Sphere eversion
≈ 3 min read
Names in other languages
No names in other languages are held for this entry yet.
Article bodies are English and are not machine-translated. A name in another language is not a translation of the article.
seen from "above" 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"). 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. This is surprising, both to non-mathematicians and to those who understand regular homotopy, and can be regarded as a veridical paradox; that is something that, while being true, on first glance seems false. More precisely, let : f\colon S^2\to \R^3 be the standard embedding; then there is a regular homotopy of immersions : f_t\colon S^2\to \R^3 such that f 0 = f and f 1 = −f.
seen from "above" 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"). 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. This is surprising, both to non-mathematicians and to those who understand regular homotopy, and can be regarded as a veridical paradox; that is something that, while being true, on first glance seems false. More precisely, let f\colon S^2\to \R^3 be the standard embedding; then there is a regular homotopy of immersions f_t\colon S^2\to \R^3 such that f 0 = f and f 1 = −f. Mathematics visualization: topology. The holiverse sphere eversion (Povray animation) The deNeve/Hills sphere eversion: video and interactive model Patrick Massot's project to formalise the proof in the Lean Theorem Prover An interactive exploration of Adam Bednorz and Witold Bednorz method of sphere eversion Outside In: A video exploration of sphere eversion, created by The Geometry Center of The University of Minnesota.
Text derived from https://en.wikipedia.org/w/index.php?oldid=1377636061 under CC BY-SA 4.0; modifications have been made.