We created möbius bands after looking at Vi Hart‘s Wind & Mr. Ug video. We checked out what happens when they are cut down the middle or when they are cut a third of the way from an edge. Wikipedia has a good article on mobius bands.
After that, I gave a short presentation on worlds shaped like spheres (like our own) or like bagels (the shape mathematicians call a “torus”). You can see some slides of that here.
We can make a map of a torus world (imagine slicing the torus and unrolling like this: https://www.youtube.com/watch?v=0H5_h-RB0T8). Such a map would have distortions (just like the maps of our sperical world), but would also work that if you traveled north out of the map, you’d come back up through the south. Same thing traveling east out of the map (but of course our regular maps of our globe work that way as well). If we imagin playing games on such a world, we have to remember this strange kind of movement. You can download the torus games made by Jeff Weeks that we played in class. Does a torus world seem far fetched? There are lots of them in the world of gaming. See http://tvtropes.org/pmwiki/pmwiki.php/Main/VideoGameGeography for a list of video game world with torus geography (for instance the game of Tetrisphere).
We watched this “Twisted Torus” video for more torus fun: https://www.youtube.com/watch?v=3_VydFQmtZ8
And we made a torus out of paper: http://www.cutoutfoldup.com/980-clifford-torus.php
Finally I showed a video about why folding a pizza slice before eating it works, using some math and physics.
As usual, you should send a response to me about this class. You can send it by email or hand it to me at the beginning of our next class. Class responses should be about 350-600 words if you are writing them, but you are welcome to do something else that responds to the class (visual, video, audio, etc). Some things to consider responding to:
- What did you think of the shapes we explored and watched videos about in class?
- What was most interesting/most confusing/most boring?
- Where have you seen or used similar shapes?
- Have you played video games that have used a torus-world? Which games?
- Explore and respond to any of the links above or below.
- Tell me anything else you need/want to tell me!
Other resources that you might like or find usefulTorus:
- Here’s the video of the sliceform torus (it has a link to the instructions): https://www.youtube.com/watch?v=-d4IQT_phlQ. Want more sliceforms?
- John Sharp wrote a book on sliceforms and started some of the current sliceform trend. He has a blog, but isn’t currently putting up anything new.
- Flickr photos of sliceforms: http://fiveprime.org/hivemind/Tags/sliceform
- Want to learn more about Klein Bottles? Want to make one? Try here. Or buy one from Acme Klein Bottles. Or just think a lot about them.
- Think some more about cutting a bagel: http://www.georgehart.com/bagel/bagel.html
- Unfolding things: http://plus.maths.org/content/os/issue27/features/mathart/index