Category Archives: Math Munch

The Rhombic Dodec, Honeycombs, and Microtone

Welcome to this week’s Math Munch! Some cool pictures, videos, and a new game this week.

A couple of week’s ago, Anna wrote about the familiar hexagonal honeycomb that bees make, but that’s not the only sort of honeycomb. Mathematically, a honeycomb is the 3D version of a tessellation. Instead of covering the plane with some kind of polygon, a honeycomb fills space with some polyhedron. The cube works. Do you think tetrahedra would work? Can you think of other shapes that might work. Can you believe this works!?! (Look at the one at the bottom of that page.)

Inside the cubic honeycomb

Truncated Octahedra

Tetradecahedra

Rhombic Dodecahedral Honeycomb

I want to introduce you to one of my new favorite “space-filling polyhedra.” Meet, the rhombic dodecahedron, which you can see packed nicely on the right or in crystal form below. (Click the crystal for a really great video by George Hart about crystals and polyhedra.)

Garnet Crystal

I’ll let this video serve as an introduction to the rhombic dodecahedron and some of its features. Plus, it gives you something to make if you’d like. You’ll just need a deck of cards, and maybe a ruler and some tape.

Pretty wonderful, am I right? Here’s a link for a simple paper net you can fold up into a rhombic dodecahedron. For the really adventurous or dexterous, here’s a how-to video for a pretty tricky origami model. And here’s two more related videos showing how one can be built from two cubes.

Stellated rhombic dodecahedral honeycomb

Here’s one final amazing fact about the rhombic dodecahedron. Its first stellation is the star form of the Yoshimoto Cube!!! (background info on stellation here) Perhaps more amazing is the fact that even this shape can stack to fill 3D space!

Microtone

But now, as promised, I present a new game. Microtone is a mindbending pathwinding game played on, you guessed it, rhombic dodecahedra. (I know.) Click to move around the shape and land on all of the X’s. To rotate the dodecahedra, click and drag on the page.

Bon appetit!

Posted by Paul Salomon. Categories: Math Munch. Tags: game, geometry, honeycomb, origami model, rhombic dodecahedron, tessellation, videos. 20 Comments

Jun. ’13

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

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

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

Andrew. Click this!

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!

Next 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.

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?!

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!

Posted by Justin Lanier. Categories: Math Munch. Tags: applet, arithmetic, big numbers, building, geometry, interactive puzzle, interview, numbers, origami, primes, puzzles, recreational mathematics, sandcastles, video. 6 Comments

Jun. ’13

Natural Geometry, Hex, and Sacred Geometry

Welcome to this week’s Math Munch!

People can be skeptical when some mathematicians and scientists talk about mathematics as the “mysterious code” that “underpins the world.” I mean, the natural world is so chaotic! But then you run across this:

Honeycombs are remarkably symmetrical. Each little cell is a perfect hexagon – and all bees build this way. Why? Because of mathematics.

NPR’s Robert Krulwich wrote about this in a recent post on his excellent science blog, Krulwich Wonders. I think the explanation is an amazing example of how the natural world often follows mathematical rules perfectly. Thousands of years ago, an ancient Roman scholar named Marcus Terrentius Varro conjectured that the hexagon is the shape that most efficiently breaks flat space up into little units – making honeycombs that hold the most amount of honey while using the least amount of wax. He couldn’t prove his idea, though. It remained a conjecture until 1999 when a mathematician named Thomas Hales finally proved it! You can read a summary of his proof here. Or, watch this snippet about bees and their hexagonal honeycombs from the BBC.

Want to learn more about hexagons? Here’s a website devoted entirely to the geometry of hexagons!

Speaking of hexagons, have you ever played the game Hex? It’s a two-player game in which players take turns claiming hexagons on a hexagonally-tiled board, trying to create a connected path from one end of the board to the other. You can play it by hand using a sheet of hexagon graph paper, or you can play against a computer online, here. Enjoy!

This stained-glass church window is an example of sacred geometry.

Bees aren’t the only animals who use symmetry in the things they make. Humans do, too – especially for spiritual purposes.

An Islamic tiling.

Humans have been in awe of the symmetrical laws that seem to govern the universe for thousands of years, and they’ve developed a type of artwork called Sacred Geometry, a way of thinking that gives spiritual significance to geometric shapes. Sacred geometry can be found in religious artwork from many different cultures, and often uses tilings of regular polygons, the Platonic solids, and interlocking circles arranged in symmetric patterns.

Mathematical artist Mark Golding has been making modern works of sacred geometry art of his own. His works are inspired by mandalas, Hindu and Buddhist spiritual symbols that represent the symmetry in the universe. The image to the right is called, “Inner Relationships.” It shows an octahedron, one of the Platonic solids, nested inside of a snub cube, which is made by chopping off the corners of a cube. I love how it demonstrates the symmetric relationships between these two shapes. If you’d like to see more of his work, check out this online gallery.

Bon appetit!

P.S. – You may have noticed a new link off to the right at the top of the page. The Math Munch Team is proud to announce that our TEDx NYED talk has been posted online!

We’re honored to have been invited to participate in this event with many other creative and accomplished educators – and we encourage you to watch the other talks from the day, too.

P.P.S. And if you’re in the mood for some more TED-style math inspiration, you might enjoy these miniTED talks about math by some of Justin’s seventh graders.

Posted by Anna Weltman. Categories: Math Munch. 41 Comments