# Mathematical Impressions, Modular Origami, and the Tenth Dimension

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!

# 3D Printer MArTH, Polyhedra, and Hart Videos

Welcome to this week’s Math Munch!

It’s my turn now to post about how much fun we had at Bridges!  One of the best parts of Bridges was seeing the art on display, both in the galleries and in the lobby where people were displaying and selling their works of art.  We spent a lot of time oogling over the 3D printed sculptures of Henry Segerman.  Henry is a research fellow at the University of Melbourne, in Australia, studying 3-dimensional geometry and topology.  The sculptures that he makes show how beautiful geometry and topology can be.

These are the sculptures that Henry had on display in the gallery at Bridges.  They won Best Use of Mathematics!  These are models of something called 4-dimensional regular polytopes.  A polytope is a geometric object with flat sides – like a polygon in two dimensions or a polyhedron in three dimensions.  4-dimensional polyhedra?  How can we see these in three dimensions?  The process Henry used to make something 4-dimensional at least somewhat see-able in three dimensions is called a stereographic projection.  Mapmakers use stereographic projections to show the surface of the Earth – which is a 3-dimensional object – on a flat sheet of paper – which is a 2-dimensional object.

A stereographic projection of the Earth.

To do a stereographic projection, you first set the sphere on the piece of paper, or plane.  It’ll touch the plane in exactly 1 point (and will probably roll around, but let’s pretend it doesn’t).  Next, you draw a straight line starting at the point at the top of the sphere, directly opposite the point set on the plane, going through another point on the sphere, and mark where that line hits the plane.  If you do that for every point on the sphere, you get a flat picture of the surface of the sphere.  The point where the sphere was set on the plane is drawn exactly where it was set – or is fixed, as mathematicians say.  The point at the top of the sphere… well, it doesn’t really have a spot on the map.  Mathematicians say that this point went to infinity.  Exciting!

A stereographic projection like this draws a 3-dimensional object in 2-dimensions.  The stereographic projection that Henry did shows a 4-dimensional object in 3-dimensions.  Henry first drew, or projected, the vertices of his 4-dimensional polytope onto a 4-dimensional sphere – or hypersphere.  Then he used a stereographic projection to make a 3D model of the polytope – and printed it out!  How beautiful!

Here are some more images of Henry’s 3D printed sculptures.  We particularly love the juggling one.

Henry will be dropping by to answer your questions! So if you have a question for him about his sculptures, the math he does, or something else, then leave it for him in the comments.

Speaking of polyhedra, check out this site of applets for visualizing polyhedra.  You can look at, spin, and get stats on all kinds of polyhedra – from the regular old cube to the majestic great stellated dodecahedron to the mindbogglingly complex uniform great rhombicosidodecahedron.  You can also practice your skills with Greek prefixes and suffixes.

Finally, two Math Munches ago, we told you about some videos made by the mathematical artist George Hart.  He’s the man who brought us the Yoshimoto cube.  And now he’s brought us… Pentadigitation.  In this video, George connects stars, knots, and rubber bands.  Enjoy watching – and trying the tricks!

Bon appetit!