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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