Sunday, February 20, 2011

Truly Pathological: Alexander's Horned Sphere

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 S2 in R3 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:
  1. Remove a radial slice of the torus.
  2. Connect a standard punctured torus to each side of the cut, interlinked with the torus on the other side.
  3. Repeat steps 1–3 on the two tori just added.
By considering only the points of the tori that are not removed at some stage, an embedding results of the sphere with a Cantor set removed. This embedding extends to the whole sphere, since points approaching two different points of the Cantor set will be at least a fixed distance apart in the construction.

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

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

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