- Alexander horned sphere
The Alexander horned sphere is one of the most famous pathological examples in
mathematics discovered in 1924 byJ. W. Alexander . It is the particular embedding of asphere in 3-dimensionalEuclidean 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 1-2 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 aCantor 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.The horned sphere, together with its inside, is a topological 3-ball, the Alexander horned ball, and so is
simply-connected , i.e., every loop can be shrunk to a point while staying inside. The exterior is "not" simply connected, unlike the exterior of the usual round sphere; a loop linking a torus in the above construction cannot be shrunk to a point without touching the horned sphere. This shows that theJordan-Schönflies theorem does not hold in three dimensions, as Alexander had originally thought. Alexander also proved that the theorem does hold in three dimensions for piecewise linear/smooth embeddings. This is one of the earliest examples where the need for distinction between thetopological category ofmanifold s, and the categories ofdifferentiable manifold s, andpiecewise linear manifold s was noticed.Now consider Alexander's horned sphere as an
embedding into the3-sphere , considered as theone-point compactification of the 3-dimensionalEuclidean space R3. The closure of the non-simply connected domain is called the solid Alexander horned sphere. Although the solid horned sphere is not amanifold ,RH Bing showed that its double (which is the 3-manifold obtained by gluing two copies of the horned sphere together along the corresponding points of their boundaries) is in fact the 3-sphere. One can consider other gluings of the solid horned sphere to a copy of itself, arising from different homeomorphisms of the boundary sphere to itself. This has also been shown to be the 3-sphere. The solid Alexander horned sphere is an example of acrumpled cube , i.e., a closed complementary domain of the embedding of a 2-sphere into the 3-sphere.One can generalize Alexander's construction to generate other horned spheres by increasing the number of horns at each stage of Alexander's construction or considering the analogous construction in higher dimensions.
Other substantially different constructions exist for constructing such "wild" spheres. Another famous example, also due to Alexander, is
Antoine's horned sphere , which is based onAntoine's necklace , a pathological embedding of theCantor set into the 3-sphere.ee also
*
Fox-Artin arc External links
* J. W. Alexander. An Example of a Simply Connected Surface Bounding a Region which is not Simply Connected. [http://www.pnas.org/ Proceedings of the National Academy of Sciences] 1924; 10(1): 8-10.
*mathworld|urlname=AlexandersHornedSphere|title=Alexander's Horned Sphere
*Zbigniew Fiedorowicz. Math 655 - Introduction to Topology. [http://www.math.ohio-state.edu/~fiedorow/math655/] - Lecture notes
* [http://www.youtube.com/watch?v=d1Vjsm9pQlc Construction of the Alexander sphere]
* [http://www.ultrafractal.com/showcase/jos/alexanders-horn.html rotating animation]
* [http://pouet.net/prod.php?which=30253 PC OpenGL demo rendering and expanding the cusp]
Wikimedia Foundation. 2010.