# Fractions, Sam Loyd, and a MArTH Journal

Welcome to this week’s Math Munch!

Check out this awesome graph:

What is it?  It’s a graph of the Farey Fractions, with the denominator of the (simplified) fraction on the vertical axis and the value of the fraction on the horizontal axis, made by mathematician and professor at Wheelock College Debra K. Borkovitz (previously).  Now, I’d never heard of Farey Fractions before I saw this image – but the graph was so cool that I wanted to learn all about them!

So, what are Farey Fractions, you ask?  Debra writes all about them and the cool visual patterns they make in this post.  To make a list of Farey Fractions you first pick a number – say, 5.  Then, you list all of the fractions between 0 and 1 whose denominators are less than or equal to the number you picked.  So, as Debra writes in her post, for 5 the list of Farey Fractions is:

As Debra writes, there are so many interesting patterns in Farey Fractions – many of which become much easier to see (literally) when you make a visualization of them.  Debra has made several awesome applets using the program GeoGebra, which she links to in her post.  (You can download GeoGebra and make applets of your own by visiting our Free Math Software page.)  These applets really show the power of using graphs and pictures to learn more about numbers.  To play with the applet that made the picture above, click here.  Check out her post to play with another applet, and to read more about the interesting patterns in Farey Fractions.

Next, check out this website devoted to the puzzles of puzzlemaster Sam Loyd.  Sam Loyd was a competitive chess player and professional puzzle-writer who lived at the end of the nineteenth century.  He wrote many puzzles that are still famous today – like the baffling Get Off the Earth puzzle.  Click the link to play an interactive version of the Get Off the Earth puzzle.

The site has links to numerous Sam Loyd puzzles.  Check out the Picture Puzzles, in which you have to figure out what object is described by the picture, or the Puzzleland Puzzles, which feature characters from the fictional place Puzzleland that Sam created.

Snow MArTH, made by MArTHist Eva Hild and others at a snow sculpture event in Colorado. From the Spring, 2011 Hyperseeing.

Finally, take a look at some of the beautiful pictures and fascinating articles in this journal about mathematical art (a.k.a., MArTH) called Hyperseeing.  Hyperseeing is edited by mathematicians and artists Nat Friedman and Ergun Akleman.  Hyperseeing is published by the International Society of the Arts, Mathematics, and Architecture, which Nat founded to encourage education connecting the arts, architecture, and math – which we here at Math Munch love!  In one of his articles, Nat defines hyperseeing as, “Interdisciplinary education… concerned with seeing from multiple viewpoints in a very general sense.  Hyperseeing is a more complete way of seeing.”

There are so many beautiful images to look at and interesting articles to read in Hyperseeing.  Among other things, each edition of Hyperseeing features a mathematical comic by Ergun.  Here are some of my favorite Hyperseeings from the archives:

 This edition of Hyperseeing features art made from Latin Squares and “organic geometry” art, among many other things. This edition of Hyperseeing features crocheted hyperbolic surfaces (which we featured not long ago in this Math Munch!) and sculptures made with a 3-D printer, among many other things. This is the first edition of Hyperseeing. In it, Nat describes the mission of Hyperseeing and the International Society of the Arts, Mathematics, and Architecture.

Bon appetit!

P.S. – You may have noticed a new thing to click off to the right, below the Favorite Munches.  This is our For Teachers section.  The Math Munch team has put together several pages to describe how we use Math Munch in our classes and give suggestions for how you might use it, too.  Teachers and non-teachers alike may want to check out our new Why Math Munch? page, which gives our mission statement.

P.P.S. – The Math Munch team is going to Bridges on Thursday!  Maybe we’ll see you there.

# Alphametics, Hyperbolic Crochet, and a Puzzle Contest

Welcome to the first Math Munch of December!

Did you know that SEND + MORE = MONEY?  Or that DOUBLE + DOUBLE + TOIL = TROUBLE?  It does if you replace the letters with the appropriate digits!  These very clever puzzles, where the digits in numbers of addition, subtraction, or multiplication problems are replaced by letters in words, are called alphametics (or sometimes cryptarithms).  Mathematician, software engineer, and writer Mike Keith calls them the “most elegant of puzzles” on his page devoted to some alphametics he’s found and created.  Check out the “doubly-true” alphametics – puzzles where the words are numbers – and Mike’s alphametic poetry.  In this poem, written in what Mike calls “Strict Alphametish,” the last word in each line is the sum of the previous words in that line!  Wow!

Next, take a look at these cool objects!

If you draw a line on a hyperbolic plane and a point not on that line, you can make an infinite number of lines parallel to the first line through the point.

These are models of hyperbolic planes crocheted by Cornell University mathematician and artist Daina Taimina.  A hyperbolic plane is a surface that is kind of like the opposite of a sphere: on a sphere, the surface always curves in towards itself, but on a hyperbolic plane, the surface always curves away from itself.

Before Daina figured out how to crochet a hyperbolic plane, mathematicians had no durable, easy-to-use models of this very important geometric object!  But now, anyone with a little crocheting skill (or a willingness to learn!) can make a hyperbolic plane!  Here are instructions on how to crochet your very own hyperbolic plane, and here’s a link to Daina’s blog.

By the way, our favorite mathematical doodler Vi Hart also makes models of hyperbolic planes out of balloons.

Finally, do you like to play with Rubik’s Cubes, stacking puzzles, or other physical math puzzles?  Think you could make one of your own?   These are some of the entries in the 2011 Nob Yoshigahara Puzzle Design Competition.  Here are the winners!  The designer of the first-place puzzle won this cool trophy!

Bon Appetit!