Theorem

Smale's Theorem (Sphere Eversion)

Topology

Smale's Theorem on Sphere Eversion is a result of differential topology showing that an ordinary sphere can be turned inside out through a continuous, smooth motion in three-dimensional space without ever cutting, tearing, or creasing its surface, a process called sphere eversion. The result is considered a veridical paradox, since it is true despite seeming, on first encounter, to contradict intuition about surfaces; it follows from Stephen Smale's more general classification of immersions up to regular homotopy.

Facts

Statement

A sphere can be turned inside out through a continuous, smooth motion in three dimensional space, allowing self intersections, without cutting, tearing, or creasing its surface. 1

Proof Year

1958 1

Classification

Statement Form

Existence Theorem 1

Connections

Has Statement Form

Existence Theorem

Entity-backed identity for the statement-form enum value this theorem 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 statement-form fact itself stays on the theorem unchanged.

In Branch

Differential Topology

Source Sphere eversion, Wikipedia

Sources

1. Sphere eversion, Wikipedia

Introduction
It is possible to smoothly and continuously turn a sphere inside out in this way (allowing self-intersections of the sphere's surface) without cutting or tearing it or creating any crease.
History
An existence proof for crease-free sphere eversion was first created by Stephen Smale (1958).
In Branch: Differential Topology, Lead sentence
In differential topology, sphere eversion is a theoretical process of turning a sphere inside out in a three-dimensional space (th

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Smale's Theorem (Sphere Eversion)

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