Tag Archives: infinity

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!

***

This 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.

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!

While 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.)

I’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.

While 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!

P.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!

Posted by Anna Weltman. Categories: Math Munch. Tags: art, computers, escher, games, history, infinity, interview, math anxiety, math encyclopedia, paradox, puzzles, recreational mathematics, software, symmetry, tessellation, video. 16 Comments

Jul. ’13

Coasts, Clueless Puzzles, and Beach Math Art

Ah, summertime. If it’s as hot where you are as it is here in New York, I bet this beach looks great to you, too. A huge expanse of beach all to myself sounds wonderful… And that makes me wonder – how much coastline is there in the whole world?

Interestingly, the length of the world’s coastline is very much up for debate. Just check out this Wikipedia page on coastlines, and you’ll notice that while the CIA calculates the total coastline of the world to be 356,000 kilometers, the World Resources Institute measures it to be 1,634,701! What???

Measuring the length of a coastline isn’t as simple as it might seem, because of something called the Coastline Paradox. This paradox states that as the ruler you use to measure a coastline gets shorter, the length of the coastline gets longer – so that if you used very, very tiny ruler, a coastline could be infinitely long! This excellent video by Veritasium explains the problem very well:

As Vertitasium says, many coastlines are fractals, like the Koch snowflake shown at left – never-ending, infinitely complex patterns that are created by repeating a simple process over and over again. In this case, that simple process is the waves crashing against the shore and wearing away the sand and rock. If coastlines can be infinitely long when you measure them with the tiniest of rulers, how to geographers measure coastline? By choosing a unit of measurement, making some approximations, and deciding what is worth ignoring! And, sometimes, agreeing to disagree.

Need something to read at the beach, and maybe something puzzle-y to ponder? Check out this interesting article by four mathematicians and computer scientists, including James Henle, a professor in Massachusetts. They’ve invented a Sudoku-like puzzle they call a “Clueless Puzzle,” because, unlike Sudoku, their puzzle never gives any number clues.

How does this work? These puzzles use shapes instead of numbers to provide clues. Here’s an example from the paper: Place the numbers 1 through 6 in the cells of the figure at right so that no digit appears more than once in a row or column AND so that the numbers in each region add to the same sum. The paper not only walks you through the solution to this problem, but also talks about how the mathematicians came up with the idea for the puzzles and studied them mathematically. It’s very interesting – I recommend you read it!

Finally, if you’re not much of a beach reader, maybe you’d like to make some geometrically-inspired beach art! Check out this land art by artist Andy Goldsworthy:

Or make one of these!

Happy summer, and bon appetit!

Posted by Anna Weltman. Categories: Math Munch. Tags: art, article, building, fractals, infinity, measurement, numbers, puzzles, recreational mathematics, video. 29 Comments

Mar. ’13

Maths Ninja, Folding Fractals, and Pi Fun

Welcome to this week’s Math Munch!

First up, have you ever been stuck on a gnarly math problem and wished that a math ninja would swoop in and solve the problem before it knew what hit it? Have you ever wished that you had a math dojo who would impart wisdom to you in cryptic but, ultimately, extremely timely and useful ways? Well, meet Colin Beverige, a math (or, as he would say, maths) tutor from England who writes a fun blog called Flying Colours Maths. On his blog, he publishes a weekly series called, “Secrets of the Mathematical Ninja,” in which the mathematical ninja (maybe Colin himself? He’s too stealthy to tell) imparts nuggets of sneaky wisdom to help you take down your staunchest math opponent.

For example, you probably know the trick for multiplying by 9 using your fingers – but did you know that there’s a simple trick for dividing by 9, too? Ever wondered how to express thirteenths as decimals, in your head? (Probably not, but maybe you’re wondering now!) Want to know how to simplify fractions like a ninja? Well, the mathematical ninja has the answers – and some cute stories, too. Check it out!

A picture of a Julia set.

Next, I find fractals fascinating, but – I’ll admit it – I don’t know much about them. I do know a little about the number line and graphing, though. And that was enough to learn a lot more about fractals from this excellent post on the blog Hackery, Math, and Design by Steven Wittens. In the post How to Fold a Julia Fractal, Steven describes how the key to understanding fractals is understanding complex numbers, which are the numbers we get when we combine our normal numbers with imaginary numbers.

Now, I think imaginary numbers are some of the most interesting numbers in mathematics – not only because they have the enticing name “imaginary,” but because they do really cool things and have some fascinating history behind them. Steven does a really great job of telling their history and showing the cool things they do in this post. One of the awesome things that imaginary numbers do is rotate. Normal numbers can be drawn on a line – and multiplying by a negative number can be thought of as changing directions along the number line. Steven uses pictures and videos to show how multiplying by an imaginary number can be thought of as rotating around a point on a plane.

A Julia set in the making.

The Julia set fractal is generated by taking complex number points and applying a function to them that squares each point and adds some number to it. The fractal is the set of points that don’t get infinitely larger and larger as the function is applied again and again. Steven shows how this works in a series of images. You can watch the complex plane twist around on itself to make the cool curves and figures of the Julia set fractal.

Steven’s blog has many more interesting posts. Check out another of my favorites, To Infinity… and Beyond! for an exploration of another fascinating, but confusing, topic – infinity.

Finally, a Pi Day doesn’t go by without the mathematicians and mathematical artists of the world putting out some new Pi Day videos! Pi Day was last Thursday (3/14, of course). Here’s a video from Numberphile in which Matt Parker calculates pi using pies!

In this video, also from Numberphile, shows how you only need 39 digits of pi to make really, really accurate measurements for the circumference of the observable universe:

Finally, it wouldn’t be Pi Day without a pi video from Vi Hart. Here’s her contribution for this year:

Bon appetit!

Posted by Anna Weltman. Categories: Math Munch. Tags: computers, fractals, imaginary numbers, infinity, math help, numbers, pi, Vi Hart, video. 18 Comments