Here’s a puzzle for you: You have 12 weights, 11 of which weigh the same amount and 1 of which is different. Luckily you also have a balance, but you’re only allowed to use it three times. Can you figure out which weight is the different weight?

You certainly can! I won’t tell you how, but you can figure it out for yourself while playing this interactive weight game. This puzzle is tricky, but definitely fun. If one weight puzzle isn’t enough for you, you’re in luck– there are many, many variations! Check out this site to try a similar puzzle with nine weights, ten weights, and 27 weights.

My solution to the Circle Pack 2 challenge. Can you do it in only 5 moves?

Next up, if you like drawing challenges, this is the game for you. Check out this crazy geometry game, in which you have to draw different shapes (like perfect equilateral triangles, squares, pentagons, and groups of circles of particular sizes) using only circles and straight lines! Here’s my solution to one of the challenges, the Circle Pack 2. See the two smaller circles inside of the larger middle circle? That’s what I wanted to draw– but I had to make all of those other circles and lines to get there! I did the Circle Pack 2 challenge in 8 moves, but apparently there’s a way to do it in only 5…

Finally, it’s pumpkin season again! Every year I scour the internet for new math-y ways to carve pumpkins. We’re all in luck this year– because I found great instructions for how to carve pumpkin polyhedra from Math Craft! Check out this site to learn how to carve all the basics– tetrahedra, cubes, octahedra, dodecahedra, and (my favorite) icosahedra– and a bonus polyhedron, the truncated icosahedron (also know as the soccer ball).

Pumpkin Platonic polyhedra!

Don’t forget to make pi with the leftover pumpkin! Oh, and, bon appetit!

You might remember our post on Tilman Zitzmann’s project called Geometry Daily. If you haven’t seen it before, go check it out now! It will help you to appreciate Lawrie Cape’s work, which both celebrates and extends the Geometry Daily project. Lawrie’s project is called Tangent Spaces. He makes Tilman’s geometry sketches move!

A box of rays, by Tilman

A box of rays, by Lawrie

Not only do Lawrie’s sketches move, they’re also interactive—you can click on them, and they’ll move in response. All kinds of great mathematical questions can come up when you set a diagram in motion. For instance, I’m wondering what moon patterns are possible to make by dragging my mouse around—and if any are impossible. What questions come up for you as you browse Tangent Spaces?

Next up, Dorry Segev and Sommer Gentry are a doctor and a mathematician. They collaborated on a new system to help sick people get kidney transplants. They are also dance partners and husband and wife. This video shares their amazing, mathematical, and very human story.

Dorry and Sommer’s work involves building graphs, kind of like the game that Paul posted about last week. Thinking about the two of them together has been fun for me. You can read more about the life-saving power of Kidney Paired Donation on optimizedmatch.com.

Last up this week, here’s some very fresh math—discovered in the last 24 hours! Joe O’Rourke is one of my favorite mathematicians. (previously) Joe recently asked whether a golyhedron exists. What’s a golyhedron? It’s the 3D version of a golygon. What’s a golygon? Glad you asked. It’s a grid polygon that has side lengths that grow one by one, from 1 up to some number. Here, a diagram will help:

The smallest golygon. It has sides of lengths 1 through 8.

A golyhedron is like this, but in 3D: a grid shape that has one face of each area from 1 up to some number. After tinkering around some with this new shape idea, Joe conjectured that no golyhedra exist. It’s kind of like coming up with the idea of a unicorn, but then deciding that there aren’t any real ones. But Joseph wasn’t sure, so he shared his golyhedron shape idea on the internet at MathOverflow. Adam P. Goucher read the post, and decided to build a golyhedron himself.

And he found one!

The first ever golyhedron, by Adam P. Goucher

Adam wrote all about the process of discovering his golyhedron in this blog post. I recommend it highly.

And the story and the math don’t stop there! New questions arise—is this the smallest golyhedron? Are there types of sequences of face sizes that can’t be constructed—for instance, what about a sequence of odd numbers? Curious and creative people, new discoveries, and new questions—that’s how math grows.

If this story was up your alley, you might enjoy checking out the story of holyhedra in this previous post.

Earlier this month, neuroscientists Stan Schein and James Gayed announced the discovery of a new class of polyhedra. We’ve often posted about Platonic solids here on Math Munch. The shapes that Stan and James found have the same symmetries as the icosahedron and dodecahedron, and they also have all equal edge lengths.

One of Stan and James’s shapes, made of equilateral pentagons and hexagons.

These new shapes are examples of fullerenes, a kind of shape named after the geometer, architect, and thinker Buckminster Fuller. In the 1980s, chemists discovered that molecules made of carbon can occur in polyhedral shapes, both in the lab and in nature. Stan and James’s new fullerenes are modifications of some existing shapes first described in 1937 by Michael Goldberg. The faces of Goldberg’s shapes were warped, not flat, and Stan and James showed that flattening can be achieved—thus turning Goldberg’s shapes into true polyhedra—while also having all equal edge lengths. There’s great coverage of Stan and James’s discovery in this article at Science News and a fascinating survey of the media’s coverage of the discovery by Adam Lore on his blog. Adam’s post includes an interview with Stan!

Next up—how much fun is it to find a fractal that’s new to you? That happened to me recently when I ran across the Fibonacci word fractal.

A portion of a Fibonacci word curve.

Fibonacci “words”—really just strings of 0’s and 1’s—are constructed kind of like the numbers in the Fibonacci sequence. Instead of adding numbers previous numbers to get new ones, we link up—or “concatenate”—previous words. The first few Fibonacci words are 1, 0, 01, 010, 01001, and 01001010. Do you see how new words are made out of the two previous ones?

Here’s a variety of images of Fibonacci word fractals, and you can find more details about the fractal in this article. The infinite Fibonacci word has an entry at the OEIS, and you can find a Fibonacci word necklace on Etsy. Dale Gerdemann, a linguist at the University of Tübingen, has a whole series of videos that show off patterns created out of Fibonacci words. Here is one of my favorites:

Last but not least this week, check out this groovy applet!

Lucas’s applet showing the relationship between epicycles and Fourier series

A basic layout of Ptolemy’s model, including epicycles.

Sometime around the year 200 AD, the astronomer Ptolemy proposed a way to describe the motion of the sun, moon, and planets. Here’s a video about his ideas. Ptolemy relied on many years of observations, a new geometrical tool we call “trigonometry”, and a lot of ingenuity. He said that the sun, moon, and planets move around the earth in circles that moved around on other circles—not just cycles, but epicycles. Ptolemy’s model of the universe was incredibly accurate and was state-of-the-art for centuries.

Joseph Fourier

In 1807, Joseph Fourier turned the mathematical world on its head. He showed that periodic functions—curves with a repeated pattern—can be built by adding together a very simple class of curves. Not only this, but he showed that curves created in this way could have breaks and gaps even though they are built out of continuous curves called “sine” and “cosine”. (Sine and cosine are a part of the same trigonometry that Ptolemy helped to found.) Fourier series soon became a powerful tool in mathematics and physics.

A Fourier series that converges to a discontinuous function.

And then in the early 21st century Lucas Vieira created an applet that combines and sets side-by-side the ideas of Ptolemy and Fourier. And it’s a toy, so you can play with it! What cool designs can you create? We’ve featured some of Lucas’s work in the past. Here is Lucas’s short post about his Fourier toy, including some details about how to use it.