I think this is amazing for a couple of reasons. ¬†First of all, how can an infinite number of numbers add together to make something that isn’t infinite??? ¬†Infinitely long sums, or series, that add to a finite number have a special name in mathematics: convergent series. ¬†Another famous convergent series is this one:

The second reason why I think this sum is amazing is that it adds to pi divided by four. ¬†Pi is an irrational number – meaning it cannot be written as a fraction, with whole numbers in the numerator and denominator. ¬†And yet, it’s the¬†sum of an infinite number of rational numbers.

In this video, mathematician Keith Devlin talks about this amazing series and a group of mathematical musicians (or mathemusicians) puts the mathematics to music.

This video is part of a larger work called Harmonious Equations written by Keith and the vocal group Zambra. ¬†Watch the rest of them, if you have the chance – they’re both interesting and beautiful.

Next up, Conway’s Game of Life is a cellular automaton created by mathematician John Conway. ¬†(It’s pretty fun: check out this to download the game, and this¬†Munch where we introduce it.) ¬†It’s discrete – each little unit of life is represented by a tiny square. ¬†What if the rules that determine whether a new cell is formed or the cell dies were applied to a continuous domain? ¬†Then, it would look like this:

Looks like a bunch of cells under a microscope, doesn’t it? ¬†Well, it’s also a cellular automaton, devised by mathematician Stephan Rafler from Nurnberg, Germany. ¬†In this paper, Stephan describes the mathematics behind the model. ¬†If you’re curious about how it works, check out these slides that compare the new continuous version to Conway’s model.

Finally, I just got a pumpkin. ¬†What should I carve in it? ¬†I spent some time browsing the web for great mathematical pumpkin carvings. ¬†Here’s what I found.

A pumpkin carved with a portion of Escher’s Circle Limit.

A pumpkin tiled with a portion of Penrose tiling.

A dodecapumpkin from Vi Hart.

I’d love to hear any suggestions you have for how I should make my own mathematical pumpkin carving! ¬†And, if you carve a pumpkin in a cool math-y way, send a picture over to MathMunchTeam@gmail.com!

Bon appetit!