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

Smale's Theorem (Sphere Eversion)

1 entry

Branch

Nationality / Culture

Gender

Stephen Smale

Smale's Theorem (Sphere Eversion)

Connections

Has Statement Form

Existence Theorem

In Branch

Differential Topology

Source Sphere eversion, Wikipedia

Principal Result Of

Stephen Smale

Source Wikipedia: Stephen Smale

Proved By

Stephen Smale

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
Proved By: Stephen Smale, Lead
An existence proof for crease-free sphere eversion was first created by Stephen Smale (1958).
In Group: Geometric and Differential Topology, Lead paragraph
In differential topology, sphere eversion is a theoretical process of turning a sphere inside out in a three-dimensional space (the word eversion means "turning inside out").

View the Source

Wikipedia: Stephen Smale

Principal Result Of: Stephen Smale, Article text

Quote, Principal Result Of: Stephen Smale, Article text

Smale's theorem has been reproved and extended a few times, notably to higher dimensions in the form of the Smale conjecture, as well as to other topological types.

Smale's Theorem (Sphere Eversion)

Stephen Smale

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