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EXERCISE: Can you describe it?
Jun. 4th, 2008 10:07 am![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
mbarker.livejournal.comoriginal posting: Mon, 16 Jun 2003 22:40:33 -0400
Since we've had mention of it...
(aka Party Tricks for Mathematicians? :-)
Try describing a mobius strip in words. In particular, can you describe what is so different about it so that Joe (or Jane) Random "gets it"?
Personally, I end up with an operational definition of how to make one, and some suggested exercises to help them explore their creation. And then I have to hope that the light dawns.
(For those who may not know what we are talking about. Take a strip of paper long enough to make a loop. When you bring the two ends together, instead of simply fastening them together directly, take one and turn it over so that there is a single half-twist in the loop. Now fasten the ends together with glue or tape. Tada! You have made a mobius strip.
What's so special about it? Try this. Pinch the edge of the strip between your left thumb and first finger. Now put your right first finger beside that spot, and slide it along the edge. Watch where it goes. Did you move your left hand, or did the right finger slip? How did it get to that edge? And then, how did it get back to the right edge?
For a real thrill, take your scissors and cut the strip in half the long way, carefully. What happened? For bonus points, try triple and quintuple twists. Or, with some care, try splitting the mobius strip into three pieces instead of two.:-)
A simple surface with one edge and one side. A normal loop has two edges and two sides (inside and outside). You can't get to the other side without crossing an edge. But the mobius only has one edge and one side! In making that little half-twist, you threw away an edge and a side. This makes you a major topological magician.
Now, for your next trick, take a simple tube (yes, a straw will do). Start to loop it, but instead of connecting the sides simply, give it a little twist through the Euclidean fourth dimension, and slide that outside up against the inside. Voila! A Klien bottle. One surface, and an ant that starts to crawl along on the outside will soon be inside. Perhaps not the best bottle for holding liquids, but great for capturing imaginations and djinn.
If you like that, we'll show you how to make a tessaract next. Count the boxes!
Since we've had mention of it...
(aka Party Tricks for Mathematicians? :-)
Try describing a mobius strip in words. In particular, can you describe what is so different about it so that Joe (or Jane) Random "gets it"?
Personally, I end up with an operational definition of how to make one, and some suggested exercises to help them explore their creation. And then I have to hope that the light dawns.
(For those who may not know what we are talking about. Take a strip of paper long enough to make a loop. When you bring the two ends together, instead of simply fastening them together directly, take one and turn it over so that there is a single half-twist in the loop. Now fasten the ends together with glue or tape. Tada! You have made a mobius strip.
What's so special about it? Try this. Pinch the edge of the strip between your left thumb and first finger. Now put your right first finger beside that spot, and slide it along the edge. Watch where it goes. Did you move your left hand, or did the right finger slip? How did it get to that edge? And then, how did it get back to the right edge?
For a real thrill, take your scissors and cut the strip in half the long way, carefully. What happened? For bonus points, try triple and quintuple twists. Or, with some care, try splitting the mobius strip into three pieces instead of two.:-)
A simple surface with one edge and one side. A normal loop has two edges and two sides (inside and outside). You can't get to the other side without crossing an edge. But the mobius only has one edge and one side! In making that little half-twist, you threw away an edge and a side. This makes you a major topological magician.
Now, for your next trick, take a simple tube (yes, a straw will do). Start to loop it, but instead of connecting the sides simply, give it a little twist through the Euclidean fourth dimension, and slide that outside up against the inside. Voila! A Klien bottle. One surface, and an ant that starts to crawl along on the outside will soon be inside. Perhaps not the best bottle for holding liquids, but great for capturing imaginations and djinn.
If you like that, we'll show you how to make a tessaract next. Count the boxes!
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