topology

Klein Bottle

A Klein bottle is a surface with a very strange property. A surface is any object that is locally 2-dimensional; every part looks like a piece of the plane. A sphere and a torus are surfaces, and they have 2 sides: you can place a...

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Projective Planes

Most of you know how to make a Mobius band—take a strip of paper and glue the ends with a half-twist. This object now has the property that is has only one “side”. It also has only one edge. Well, a disc has only one...

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Sphere Eversions

If you take a loop of string in the plane and place an arrow along it pointing clockwise, is it possible to deform the string, keeping it in the plane, so that the arrow points counterclockwise, without causing any kinks in the string? A moment’s...

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Pretzel Unlinking

Imagine that each of the ropes in the two sets of links in Figure 1 are solid (with thickness) and made of very flexible and stretchy rubber. Question: is it possible to deform one set of links into the other in a continuous motion (without...

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Unbelievable Unlinking

Imagine that the two objects in Figure 1 are solid (with thickness) and made of very flexible and stretchy rubber. Question: is it possible to deform one object into the other in a continuous motion (without tearing or cutting)? Surprise answer: Yes!! Hint: it is important that...

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Rubber Bands Stuck on a Torus

Consider a rubber tire (a torus) with a hole in it. Suppose that there is a green rubber band stuck to the outside of the torus that goes through the central cylinder, and a red rubber band pasted to the inside that stretches around the...

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Connected Sums

A surface is any object which is locally like a piece of the plane. A sphere, a projective plane, a Klein bottle, a torus, a 2-holed torus are all examples of surfaces. We do not distinguish between a sphere and a deformed sphere… we say they are...

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Borsuk-Ulam Theorem

The Borsuk-Ulam theorem is another amazing theorem from topology. An informal version of the theorem says that at any given moment on the earth’s surface, there exist 2 antipodal points (on exactly opposite sides of the earth) with the same temperature and barometric pressure! More...

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Brouwer Fixed Point Theorem

One of the most useful theorems in mathematics is an amazing topological result known as the Brouwer Fixed Point Theorem. Take two sheets of paper, one lying directly above the other. If you crumple the top sheet, and place it on top of the other...

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Ham Sandwich Theorem

Here is one of my favorite theorems from topology, called the Ham Sandwich Theorem. It says: given globs of ham, bread, and cheese (in any shape), placed any way you like, there exists one flat slice of a knife (a plane) that will bisect each of...

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