page_title: Sphere eversion
hierarchical_section_title: Sphere eversion
caption_attribution_description: English: Central part of sphere eversion (turning inside out) process flattened with folds along a square, cut out of doubly colored paper (red/green). The depicted part is homeomorphic to a cylinder/collar/bottleneck, to be glued to two spheres opposite oriented. The middle stage is the Morin surface, with both sides equally privileged. The left stage can be easily unfolded to a green (outer side) sphere, the right - red. The middle stage has removed four flat triangles to see the folds underneath, the triangles can be shifted above of beneath the surface to proceed to the left/right. Note the quadruple intersection in the center, double intersections along diagonals and fold annihilation at the vertices of the inner square (midpoints of sides of the outer square).English: Central part of sphere eversion (turning inside out) process flattened with folds along a square, cut out of doubly colored paper (red/green). The depicted part is homeomorphic to a cylinder/collar/bottleneck, to be glued to two spheres opposite oriented. The middle stage is the Morin surface, with both sides equally privileged. The left stage can be easily unfolded to a green (outer side) sphere, the right - red. The middle stage has removed four flat triangles to see the folds underneath, the triangles can be shifted above of beneath the surface to proceed to the left/right. Note the quadruple intersection in the center, double intersections along diagonals and fold annihilation at the vertices of the inner square (midpoints of sides of the outer square).
context_page_description: In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space. Remarkably, it is possible to smoothly and continuously turn a sphere inside out in this way 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
  be the standard embedding; then there is a regular homotopy of immersions
  such that ƒ₀ = ƒ and ƒ₁ = −ƒ.
context_section_description: In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space. (The word eversion means "turning inside out".) Remarkably, it is possible to smoothly and continuously turn a sphere inside out in this way (with possible self-intersections) 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
  be the standard embedding; then there is a regular homotopy of immersions
  such that ƒ₀ = ƒ and ƒ₁ = −ƒ.
caption: paper sphere eversion and Morin surface
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