Perhaps the strangest, wildest, and most fun example of how crazy Topology can get is Alexander's Horned Sphere, which shows how something not even remotely resembling a sphere, can, in fact, be treated exactly like one.

From Wiki:

In mathematics, a

**pathological**phenomenon is one whose properties are considered atypically bad or counterintuitive; the opposite is well-behaved. Often, when the usefulness of a theorem is challenged by counterexamples, defenders of the theorem argue that the exceptions are pathological. A famous case is the Alexander horned sphere, a counterexample showing that topologically embedding the sphere S

^{2}in

**R**

^{3}may fail to "separate the space cleanly", unless an extra condition of

*tameness*is used to suppress possible

*wild*behaviour.

The

**Alexander horned sphere**is one of the most famous pathological examples in mathematics, discovered in 1924 by J. W. Alexander. It is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting with a standard torus:

- Remove a radial slice of the torus.
- Connect a standard punctured torus to each side of the cut, interlinked with the torus on the other side.
- Repeat steps 1–3 on the two tori just added.

This is my favorite video of this oddity, more follow:

http://www.ultrafractal.com/showcase/jos/alexanders-horn.html

THE HORNED SPHERE FAN CLUB - Just kidding, that's The Butcher Boys, Jane Alexander (South Africa), 1985-86 |

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