Tag Archives: escher

TesselManiac, Zeno’s Paradox, and Platonic Realms

Welcome to this week’s Math Munch!

Before we begin, we’d like to thank all of you who have checked out the site in the past week. Since we’ve kicked off our share campaign, we’ve had so many new visitors and heard from many of them, too! Reading your feedback – whether a recommendation, some praise, a question, or just a brief, “Hello!” – brings us great joy and helps us to know that you all are out there.

Whether you’re a regular reader or visiting the site for the first time, we’d like to ask you for a little favor. If you see some math you like, share it with someone who you think would like it, too! Do you love the burst of excitement that you get from reading about a new mathematical idea, seeing an original piece of math artwork, or trying out a new game? Do you know someone who would love that, too? Then tell them about Math Munch – we’d love to spread the joy.

If you enjoy Math Munch, join in our “share campaign” this week.

You can read more about the share campaign here. There are lots of ways to participate, and you can let us know about your sharing through this form. We’d love to see the share total rise up to 1000 over the course of the next week.

Now for the post!

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Lee boxThis beautiful tessellated wooden box was made by computer scientist and mathematical artist Kevin Lee. I met Kevin two weeks ago at the MOVES conference (which Justin and Paul have both written about already). Kevin teaches computer science at Normandale Community College in Minnesota. He makes woodcut tessellations (which won an award for the “Best Textile, Sculpture, or Other Medium” at the Joint Mathematics Meetings art exhibition this year!). He’s also used a combination of his knowledge of computer science and his love of Escher-type tessellations to make software that helps you create tessellations. His new software, TesselManiac!, is due out soon (watch this short movie Kevin made about it for the Bridges conference), but you can download an older version of the software here and play a preview version of The Flipping Tile Game.

tesselmaniac pictures

To play this game, you must fill in an outline of a tessellation with the piece given. You can use any of four symmetry motions – translation (or shift), rotation, reflection, or glide reflection (which reflects the tile and then translates it along a line parallel to the line of reflection). You get points for each correct tile placed (and lose points if you have to delete). Translations are the simplest, and only give you 5 points each. Reflections are the most difficult – you get 20 points for each one used!

dot to dotWhile you’re downloading The Flipping Tile Game, try one of Kevin’s Dot-to-Dot puzzles. These are definitely not your typical dot-to-dot. Only the portion of the image corresponding to one tile in the tessellation is numbered. Once you figure out the shape of that single tile, you have to figure out how to number the rest of the puzzle!

Lucky for us, Kevin has agreed to answer some questions about his life and work as a math artist and computer scientist. Leave a question for Kevin here. (We’ll take questions for the next two weeks.)

tortoiseI’ve recently been thinking about a paradox that has puzzled mathematicians for centuries. Maybe you’ve heard of it. It’s one of the ancient Greek philosopher Zeno‘s paradoxes of motion, and it goes like this: Achilles (a really fast Greek hero) and a tortoise are going to run a race. Achilles agrees to give the tortoise a head-start, because the tortoise is so slow. Achilles then starts to run. But as Achilles catches up with the tortoise, the tortoise moves a little further. So the tortoise is still ahead. And as Achilles moves to catch up again, the tortoise moves even further! Sounds like Achilles will never catch up to the tortoise, let alone pass him… But that doesn’t make sense…

Will Achilles lose the race??? Check out this great video from Numberphile that explains both the paradox and the solution.

logo_PR_225_160While I was looking for information about this paradox, I stumbled across a great math resource site called Platonic Realms. The homepage of this site has a daily historical fact, mathematical quote, and puzzle.

The site hosts a math encyclopedia with explanations of all kinds of math terms and little biographies of famous mathematicians. You can also read “mini-texts” about different mathematical topics, such as this one about M. C. Escher (the inspiration behind the art at the beginning of this post!) or this one about coping with math anxiety.

I hope we here at Math Munch have given you something to tantalize your mathematical taste buds this week! If so, we’d love it if you would pass it along.

Thank you for reading, and bon appetit!

Rush hourP.S. – We’ve posted a new game, suggested to us by one of our readers! It’s an online version of Rush Hour. Check it out!

World’s Oldest Person, Graphing Challenge, and Escher Sketch

265282-jiroemon-kimura-the-world-s-oldest-living-man-celebrated-his-115th-birOn April 19th, Jiroeman Kimura celebrated his 116th birthday. He was – and still is – the world’s oldest person, and the world’s longest living man – ever. (As far as researchers know, that is. There could be a man who has lived longer that the public doesn’t know about.) The world’s longest living woman was Jeanne Calment, who lived to be 122 and a half!

Most people don’t live that long, and, obviously, only one person can hold the title of “Oldest Person in the World” at any given time. So, you may  be wondering… how often is there a new oldest person in the world? (Take a few guesses, if you like. I’ll give you the answer soon!)

stackSome mathematicians were wondering this, too, and they went about answering their question in the way they know best: by sharing their question with other mathematicians around the world! In April, a mathematician who calls himself Gugg, asked this question on the website Mathematics Stack Exchange, a free question-and-answer site that people studying math can use to share their ideas with each other. Math Stack Exchange says that it’s for “people studying math at any level.” If you browse around, you’ll see mathematicians asking for help on all kinds of questions, such as this tricky algebra problem and this problem about finding all the ways to combine coins to get a certain amount of money.  Here’s an entry from a student asking for help on trigonometry homework. You might need some specialized math knowledge to understand some of the questions, but there’s often one that’s both interesting and understandable on the list.

Anyway, Gugg asked on Math Stack Exchange, “How often does the oldest person in the world die?” and the community of mathematicians around the world got to work! Several mathematicians gave ways to calculate how often a new person becomes the oldest person in the world. You can read about how they worked it out on Math Stack Exchange, if you like, or on the Smithsonian blog – it’s a good example of how people use math to model things that happen in the world. Oh, and, in case you were wondering, a new person becomes the world’s oldest about every 0.65 years. (Is that around what you expected? It was definitely more often than I expected!)

advanced 4

Next, check out this graph! Yes, that’s a graph – there is a single function that you can make so that when you graph it, you get that.  Crazy – and beautiful! This was posted by a New York City math teacher named Michael Pershan to a site called Daily Desmos, and he challenges you to figure out how to make it!  (He challenged me, too. I worked on this for days.)

qod0nxgctfMichael made this graph using an awesome free, online graphing program called Desmos. Michael and many other people regularly post graphing challenges on Daily Desmos. Some of them are very difficult (like the one shown above), but some are definitely solvable without causing significant amounts of pain. They’re marked with levels “Basic” and “Advanced.” (See if you can spot contributions from a familiar Math Munch face…)

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Here are more that I think are particularly beautiful. If you’re feeling more creative than puzzle-solvey, try making a cool graph of your own! You can submit a graphing challenge of your own to Daily Desmos.

escher 3If you’ve got the creative bug, you could also check out a new MArTH tool that we just found called Escher Web Sketch. This tool was designed by three Swiss mathematicians, and it helps you to make intricate tessellations with interesting symmetries – like the ones made by the mathematical artist M. C. Escher. If you like Symmetry Artist and Kali, you’ll love this applet.

Be healthy and happy! Enjoy graphing and sketching! And, bon appetit!

Visualizations, Inspirations, and the Super Ultimate Graphing Challenge

Welcome to this week’s Math Munch!

Jason Davies

Meet Jason Davies, a freelance mathematician living in the UK. Growing up in Wales (one of the 4 countries of the United Kingdom) his classes were taught in Welsh. This makes Jason one of only about 611,000 people that speak the language, only 21.7% of the population of Wales! Imagine if only 1/5 of France spoke French!! These statistics are from a 2004 study, so the numbers may have changed a bit, but they still say something interesting don’t they?

Prime Seive

Jason is all about what numbers and pictures can tell us.  Since graduating from Cambridge, he’s been doing all sorts of data visualization and computer science on his own for various companies and IT firms. I originally found Jason through a link to his Prime Seive visualization, but take a look at his gallery and you’re bound to find something beautiful, interesting, interactive, and cool. I’ve linked to some of my favorites below.

Interactive Apollonian Gasket

Rhodonea Curves

Set Partitions

I asked Jason a few questions about his interest in data visualization and math in general. Here’s a tasty little excerpt:

MM: What’s the most important trait for a mathematician to have? Is there one?

JD: Persistance is always useful in maths! I think the stereotype is to be analytical and logical, but in fact there are many other traits that are highly important, for instance communication skills. Mathematics is passed on from person to person, after all, so being able to communicate ideas effectively is dynamite.

MM: Do you have a message you’d like to give to young mathematicians?

JD: The world needs you!

Read the rest in our Q&A with Jason Davies, and you can see all of our interviews on the Q&A page we’ve just created.

Up next, a beautiful and inspiring video from Spain. The video is actually called Insprations, and it comes to us from Etérea Studios, the online home of animator Cristóbal Vila. In the intro he says, “I looked into that enormous and inexhaustible source of inspiration that is Escher and tried to imagine how it could be his workplace, what things would surround an artist like him, so deeply interested in science in general and mathematics in particular.”

I’d die to have an office like this!

It gets better.  Cristóbal added a page explaining all of the wonderful maths in the video. Click to read about Platonic solids, tilings, tangrams, and various works of art by M.C. Escher.

Finally, a nifty new game that explores the relationship between graphs and different kinds of motion. Super Ultimate Graphing Challenge is a game developed by Physics teacher Matthew Blackman to help his students understand the physics and mathematics of motion. You might not understand it all when you start, but keep playing and see what you can make of it. If you need a bit of help or have something to say, post it in our comments, and we’ll happily reply.

Bon appetit!