Welcome to this week’s Math Munch!

First up, check out the latest video in George Hart‘s series called “Mathematical Impressions.” George has been making videos for “Mathematical Impressions,” which is sponsored by the Simons Foundation, since summer, when he made his video debut – so there are many videos to watch! Here’s his newest video, called, “Attesting to Atoms,” about how the geometric structure of crystals gives clues to the existence of atoms. (Click on the picture below to watch the video.)

I love how this video shows a real way in which knowledge of mathematics – which can seem very abstract at times – can help us to understand the structure of the world, which is very concrete. In this second video, one of my favorites, George talks about the reverse of that – allowing our knowledge of something concrete to help us understand abstract mathematics. This video is called, “Knot Possible.” (Again, click on the picture to watch the video!)

I could have used these words of wisdom from George when I was thinking about the problem he poses in this video: “Don’t let your knowledge of mathematics artificially limit what you think is physically possible. Quite to the contrary! Mathematics is a tool which can empower us to do amazing things that no one has ever done before.” Well said, George!

Speaking of using mathematics to do and make amazing things, check out this website of modular origami models and patterns!

This site was put together by Michal Kosmulski, who lives in Poland and works in information technology. In addition, however, he folds these amazing modular origami polyhedra, fractals, and other awesome mathematical objects! Michal’s site is full of pictures of his modular origami creations and links to patterns for how to make them yourself as well as information about the mathematics behind the objects. He has also included some useful tips on how to make the more challenging shapes.

One of my favorites is the object to the left, “Five Intersecting Tetrahedra.” I think that this structure is both beautiful and very interesting. It can be made by intersecting five tetrahedra, or triangular-based pyramids, as shown, or by making a stellation of an icosahedron. What does that mean? Well, an icosahedron is a polyhedron with twenty equilateral triangular faces. To stellate a polyhedron, you extend some element of the polyhedron – such as the faces or edges – in a symmetric way until they meet to form a new polyhedron. There are 59 possible stellations of the icosahedron! Michal has models of several of them, including the Five Intersecting Tetrahedra and the great stellated icosahedron shown below on the left. The figure on the right is called “Cube.”

Finally, all the talking about dimensions that we’ve been doing for the past few weeks reminded me of my favorite video about higher dimensions. It’s called, “Imagining the Tenth Dimension,” and it shows a way of thinking about dimensions, from the zero dimension all the way up to the tenth. I can watch this video again and again and still find it mind-blowing and fascinating.

Bon appetit!