The Alexander horned sphere is a pathological embedding of an ordinary two-dimensional sphere into three-dimensional Euclidean space, discovered by J. W. Alexander in 1924. Together with its interior it forms a topological ball that is simply connected, meaning every loop inside it can be shrunk to a point, but unlike the exterior of an ordinary round sphere, its exterior is not simply connected. This distinction shows that the Schoenflies theorem, which holds in two dimensions for any simple closed curve, fails to extend to three dimensions: no homeomorphism of three-dimensional space can straighten the horned sphere into a standard sphere.
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Is Kind Of Object
Entity-backed identity for the object-kind enum value this mathematical object already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The object-kind fact itself stays on the object unchanged.
Sources
1. Alexander's Horned Sphere, from Wolfram MathWorld
Definition
Alexander's horned sphere is the above topological structure, composed of a countable union of compact sets.
Discovery illustration caption
The horned sphere as originally drawn by Alexander (1924) is illustrated above.
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