A Klein bottle is a surface with a very strange property. A surface is any object that is locally 2-dimensional;...
Continue reading...topology
Projective Planes
Most of you know how to make a Mobius band—take a strip of paper and glue the ends with a...
Continue reading...Sphere Eversions
If you take a loop of string in the plane and place an arrow along it pointing clockwise, is it...
Continue reading...Pretzel Unlinking
Imagine that each of the ropes in the two sets of links in Figure 1 are solid (with thickness) and...
Continue reading...Unbelievable Unlinking
Imagine that the two objects in Figure 1 are solid (with thickness) and made of very flexible and stretchy rubber....
Continue reading...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...
Continue reading...Connected Sums
A surface is any object which is locally like a piece of the plane. A sphere, a projective plane, a Klein bottle,...
Continue reading...Borsuk-Ulam Theorem
The Borsuk-Ulam theorem is another amazing theorem from topology. An informal version of the theorem says that at any given...
Continue reading...Brouwer Fixed Point Theorem
One of the most useful theorems in mathematics is an amazing topological result known as the Brouwer Fixed Point Theorem....
Continue reading...Ham Sandwich Theorem
Here is one of my favorite theorems from topology, called the Ham Sandwich Theorem. It says: given globs of ham, bread,...
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