Tag Archives: origami

Origami Stars, Tessellation Stars, and Chaotic Stars

Welcome to this week’s star-studded Math Munch!

downloadModular origami stars have taken the school I teach in by storm in recent months! We love making them so much that I thought I’d share some instructional videos with you. My personal favorite is this transforming eight-pointed star. It slides between a disk with a hole the middle (great for throwing) and a gorgeous, pinwheel-like eight-pointed star. Here’s how you make one:

Another favorite is this lovely sixteen-pointed star. You can make it larger or smaller by adding or removing pieces. It’s quite impressive when completed and not that hard to make. Give it a try:

type6thContinuing on our theme of stars, check out these beautiful star tessellations. They come from a site made by Jim McNeil featuring oh-so-many things you can do with polygons and polyhedra. On this page, Jim tells you all about tessellations, focusing on a category of tessellations called star and retrograde tessellations.

type3b400px-Tiling_Semiregular_3-12-12_Truncated_Hexagonal.svgTake, for example, this beautiful star tessellation that he calls the Type 3. Jim describes how one way to make this tessellation is to replace the dodecagons in a tessellation called the 12.12.3 tessellation (shown to the left) with twelve-pointed stars. He uses the 12/5 star, which is made by connecting every fifth dot in a ring of twelve dots. Another way to make this tessellation is in the way shown above. In this tessellation, four polygons are arranged around a single point– a 12/5 star, followed by a dodecagon, followed by a 12/7 star (how is this different from a 12/5 star?), and, finally, a 12/11-gon– which is exactly the same as a dodecagon, just drawn in a different way.

I think it’s interesting that the same pattern can be constructed in different ways, and that allowing for cool shapes like stars and different ways of attaching them can open up crazy new worlds of tessellations! Maybe you’ll want to try drawing some star tessellations of your own after seeing some of these.

Screenshot 2014-05-12 10.48.46Finally, to finish off our week of everything stars, check out the star I made with this double pendulum simulator.  What’s so cool about the double pendulum? It’s a pendulum– a weight attached to a string suspended from a point– with a second weight hung off the bottom of the first. Sounds simple, right? Well, the double pendulum actually traces a chaotic path for most sizes of the weights, lengths of the strings, and angles at which you drop them. This means that very small changes in the initial conditions cause enormous changes in the path of the pendulum, and that the path of the pendulum is not a predictable pattern.

Using the simulator, you can set the values of the weights, lengths, and angles and watch the path traced on the screen. If you select “star” under the geometric settings, the simulator will set the parameters so that the pendulum traces this beautiful star pattern. Watch what happens if you wiggle the settings just a little bit from the star parameters– you’ll hardly recognize the path. Chaos at work!

Happy star-gazing, and bon appetit!

Light Bulbs, Lanterns, and Lights Out

Welcome to this week’s Math Munch!

thomas-edison

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.

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.

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.

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 Torrence and his Lights Out puzzle.

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.

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!

Partial Cubes, Open Cubes, and Spidrons

Welcome to this week’s Math Munch!

Recently the videos that Paul and I made about the Yoshimoto Cube got shared around a bit on the web. That got me to thinking again about splitting cubes apart, because the Yoshimoto Cube is made up of two pieces that are each half of a cube.

A part of Wall Drawing #601 by Sol LeWitt

A part of Wall Drawing #601
by Sol LeWitt

A friend of mine once shared with me some drawings of cubes by the artist Sol LeWitt. The cubes were drawn as solid objects, but parts of them were cut away and removed. It was fun trying to figure out what fraction of a cube remained.

On the web, I found a beautiful image that Sol made called Wall Drawing #601. In the clipping of it to the left, I see 7/8 of a cube and 3/4 of a cube. Do you? You can view the whole of this piece by Sol on the website of the Greater Des Moines Public Art Foundation.

The Cube Vinco by Vaclav Obsivac.

The Cube Vinco by Vaclav Obsivac.

There are other kinds of objects that break a cube into pieces in this way, like this tricky puzzle by Vaclav Obsivac and this “shaved” Rubik’s cube modification. Maybe you’ll design a cube dissection of your own!

As I further researched Sol LeWitt’s art, I found that he had investigated partial cubes in other ways, too. My favorite of Sol’s tinkerings is the sculpture installation called “Variations of Incomplete Cubes“. You can check out this piece of artwork on the SFMOMA site, as well as in the video below.

In the video, a diagram appears that Sol made of all of the incomplete open cubes. He carefully listed out and arranged these pictures to make sure that he had found them all—a very mathematical task. It reminds me of the list of rectangle subdivisions I wrote about in this post.

sollewitt_variationsonincompleteopencubes_1974

Sol’s diagram got me to thinking and making: what other shapes might have interesting “incomplete open” variations? I started working on tetrahedra. I think I might try to find and make them all. How about you?

Two open tetrahedra I made. Can you find some more?

Two open tetrahedra I made. Can you find some more?

Finally, as I browsed Google Images for “half cube”, one image in particular jumped out at me.

half-cube-newnweb

What are those?!?!

Dániel's original spidron from 1979

Dániel’s original spidron from 1979

These lovely rose-shaped objects are called spidrons—or more precisely, they appear to be half-cubes built out of fold-up spidrons. What are spidrons? I had never heard of them, but there’s one pictured to the right and they have their own Wikipedia article.

The first person who modeled a spidron was Dániel Erdély, a Hungarian designer and artist. Dániel started to work with spidrons as a part of a homework assignment from Ernő Rubik—that’s right, the man who invented the Rubik’s cube.

A cube with spidron faces.

A cube with spidron faces.

Two halves of an icosahedron.

Two halves of an icosahedron.

A hornflake.

A hornflake.

Here are two how-to videos that can help you to make a 3D spidron—the first step to making lovely shapes like those pictured above. The first video shows how to get set up with a template, and the second is brought to you by Dániel himself! Watching these folded spidrons spiral and spring is amazing. There’s more to see and read about spidrons in this Science News article and on Dániel’s website.

And how about a sphidron? Or a hornflake—perhaps a cousin to the flowsnake? So many cool shapes!

To my delight, I found that Dániel has created a video called Yoshimoto Spidronised—bringing my cube splitting adventure back around full circle. You’ll find it below. Bon appetit!

Reflection Sheet – Partial Cubes, Open Cubes, and Spidrons

Andrew Hoyer, Cameron Browne, & Sphere Inversion

Welcome to this week’s Math Munch! Fractals, origami, math art, games, and a mind-bending video are all ahead, so let’s get into it.

Andrew Hoyer

Andrew Hoyer

First up, let’s take a look at the work of Andrew Hoyer.  According to his website, he’s a “software engineer in his mid-twenties living it up in sometimes sunny San Francisco.” I came across his work when I found his beautiful and completely engaging introduction to simple fractals.  (Go on! Click. Then read, experiment and play!)

Cantor Set

A Cantor set

At the bottom of that page, Andrew links to a wonderful, long list of fractals, arranged by Hausdorff dimension, which is a way of measuring fractals as being something like 2.5 dimensions.  A line is 1 dimensional.  A plane is 2D, and you can find many fractals with dimension in between!!  Weird, right?

I was also really pleased to find Andrew’s Instagram feed, which features some of his beautiful origami creations.  Andrew’s agreed to answer your questions for an upcoming Q&A, so ask away!

Compound of 5 tetrahedra Truncated Icosahedron Cube Dodecahedron
Cameron Browne

Cameron Browne

Up next, meet Cameron Browne. He’s an accomplished researcher who designs and studies games. Take a look at the many many games Cameron has created. The rules and descriptions are there, and Cameron sent along links to playable versions of a few, which you can find by clicking the pictures below.  For the third one, you’ll need to search for “Margo” or “Spargo.” For his research, Cameron investigates the possibilities of artificial intelligence, and how a computer can be used to generate games and puzzles.

Yavalath

Yavalath
description

Margo and Spargo

Margo and Spargo
description
description

Cameron is also an artist, and he has a page full of his graphic designs.  I found Cameron through his page of Truchet curves.  I love the way his pages are full of diagrams and just enough information to start making sense of things, even if it’s not perfectly clear.  Cameron also has MANY pages of wonderful fractal-ish graphics: Impossible Fractals, Cantor Knots, Fractal Board Games, Woven Horns, Efficient Trees, and on and on…  And he has agreed to do a Q&A with us, so please, submit a question. What are you wondering?

A Cantor Knot

A Cantor Knot

A Truchet curve "Mona Lisa"

A Truchet curve “Mona Lisa”

An "impossible" fractal

An “impossible” fractal

And, as if that wasn’t enough mathy awesomeness, check out this video about turning a sphere inside out.  A bit of personal history, I actually used this video  (though it was only on VHS back then, checked out from the library) as part of the research for my independent research project during my senior year of college.  It gets pretty tricky, but if you watch it all the way through it starts to make some sense.

Have a great week.  Bon appetit!

Reflection sheet – Andrew Hoyer, Cameron Browne, & Sphere Inversion

MoMA, Pop-Up Books, and A Game of Numbers

Welcome to this week’s Math Munch!

Thank you so much to everyone who participated in our Math Munch “share campaign” over the past two weeks. Over 200 shares were reported and we know that even more sharing happened “under the radar”. Thanks for being our partners in sharing great math experiences and curating the mathematical internet.

Of course, we know that the sharing will continue, even without a “campaign”. Thanks for that, too.

All right, time to share some math. On to the post!

N_JoshiTo kick things off, you might like to check out our brand-new Q&A with Nalini Joshi. A choice quote from Nalini:

In contrast, doing math was entirely different. After trying it for a while, I realized that I could take my time, try alternative beginnings, do one step after another, and get to glimpse all kinds of possibilities along the way.

By Philippe Decrauzat.

By Philippe Decrauzat.

I hope the math munches I share with you this week will help you to “glimpse all kinds of possibilities,” too!

Recently I went to the Museum of Modern Art (MoMA) in New York City. (Warning: don’t confuse MoMA with MoMath!) On display was an exhibit called Abstract Generation. You can view the pieces of art in the exhibit online.

As I browsed the galley, the sculptures by Tauba Auerbach particularly caught my eye. Here are two of the sculptures she had on display at MoMA:

CRI_244599 CRI_244605

Just looking at them, these sculptures are definitely cool. However, they become even cooler when you realize that they are pop-up sculptures! Can you see how the platforms that the sculptures sit on are actually the covers of a book? Neat!

Here’s a video that showcases all of Tauba’s pop-ups in their unfolding glory. Why do you think this series of sculptures is called [2,3]?

This idea of pop-up book math intrigued me, so I started searching around for some more examples. Below you’ll find a video that shows off some incredible geometric pop-ups in action. To see how you can make a pop-up sculpture of your own, check out this how-to video. Both of these videos were created by paper engineer Peter Dahmen.

Taura Auerbach.

Tauba Auerbach.

Tauba got me thinking about math and pop-up books, but there’s even more to see and enjoy on her website! Tauba’s art gives me new ways to connect with and reimagine familiar structures. Remember our post about the six dimensions of color? Tauba created a book that’s a color space atlas! The way that Tauba plays with words in these pieces reminds me both of the word art of Scott Kim and the word puzzles of Douglas Hofstadter. Some of Tauba’s ink-on-paper designs remind me of the work of Chloé Worthington. And Tauba’s piece Componants, Numbers gives me some new insight into Brandon Todd Wilson’s numbers project.

0108 MM MM-Tauba-Auerbach-large

This piece by Tauba is a Math Munch fave!

For me, both math and art are all about playing with patterns, images, structures, and ideas. Maybe that’s why math and art make such a great combo—because they “play” well together!

Speaking of playing, I’d like to wrap up this week’s post by sharing a game about numbers I ran across recently. It’s called . . . A Game of Numbers! I really like how it combines the structure of arithmetic operations with the strategy of an escape game. A Game of Numbers was designed by a software developer named Joseph Michels for a “rapid” game competition called Ludum Dare. Here’s a Q&A Joseph did about the game.

A Game of Numbers.

A Game of Numbers.

If you enjoy A Game of Numbers, maybe you’ll leave Joseph a comment on his post about the game’s release or drop him an email. And if you enjoy A Game of Numbers, then you’d probably enjoy checking out some of the other games on our games page.

Bon appetit!

PS Tauba also created a musical instrument called an auerglass that requires two people to play. Whooooooa!

Reflection Sheet – MoMA, Pop-Up Books, and A Game of Numbers

Prime Gaps, Mad Maths, and Castles

Welcome to this week’s Math Munch!

It has been a thrilling last month in the world of mathematics. Several new proofs about number patterns have been announced. Just to get a flavor for what it’s all about, here are some examples.

I can make 15 by adding together three prime numbers: 3+5+7. I can do this with 49, too: 7+11+31. Can all odd numbers be written as three prime numbers added together? The Weak Goldbach Conjecture says that they can, as long as they’re bigger than five. (video)

11 and 13 are primes that are only two apart. So are 107 and 109. Can we find infinitely many such prime pairs? That’s called the Twin Prime Conjecture. And if we can’t, are there infinitely many prime pairs that are at most, say, 100 apart? (video, with a song!)

Harald Helfgott

Harald Helfgott

Yitang "Tom" Zhang

Yitang “Tom” Zhang

People have been wondering about these questions for hundreds of years. Last month, Harald Helfgott showed that the Weak Goldbach Conjecture is true! And Yitang “Tom” Zhang showed that there are infinitely many prime pairs that are at most 70,000,000 apart! You can find lots of details about these discoveries and links to even more in this roundup by Evelyn Lamb.

What’s been particularly fabulous about Tom’s result about gaps between primes is that other mathematicians have started to work together to make it even better. Tom originally showed that there are an infinite number of prime pairs that are at most 70,000,000 apart. Not nearly as cute as being just two apart—but as has been remarked, 70,000,000 is a lot closer to two than it is to infinity! That gap of 70,000,000 has slowly been getting smaller as mathematicians have made improvements to Tom’s argument. You can see the results of their efforts on the polymath project. As of this writing, they’ve got the gap size narrowed down to 12,006—you can track the decreasing values down the page in the H column. So there are infinitely many pairs of primes that are at most 12,006 apart! What amazing progress!

Two names that you’ll see in the list of contributors to the effort are Andrew Sutherland and Scott Morrison. Andrew is a computational number theorist at MIT and Scott has done research in knot theory and is at the Australian National University. They’ve improved arguments and sharpened figures to lower the prime gap value H. They’ve contributed by doing things like using a hybrid Schinzel/greedy (or “greedy-greedy”) sieve. Well, I know what a sieve is and what a greedy algorithm is, but believe me, this is very complicated stuff that’s way over my head. Even so, I love getting to watch the way that these mathematicians bounce ideas off each other, like on this thread.

Andrew Sutherland

Andrew Sutherland

Click through to see Andrew next to an amazing Zome creation!

Andrew. Click this!

Scott Morrison

Scott Morrison

Andrew and Scott have agreed to answer some of your questions about their involvement in this research about prime gaps and their lives as mathematicians. I know I have some questions I’m curious about! You can submit your questions in the form below:

I can think of only two times in my life where I was so captivated by mathematics in the making as I am by this prime gaps adventure. Andrew Wiles’s proof of Fermat’s Last Theorem was on the fringe of my awareness when it came out in 1993—its twentieth anniversary of his proof just happened, in fact. The result still felt very new and exciting when I read Fermat’s Enigma a couple of years later. Grigori Perelman’s proof of the Poincare Conjecture made headlines just after I moved to New York City seven years ago. I still remember reading a big article about it in the New York Times, complete with a picture of a rabbit with a grid on it.

This work on prime gaps is even more exciting to me than those, I think. Maybe it’s partly because I have more mathematical experience now, but I think it’s mostly because lots of people are helping the story to unfold and we can watch it happen!

fig110u2bNext up, I ran across a great site the other week when I was researching the idea of a “cut and slide” process. The site is called Mad Maths and the page I landed on was all about beautiful dissections of simple shapes, like circles and squares. I’ve picked out one that I find especially charming to feature here, but you might enjoy seeing them all. The site also contains all kinds of neat puzzles and problems to try out. I’m always a fan of congruent pieces problems, and these paper-folding puzzles are really tricky and original. (Or maybe, origaminal!) You’ll might especially like them if you liked Folds.

Christian's applet displaying the original four-room castle.

Christian’s applet displaying the original four-room castle.

Finally, we previously posted about Matt Parker’s great video problem about a princess hiding in a castle. Well, Christian Perfect of The Aperiodical has created an applet that will allow you to explore this problem—plus, it’ll let you build and try out other castles for the princess to hide in. Super cool! Will I ever be able to find the princess in this crazy star castle I designed?!

Crazy star castle!

My crazy star castle!

And as summer gets into full swing, the other kind of castle that’s on my mind is the sandcastle. Take a peek at these photos of geometric sandcastles by Calvin Seibert. What shapes can you find? Maybe Calvin’s creations will inspire your next beach creation!

Bon appetit!

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Folds, GIMPS, and More Billiards

Welcome to this week’s Math Munch!

First up, we’ve often featured mathematical constructions made of origami. (Here are some of those posts.) Origami has a careful and peaceful feel to it—a far cry from, say, the quick reflexes often associated with video games. I mean, can you imagine an origami video game?

heartfolds

One of Fold’s many origami puzzles.

Well, guess what—you don’t have to, because Folds is just that! Folds is the creation of Bryce Summer, a 21-year-old game designer from California. It’s so cool. The goal of each level of its levels is simple: to take a square piece of paper and fold it into a given shape. The catch is that you’re only allowed a limited number of folds, so you have to be creative and plan ahead so that there aren’t any loose ends sticking out. As I’ve noted before, my favorite games often require a combo of visual intuition and careful thinking, and Folds certainly fits the bill. Give it a go!

Once you’re hooked, you can find out more about Bryce and how he came to make Folds in this awesome Q&A. Thanks so much, Bryce!

gimpsNext up, did you know that a new largest prime number was discovered less than a month ago? It’s very large—over 17 million digits long! (How many pages would that take to print or write out?) That makes it way larger than the previous record holder, which was “only” about 13 million digits long. Here is an article published on the GIMPS website about the new prime number and about the GIMPS project in general.

What’s GIMPS you ask? GIMPS—the Great Internet Mersenne Primes Search—is an example of what’s called “distributed computing”. Testing whether a number is prime is a simple task that any computer can do, but to check many or large numbers can take a lot of computing time. Even a supercomputer would be overwhelmed by the task all on its own, and that’s if you could even get dedicated time on it. Distributed computing is the idea that a lot of processing can be accomplished by having a lot of computers each do a small amount of work. You can even sign up to help with the project on your own computer. What other tasks might distributed computing be useful for? Searching for aliens, perhaps?

GIMPS searches only for a special kind of prime called Mersenne primes. These primes are one less than a power of two. For instance, 7 is a Mersenne prime, because it’s one less that 8, which is the third power of 2. For more on Mersenne primes, check out this video by Numberphile.

Finally, we’ve previously shared some resources about the math of billiards on Math Munch. Below you’ll find another take on bouncing paths as Michael Moschen combines the math of billiards with the art of juggling.

So lovely. For more on this theme, here’s a second video to check out.

Bon appetit!