Data Portal @ linkeddatafragments.org

Matches in DBpedia 2014 for { <http://dbpedia.org/resource/Alexander_horned_sphere> ?p ?o. }

• Alexander_horned_sphere abstract "The Alexander horned sphere is a wild embedding of a sphere into space, discovered by J. W. Alexander (1924). 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 1–2 on the two tori just added ad infinitum.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. 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 the Jordan–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 the topological category of manifolds, and the categories of differentiable manifolds, and piecewise linear manifolds was noticed.Now consider Alexander's horned sphere as an embedding into the 3-sphere, considered as the one-point compactification of the 3-dimensional Euclidean space R3. The closure of the non-simply connected domain is called the solid Alexander horned sphere. Although the solid horned sphere is not a manifold, R. H. 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 a crumpled 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 example, also found by Alexander, is Antoine's horned sphere, which is based on Antoine's necklace, a pathological embedding of the Cantor set into the 3-sphere.".
• Alexander_horned_sphere thumbnail Alexander_horned_sphere.png?width=300.
• Alexander_horned_sphere wikiPageExternalLink prod.php?which=30253.
• Alexander_horned_sphere wikiPageExternalLink math655.
• Alexander_horned_sphere wikiPageExternalLink alexanders-horn.html.
• Alexander_horned_sphere wikiPageExternalLink watch?v=d1Vjsm9pQlc.
• Alexander_horned_sphere wikiPageID "685665".
• Alexander_horned_sphere wikiPageRevisionID "571752668".
• Alexander_horned_sphere authorlink "James Waddell Alexander II".
• Alexander_horned_sphere first "J. W.".
• Alexander_horned_sphere hasPhotoCollection Alexander_horned_sphere.
• Alexander_horned_sphere last "Alexander".
• Alexander_horned_sphere title "Alexander's Horned Sphere".
• Alexander_horned_sphere urlname "AlexandersHornedSphere".
• Alexander_horned_sphere year "1924".
• Alexander_horned_sphere subject Category:Fractals.
• Alexander_horned_sphere subject Category:Geometric_topology.
• Alexander_horned_sphere subject Category:Topology.
• Alexander_horned_sphere type Abstraction100002137.
• Alexander_horned_sphere type Cognition100023271.
• Alexander_horned_sphere type Form105930736.
• Alexander_horned_sphere type Fractal105931152.
• Alexander_horned_sphere type Fractals.
• Alexander_horned_sphere type PsychologicalFeature100023100.
• Alexander_horned_sphere type Structure105726345.
• Alexander_horned_sphere comment "The Alexander horned sphere is a wild embedding of a sphere into space, discovered by J. W. Alexander (1924).".
• Alexander_horned_sphere label "Alexander horned sphere".
• Alexander_horned_sphere label "Esfera cornuda de Alexander".
• Alexander_horned_sphere label "Rogata sfera Alexandera".
• Alexander_horned_sphere label "Sfera di Alexander".
• Alexander_horned_sphere label "Sphère cornue d'Alexander".
• Alexander_horned_sphere label "Дикая сфера".
• Alexander_horned_sphere label "كرة ألكسندر القرنية".
• Alexander_horned_sphere sameAs Esfera_cornuda_de_Alexander.
• Alexander_horned_sphere sameAs Sphère_cornue_d'Alexander.
• Alexander_horned_sphere sameAs Sfera_di_Alexander.
• Alexander_horned_sphere sameAs 알렉산더의_뿔_달린_구.
• Alexander_horned_sphere sameAs Rogata_sfera_Alexandera.
• Alexander_horned_sphere sameAs m.032wdw.
• Alexander_horned_sphere sameAs Q1361755.
• Alexander_horned_sphere sameAs Q1361755.
• Alexander_horned_sphere sameAs Alexander_horned_sphere.
• Alexander_horned_sphere wasDerivedFrom Alexander_horned_sphere?oldid=571752668.
• Alexander_horned_sphere depiction Alexander_horned_sphere.png.
• Alexander_horned_sphere isPrimaryTopicOf Alexander_horned_sphere.