# Making Pi, Transcending Pi, and Cookies

Welcome to this week’s Math Munch– and happy Pi Day!

What does pi look like? The first 10,000 digits of pi, each digit 0 through 9 assigned a different color.

You probably know some pretty cool things about the number pi. Perhaps you know that pi has quite a lot to do with circles. Maybe you know that the decimal expansion for pi goes on and on, forever and ever, without repeating. Maybe you know that it’s very likely that any string of numbers– your birthday, phone number, all the birthdays of everyone you know listed in a row, followed by all their phone numbers, ANYTHING– can be found in the decimal expansion of pi.

But did you know that pi can be approximated by dropping needles on a piece of paper? Well, it can! If you drop a needle again and again on a lined piece of paper, and the needle is the same length as the distance between the lines, the probably that the needle lands on a line is two divided by pi. This experiment is called Buffon’s needle, after the French naturalist Buffon.

If the angle the needle makes with the lines is in the gray area (like the red needle’s angle is), it will cross the line. If the angle isn’t, it won’t. The possible angles trace out a circle. The closer the center of the needle (or center of the circle) is to the line, the larger the gray area– and the higher the probability of the needle hitting the line.

This may seem strange to you– but if you think about how the needle hitting a line has a lot to do with the distance between the middle of the needle and the nearest line and the angle it makes with the lines, maybe you’ll start to think about circles… and then you’ll get a clue about the connection between this experiment and pi. Working out this probability exactly requires some pretty advanced mathematics. (Feeling ambitious? Read about the calculation here.) But, you can get some great experimental results using this Buffon’s needle applet.

Click on the picture to try the applet.

I had the applet drop 500 needles. Then, the applet used the fact that the probability of the needle hitting a line should be two divided by pi and the probability it measured to calculate an approximation for pi. It got… well, you can see in the picture. Pretty close, right?

Here’s another thing you might not know: pi is a transcendental number. Sounds trippy– but, like some other famous numbers with letter names, like e, pi can never be the solution to an algebraic equation involving whole numbers. That means that no matter what equation you give me– no matter how large the exponent, how many negatives you toss in, how many times you multiply or divide by a whole number– pi will never, ever be a solution. Maybe this doesn’t sound amazing to you. If not, check out this video from Numberphile about transcendental numbers. Numbers like pi and e don’t do mathematical things we’re used to numbers doing… and it’s pretty weird.

Still curious about transcendental numbers? Here’s a page listing the fifteen most famous transcendental numbers. My favorite? Definitely the fifth, Liouville’s number, which has a 1 in each consecutive factorial numbered place.

Finally, maybe you don’t like pi. Maybe you like cookies instead. Lucky for you, you can do many mathematical things with cookies, too. Like make cookie tessellations! This mathematical artist and baker made cookie cutters in the shapes of tiles from Escher tessellations and used them to make mathematical cookie puzzles. Beautiful, and certainly delicious.

If you happen to have a 3D printer, you can make your own Escher cookie cutters. Here’s a link to print out the lizard cutter. If you don’t have a 3D printer, you could try printing out a 2D image of an Escher tessellation and tracing a tile onto a sheet of paper. Cut out the tile, roll out your dough, and slice around the outside of the tile to make your cookies. If you do it right, you shouldn’t have to waste any dough…

Here’s hoping you eat some pi or cookies on pi day! Bon appetit!

# Light Bulbs, Lanterns, and Lights Out

Welcome to this week’s Math Munch!

Edison with his light bulb.

On this day in 1880, Thomas Edison was given a patent for his most famous bright idea—the light bulb.

Edison once said, “Genius is one per cent inspiration, ninety-nine per cent perspiration”—a good reminder that putting in some work is important both in math and in life. He also said, “We don’t know a millionth of one percent about anything.” A humbling thought. Also, based on that quote, it sounds like Edison might have had a use for permilles or even permyraids in addition to percents!

Mike’s octahedron-in-a-light-buld.

In celebration of this illustrious anniversary, I’d like to share some light mathematical fare relating to, well, light bulbs. For starters, J. Mike Rollins of North Carolina has created each of the Platonic solids inside of light bulbs, ship-in-a-bottle style. Getting just the cube to work took him the better part of twelve hours! Talk about perspiration. Mike has also made a number of lovely Escher-inspired woodcuts. Check ’em out!

Evelyn’s Schwartz lantern.

Next up is a far-out example from calculus that’s also a good idea for an art project. It’s called the Schwartz lantern. I found out about this amazing object last fall when Evelyn Lamb tweeted and blogged about it.

The big idea of calculus is that we can find exact answers to tough problems by setting up a pattern of approximations that get better and better and then—zoop! take the process to its logical conclusion at infinity. But there’s a catch: you have to be careful about how you set up your pattern!

A “nicely” triangulated cylinder.

For example, if you take a cylinder and approximate its surface with a bunch of triangles carefully, you’ll end up with a surface that matches the cylinder in shape and size. But if you go about the process in a different way, you can end up with a surface that stays right near the cylinder but that has infinite area. That’s the Schwartz lantern, first proposed by Karl Hermann Amandus Schwarz of Cauchy-Schwartz fame. The infinite area happens because of all the crinkles that this devilish pattern creates. For some delightful technical details about the lantern’s construction, check out Evelyn’s post and this article by Conan Wu.

Maybe you’ll try folding a Schwartz lantern of your own. There’s a template and instructions on Conan’s blog to get you started. You’ll be glowing when you finish it up—especially if you submit a photo of it to our Readers’ Gallery. Even better, how about a video? You could make the internet’s first Schwartz lantern short film!

Robert and his Lights Out puzzle.

At the MOVES Conference last fall, Bruce Torrence of Randolf-Macon College gave a talk about the math of Lights Out. Lights Out is a puzzle—a close relative of Ray Ray—that’s played on a square grid. When you push one of the buttons in the grid it switches on or off, and its neighbors do, too. Bruce and his son Robert created an extension of this puzzle to some non-grid graphs. Here’s an article about their work and here’s an applet on the New York Times website where you can play Lights Out on the Peterson graph, among others. You can even create a Lights Out puzzle of your own! If it’s more your style, you can try a version of the original game called All Out on Miniclip.

The original Lights Out handheld game from 1995.

There’s a huge collection of Lights Out resources on Jaap’s Puzzle Page (previously), including solution strategies, variations, and some great counting problems. Lights Out and Ray Ray are both examples of what’s called a “sigma-plus game” in the mathematical literature. Just as a bonus, there’s this totally other game called Light Up. I haven’t solved a single puzzle yet, but my limitations shouldn’t stop you from trying. Perspiration!

All this great math work might make you hungry, so…bon appetit!

# Isomorphisms in Five, Parquet Deformations, and POW!

Welcome to this week’s Math Munch!

Here’s a catchy little video. It’s called “Isomorphisms in Five.” Can you figure out why? The note posted below the video says:

An isomorphism is an underlying structure that unites outwardly different mathematical expressions. What underlying structure do these figures share? What other isomorphisms of this structure will you discover?

One of the reasons I LOVE this video is because I really like how the shapes change with the music– which is played in a very interesting time signature. I also love how you can learn a lot about the different growing shape patterns by comparing them. Watch how they grow as the video flips from pattern to pattern. What do you notice? What does the music tell you about their growth?

This video is by a math educator from North Carolina named Stuart Jeckel. The only thing written about him on his “About” page is, “The Art of Math”– so he’s a bit of a mystery! He has three more beautiful videos, all of which present little puzzles for you to solve. Check them out!

(Five-four isn’t a common time-signature for music, but it makes some great pieces. Check out this particularly awesome one. Anyone want to try making a growing shape pattern video to this tune?)

Here is an example of one of my favorite types of geometric patterns– the parquet deformation. To make one, you start with a tessellation. Then you change it- very gradually- until you’ve made a completely different tessellation that’s connected by many tiny steps to the original one.

I love to draw them. It’s challenging, but full of surprises. I never know what it’s going to look like in the end.

Want to try making your own? Check out this site by the professors/architects Tuğrul Yazar and Serkan Uysal. They had one of their classes map out how some different parquet deformations are made. They mostly used computers, but you could follow their instructions by hand, if you like. The image above is a map for the first deformation I showed.

Click on this link to see some awesome deformations made out of tiles. Aren’t they beautiful? And here’s one made by mathematical artist Craig Kaplan. It has a great fractal quality to it:

Finally, here’s something I’ve been meaning to share with you for ages! Do you ever crave a good puzzle and aren’t sure where to find one? Look no farther than the Saint Ann’s School Problem of the Week! Each week, math teacher Richard Mann writes a new awesome problem and posts it on this website. Here’s this week’s problem:

For November 26, 2013– In the picture below, find the shaded right triangle marked A, the equilateral triangle marked B and the striped regular hexagon marked C. Six students make the following statements about the picture below: Anne says “I can find an equilateral triangle three times the area of B.”  Ben says I can find an equilateral triangle four times the area of B.” Carol says, “I can find a find a right triangle triple the area of A.” Doug says, “I can find a right triangle five times the area of A.” Eloise says, “I can find a regular hexagon double the area of C.” Frank says, “I can find a regular hexagon three times the area of C.” Which students are undoubtedly mistaken?

If you solve this week’s problem, send us a solution!

Bon appetit!
The text passage contains multiple mathematical topics and examples, but does not contain a single cohesive mathematical question or problem for the reader to solve. It describes various mathematical concepts, puzzles, and activities, but does not pose a specific question with a clear answer.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Making Pi, Transcending Pi, and Cookies

Welcome to this week’s Math Munch– and happy Pi Day!

What does pi look like? The first 10,000 digits of pi, each digit 0 through 9 assigned a different color.

You probably know some pretty cool things about the number pi. Perhaps you know that pi has quite a lot to do with circles. Maybe you know that the decimal expansion for pi goes on and on, forever and ever, without repeating. Maybe you know that it’s very likely that any string of numbers– your birthday, phone number, all the birthdays of everyone you know listed in a row, followed by all their phone numbers, ANYTHING– can be found in the decimal expansion of pi.

But did you know that pi can be approximated by dropping needles on a piece of paper? Well, it can! If you drop a needle again and again on a lined piece of paper, and the needle is the same length as the distance between the lines, the probably that the needle lands on a line is two divided by pi. This experiment is called Buffon’s needle, after the French naturalist Buffon.

If the angle the needle makes with the lines is in the gray area (like the red needle’s angle is), it will cross the line. If the angle isn’t, it won’t. The possible angles trace out a circle. The closer the center of the needle (or center of the circle) is to the line, the larger the gray area– and the higher the probability of the needle hitting the line.

This may seem strange to you– but if you think about how the needle hitting a line has a lot to do with the distance between the middle of the needle and the nearest line and the angle it makes with the lines, maybe you’ll start to think about circles… and then you’ll get a clue about the connection between this experiment and pi. Working out this probability exactly requires some pretty advanced mathematics. (Feeling ambitious? Read about the calculation here.) But, you can get some great experimental results using this Buffon’s needle applet.

Click on the picture to try the applet.

I had the applet drop 500 needles. Then, the applet used the fact that the probability of the needle hitting a line should be two divided by pi and the probability it measured to calculate an approximation for pi. It got… well, you can see in the picture. Pretty close, right?

Here’s another thing you might not know: pi is a transcendental number. Sounds trippy– but, like some other famous numbers with letter names, like e, pi can never be the solution to an algebraic equation involving whole numbers. That means that no matter what equation you give me– no matter how large the exponent, how many negatives you toss in, how many times you multiply or divide by a whole number– pi will never, ever be a solution. Maybe this doesn’t sound amazing to you. If not, check out this video from Numberphile about transcendental numbers. Numbers like pi and e don’t do mathematical things we’re used to numbers doing… and it’s pretty weird.

Still curious about transcendental numbers? Here’s a page listing the fifteen most famous transcendental numbers. My favorite? Definitely the fifth, Liouville’s number, which has a 1 in each consecutive factorial numbered place.

Finally, maybe you don’t like pi. Maybe you like cookies instead. Lucky for you, you can do many mathematical things with cookies, too. Like make cookie tessellations! This mathematical artist and baker made cookie cutters in the shapes of tiles from Escher tessellations and used them to make mathematical cookie puzzles. Beautiful, and certainly delicious.

If you happen to have a 3D printer, you can make your own Escher cookie cutters. Here’s a link to print out the lizard cutter. If you don’t have a 3D printer, you could try printing out a 2D image of an Escher tessellation and tracing a tile onto a sheet of paper. Cut out the tile, roll out your dough, and slice around the outside of the tile to make your cookies. If you do it right, you shouldn’t have to waste any dough…

Here’s hoping you eat some pi or cookies on pi day! Bon appetit!

# Light Bulbs, Lanterns, and Lights Out

Welcome to this week’s Math Munch!

Edison with his light bulb.

On this day in 1880, Thomas Edison was given a patent for his most famous bright idea—the light bulb.

Edison once said, “Genius is one per cent inspiration, ninety-nine per cent perspiration”—a good reminder that putting in some work is important both in math and in life. He also said, “We don’t know a millionth of one percent about anything.” A humbling thought. Also, based on that quote, it sounds like Edison might have had a use for permilles or even permyraids in addition to percents!

Mike’s octahedron-in-a-light-buld.

In celebration of this illustrious anniversary, I’d like to share some light mathematical fare relating to, well, light bulbs. For starters, J. Mike Rollins of North Carolina has created each of the Platonic solids inside of light bulbs, ship-in-a-bottle style. Getting just the cube to work took him the better part of twelve hours! Talk about perspiration. Mike has also made a number of lovely Escher-inspired woodcuts. Check ’em out!

Evelyn’s Schwartz lantern.

Next up is a far-out example from calculus that’s also a good idea for an art project. It’s called the Schwartz lantern. I found out about this amazing object last fall when Evelyn Lamb tweeted and blogged about it.

The big idea of calculus is that we can find exact answers to tough problems by setting up a pattern of approximations that get better and better and then—zoop! take the process to its logical conclusion at infinity. But there’s a catch: you have to be careful about how you set up your pattern!

A “nicely” triangulated cylinder.

For example, if you take a cylinder and approximate its surface with a bunch of triangles carefully, you’ll end up with a surface that matches the cylinder in shape and size. But if you go about the process in a different way, you can end up with a surface that stays right near the cylinder but that has infinite area. That’s the Schwartz lantern, first proposed by Karl Hermann Amandus Schwarz of Cauchy-Schwartz fame. The infinite area happens because of all the crinkles that this devilish pattern creates. For some delightful technical details about the lantern’s construction, check out Evelyn’s post and this article by Conan Wu.

Maybe you’ll try folding a Schwartz lantern of your own. There’s a template and instructions on Conan’s blog to get you started. You’ll be glowing when you finish it up—especially if you submit a photo of it to our Readers’ Gallery. Even better, how about a video? You could make the internet’s first Schwartz lantern short film!

Robert and his Lights Out puzzle.

At the MOVES Conference last fall, Bruce Torrence of Randolf-Macon College gave a talk about the math of Lights Out. Lights Out is a puzzle—a close relative of Ray Ray—that’s played on a square grid. When you push one of the buttons in the grid it switches on or off, and its neighbors do, too. Bruce and his son Robert created an extension of this puzzle to some non-grid graphs. Here’s an article about their work and here’s an applet on the New York Times website where you can play Lights Out on the Peterson graph, among others. You can even create a Lights Out puzzle of your own! If it’s more your style, you can try a version of the original game called All Out on Miniclip.

The original Lights Out handheld game from 1995.

There’s a huge collection of Lights Out resources on Jaap’s Puzzle Page (previously), including solution strategies, variations, and some great counting problems. Lights Out and Ray Ray are both examples of what’s called a “sigma-plus game” in the mathematical literature. Just as a bonus, there’s this totally other game called Light Up. I haven’t solved a single puzzle yet, but my limitations shouldn’t stop you from trying. Perspiration!

All this great math work might make you hungry, so…bon appetit!

# Isomorphisms in Five, Parquet Deformations, and POW!

Welcome to this week’s Math Munch!

Here’s a catchy little video. It’s called “Isomorphisms in Five.” Can you figure out why? The note posted below the video says:

An isomorphism is an underlying structure that unites outwardly different mathematical expressions. What underlying structure do these figures share? What other isomorphisms of this structure will you discover?

One of the reasons I LOVE this video is because I really like how the shapes change with the music– which is played in a very interesting time signature. I also love how you can learn a lot about the different growing shape patterns by comparing them. Watch how they grow as the video flips from pattern to pattern. What do you notice? What does the music tell you about their growth?

This video is by a math educator from North Carolina named Stuart Jeckel. The only thing written about him on his “About” page is, “The Art of Math”– so he’s a bit of a mystery! He has three more beautiful videos, all of which present little puzzles for you to solve. Check them out!

(Five-four isn’t a common time-signature for music, but it makes some great pieces. Check out this particularly awesome one. Anyone want to try making a growing shape pattern video to this tune?)

Here is an example of one of my favorite types of geometric patterns– the parquet deformation. To make one, you start with a tessellation. Then you change it- very gradually- until you’ve made a completely different tessellation that’s connected by many tiny steps to the original one.

I love to draw them. It’s challenging, but full of surprises. I never know what it’s going to look like in the end.

Want to try making your own? Check out this site by the professors/architects Tuğrul Yazar and Serkan Uysal. They had one of their classes map out how some different parquet deformations are made. They mostly used computers, but you could follow their instructions by hand, if you like. The image above is a map for the first deformation I showed.

Click on this link to see some awesome deformations made out of tiles. Aren’t they beautiful? And here’s one made by mathematical artist Craig Kaplan. It has a great fractal quality to it:

Finally, here’s something I’ve been meaning to share with you for ages! Do you ever crave a good puzzle and aren’t sure where to find one? Look no farther than the Saint Ann’s School Problem of the Week! Each week, math teacher Richard Mann writes a new awesome problem and posts it on this website. Here’s this week’s problem:

For November 26, 2013– In the picture below, find the shaded right triangle marked A, the equilateral triangle marked B and the striped regular hexagon marked C. Six students make the following statements about the picture below: Anne says “I can find an equilateral triangle three times the area of B.”  Ben says I can find an equilateral triangle four times the area of B.” Carol says, “I can find a find a right triangle triple the area of A.” Doug says, “I can find a right triangle five times the area of A.” Eloise says, “I can find a regular hexagon double the area of C.” Frank says, “I can find a regular hexagon three times the area of C.” Which students are undoubtedly mistaken?

If you solve this week’s problem, send us a solution!

Bon appetit!
user
The text passage contains multiple mathematical topics and examples, but does not contain a single cohesive mathematical question or problem for the reader to solve. It describes various mathematical concepts, puzzles, and activities, but does not pose a specific question with a clear answer.

TYPE: NO_QUESTION