Tag Archives: knots

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.



Margo and Spargo

Margo and Spargo

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

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.)

Atoms 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!)

Knot 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!

sierpinski-tetrahedron-tri-2Speaking 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.

fit-five-intersecting-tetrahedra-60deg-2One 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!

Knots, Torus Games, and Bagels

Welcome to this week’s Math Munch!

The things we have lined up for you this week have to do with a part of math called topology.  Topology is like geometry in many ways, except the shapes you study aren’t rigid.  Instead, you can twist, stretch, squish, and generally deform them in any way you like, so long as you don’t rip any holes or attach things that weren’t already attached.  One of the reasons why topology is interesting is that you get to play with new and fascinating shapes, like…

… knots!  This nifty site, Knot Theory Online, is full of interesting information about the study of mathematical knots and its history and applications.  For some basic information, check out the introduction to knots page.  It talks about what a knot is, mathematically speaking, and some ways that mathematicians answer the most important question in knot theory: is this knot the unknot?  The site also has some fun games in which you can play with transforming one knot into another.  Here’s my favorite: The Hunt for the Elusive Trefoil Knot.

Knots can also be works of art – and this site, Knot Plot, showcases artistic knots at their best.  Here are some images of beautiful decorative knots.

A really cool thing about knot theory is that it is a relatively new area of mathematical research – which means that there are many unsolved knot theory problems that a person without a lot of math training could attempt!  Here’s a page of “approachable open problems in knot theory,” compiled by knot theorist and Williams College professor Colin Adams.

One of a topologist’s favorite objects to study is one that you might encounter at breakfast – the torus, or donut (or bagel).  To get a sense for what makes a torus topologically interesting and for what life might be like if you lived on a torus (instead of a sphere, a topologically different surface), check out Torus Games.  Torus Games was created by mathematician Jeff Weeks.  You can play games that you’d typically play on a plane, in flat space – such as Tic-Tac-Toe, chess, and pool – but on a torus (or a Klein bottle) instead!

A maze – on a torus!

By the way, you can find Torus Games and other cool, free, downloadable math software on our new page – Free Math Software.  You’ll find links to other software that we love to use – such as Scratch and GeoGebra, and another program by Jeff Weeks called Curved Spaces.

All this talk of tori making you hungry?  Go get your own tasty torus (bagel), and try this fun trick to slice your bagel into two linked halves.  This topologically delicious breakfast problem was created by mathematical artist George Hart.

Bon appetit!  (Literally, this time.)