Tag Archives: geometry

Platonic Terrariums, Geometric Decor, and Multiplying Polyhedra

Welcome to this week’s Math Munch! We’ve got some beautiful geometric objects meant to house a plant or decorate your home, as well as a really clever kind of “multiplication chart” relating the Platonic solids to each other.

Icosahedron Terrarium

First up, let’s take a look at some gorgeous glass terrarium models of the Platonic solids. We don’t usually share products here on Math Munch, because we want to make sure you can enjoy the math for free, but these are so beautiful I just had to show you. I’m a sucker for spherical symmetry!

The Turning Triangles Terrarium actually sits on my mantle at home. It’s 20 pieces of triangular glass (with one hinged pane) coming together to make an icosahedron home for a little plant.

Octahedron Terrariums

Above you can see a spread of octahedron terrariums, which will have to be my next purchase. Does $29 seem like a lot for one of those? I was kind of shocked to see prices for other ones that are about 4 times that much. Take a look at the dodecahedron and cube terrariums below. They’re over $100 each, but man are they cool!?

Dodecahedron and Cube Terrariums

I love how they stood the cube up on its corner. Did you ever think about how cutting off the corner of a cube creates a little triangle?

Speaking of cutting off corners, that’s called “truncation.” I bet you never realized the soccer ball pattern is a truncated icosahedron. Well it is! And West Elm is selling a pair of really beautiful truncated polyhedra made of Capiz shells. Below are the corner-cut versions of the icosahedron and dodecahedron.

Capiz Shell Truncated Polyhedra

OK, just a couple more. First, I love the blue and white of these two shapes. One correction: the seller calls them an “octahedron”, but they have more than 8 faces. These are actually cuboctahedra. (Can you figure out how many sides they do have?)

Metal Icosidodecahedra

And lastly, the really cool, metal rhombicosidodecahedron. This is the shape that is used for the Zome construction kit. Check out this video showing a project we did last year. In short, we made a really big version of this out of lots of little ones.

If you end up buying one of these decorative sculptures, let us know. We’d love to see a picture of it in your house.

Finally, this is a really incredible image I found on Pinterest. Can you tell what’s going?

A Platonic solid “multiplication” chart

It’s set up like a multiplication chart, with the Platonic solids along the top and left edges. In the middle, we get a picture showing how the two shapes might be related to each other. I could (and have) stared at this for hours!

In the A1 position, for example, we have a picture showing that the tetrahedron is the dual of the tetrahedron. That means, when you connect the centers of the faces on the tetrahedron, you get another tetrahedron!

B3 shows that the octhahedron is the dual of the cube. E4 shows that the icosahedron is the dual of the dodecahedron. B2 appears to be a hypercube, and B1 shows the way that a tetrahedron can be made by connecting alternating corners of a cube. It’s a fascinating chart, and I hope you’ll take some time to check it out. Can you figure out what’s going on in C3?

I would love to know where this image came from, but I can’t find anything about it. If you know anything about the origin of the chart, please let us know.

Well that’s it. I hope you found something juicy. Bon appetit!

Posted by Paul Salomon. Categories: Math Munch. Tags: geometry, multiplication chart, Platonic solids, polyhedra. 9 Comments

Nov. ’13

A Periodic Table, Linkages, and Dance Squared

Welcome to this week’s Math Munch!

I like finding new ways of organizing information. That’s part of why I enjoy this Periodic Table of Mathematicians.

The letters in the table are the abbreviations of the chemical elements—like gold, helium, and iron—that are found on the usual periodic table. With a little creativity, they can also be abbreviations for the names of a bunch of celebrated mathematicians. Clicking on a square brings up the mathematician’s biography. I like guessing who might pop up!

The table was created by Erich Friedman, a mathematician who works at Stetson University in Florida. We’ve previously shared Erich’s holiday puzzles (here) and weight puzzles (here) and monthly research contest (here), but there’s even more to explore on his site. I’m partial to his Packing Center, which shows the best ways that have been found to pack shapes inside of other shapes. You might also enjoy his extensive listing of What’s Special About This Number?—a project in the same spirit as Tanya Khovanova’s Number Gossip.

A dense packing of 26 squares within a square that Erich discovered.

I wonder what a multiplicative persistence is?

Next up, another Erik—Erik Demaine, whose work we’ve also often featured. What does he have for us this time? Some fantastic uncurling linkages, that’s what!

In 2000, Erik worked with Robert Connelly and Günter Rote to show that any wound-up 2D shape made of hinged sticks can be unwound without breaking, crossing, or lifting out of the plane. In the end, the shape must be convex, so that it doesn’t have any dents in it. For a while Erik and his colleagues thought that some linkages might be “locked” and unwinding some of the examples they created took months. You can find some great animations shared on the webpage that describes their result that locked linkages don’t in fact exist.

One thing that amazes me about Erik’s mathematical work is how young the problems are that he works on and solves. You might think a problem that can be put in terms of such simple ideas would have been around for a while, but in fact this problem of unwinding linkages was first posed only in the 1970s! It just goes to show that there are new simple math problems just waiting to be invented all the time.

Finally, I was so glad to run across this short film called Dance Squared. It was made by René Jodoin, a Canadian director and producer. Check out how much René expresses with just a simple square!

There’s a wonderful celebration of René titled When I Grow Up I Want To Be René Jodoin—written back in 2000 when René was “only” 80 years old. Now here’s 92! Making math is for people of all ages. You might also enjoy watching René’s Notes on a Triangle.

Bon appetit!

Reflection Sheet – A Periodic Table, Linkages, and Dance Squared

Posted by Justin Lanier. Categories: Math Munch. Tags: animations, art, discrete math, geometry, mathematicians, numbers, packing, recreational mathematics, spirals, video. 12 Comments

Nov. ’13

Numenko, Turning Square, and Toilet Paper

Welcome to this week’s Math Munch!

Have you ever played Scrabble or Bananagrams? Can you imagine versions of these games that would use numbers instead of letters?

Meet Tom Lennett, who imagined them and then made them!

Tom playing Numenko with his grandkids.

Numemko is a crossnumber game. Players build up number sentences, like 4×3+8=20, that cross each other like in a crossword puzzle. There is both a board game version of Numenko (like Scrabble) and a bag game version (like Banagrams). Tom invented the board game years ago to help his daughter get over her fear of math. He more recently invented the bag game for his grandkids because they wanted a game to play where they didn’t have to wait their turn!

The Multichoice tile.

One important feature of Numenko is the Multichoice tile. Can you see how it can represent addition, subtraction, multiplication, division, or equality?

How would you like to have a Numenko set of your own? Well, guess what—Tom holds weekly Numenko puzzle competitions with prizes! You can see the current puzzle on this page, as well as the rules. Here’s the puzzle at the time of this post—the week of November 3, 2013.

Can you replace the Multichoice tiles to create a true number sentence?

Challenge: replace the Multichoice tiles to create a true number sentence.

I can assure you that it’s possible to win Tom’s competitions, because one of my students and I won Competition 3! I played my first games of Numenko today and really enjoyed them. I also tried making some Numenko puzzles of my own; see the sheet at the bottom of this post to see some of them.

Tom in 1972.

In emailing with Tom I’ve found that he’s had a really interesting life. He grew up in Scotland and left school before he turned 15. He’s been a football-stitcher, a barber, a soldier, a distribution manager, a paintball site operator, a horticulturist, a property developer, and more. And, of course, also a game developer!

Do you have a question you’d like to ask Tom? Send it in through the form below, and we’ll try to include it in our upcoming Q&A!

The level editor.

Say, do you like Bloxorz? I sure do—it’s one of my favorite games! So imagine my delight when I discovered that a fan of the game—who goes by the handle Jz Pan—created an extension of it where you can make your own levels. Awesome, right? It’s called Turning Square, and you can download it here.

(You’ll need to uncompress the file after downloading, then open TurningSquare.exe. This is a little more involved than what’s usual here on Math Munch, but I promise it’s worth it! Also, Turning Square has only been developed for PC. Sorry, Mac fans.)

The level I made!

But wait, there’s more! Turning Square also introduces new elements to Bloxorz, like slippery ice and pyramids you can trip over. It has a random level generator that can challenge you with different levels of difficulty. Finally, Turning Square includes a level solver—it can determine whether a level that you create is possible or not and how many steps it takes to complete.

Jz Pan is from China and is now a graduate student at the Chinese Academy of Sciences, majoring in mathematics and studying number theory. Jz Pan made Turning Square in high school, back in 2008.

Jz Pan has agreed to answer some of your questions! Use the form below to send us some.

If you make a level in Turning Square that you really like, email us the .box file and we can share it with everyone through our new Readers’ Gallery! Here is my level from above, if you want to try it out.

Jz Pan has also worked on an even more ambitious extension of Bloxorz called Turning Polyhedron. The goal is the same, but like the game Dublox, the shape that you maneuver around is different. Turning Polyhderon features several different shapes. Check out this video of it being played with a u-polyhedron!

And if you think that’s wild, check out this video with multiple moving blocks!

Last up this week, have you ever heard that it’s impossible to fold a piece of paper in half more than eight times? Or maybe it’s seven…? Either way, it’s a “fact” that seems to be common knowledge, and it sure seems like it’s true when you try to fold up a standard sheet of paper—or even a jumbo sheet of paper. The stack sure gets thick quickly!

Britney and her 11th fold.

Well, here’s a great story about a teenager who decided to debunk this “fact” with the help of some math and some VERY big rolls of toilet paper. Her name is Britney Gallivan. Back in 2001, when she was a junior in high school, Britney figured out a formula for how much paper she’d need in order to fold it in half twelve times. Then she got that amount of paper and actually did it!

Due to her work, Britney has a citation in MathWorld’s article on folding and even her own Wikipedia article. After high school, Britney went on to UC Berkeley where she majored in Environmental Science. I’m trying to get in touch with Britney for an interview—if you have a question for her, hold onto it, and I’ll keep you posted!

EDIT: I got in touch with Britney, and she’s going to do an interview!

A diagram that illustrates how Britney derived her equation.

The best place to read more about Britney’s story in this article at pomonahistorical.org—the historical website of Britney’s hometown. Britney’s story shows that even when everyone else says that something’s impossible, that doesn’t mean you can’t be the one to do it. Awesome.

I hope you enjoy trying some Numenko puzzles, tinkering with Turning Square, and reading about Britney’s toilet paper adventure.

Bon appetit!

PS Want to see a video of some toilet-paper folding? Check out the very first “family math” video by Mike Lawler and his kids.

Reflection Sheet – Numenko, Turning Square, and Toilet Paper

Posted by Justin Lanier. Categories: Math Munch. Tags: algebra, arithmetic, big numbers, board game, contest, diversity, exponential, games, geometry, interview, numbers, polyhedra, puzzles, recreational mathematics, scrabble, software, video. 5 Comments