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Fifty Years of Sphere Eversions
- 1. 50 years of sphere eversions Tony Phillips Stony Brook, 2016 Second edition, augmented
- 2. Stephen Smale - 1959 TAMS 90 281-290
- 14. Stephen Smale - 1959 TAMS 90 281-290
- 15. Arnold Shapiro - 1960 Shapiro’s eversion was only generally known much later. He died in 1962, but had told his construction to Bernard Morin. Morin published it with George Francis in 1980. “Arnold Shapiro’s eversion of the sphere.” Math Intell 2 200-203 drawing by George Francis
- 16. Nicolaas Kuiper -1961 Kuiper in 1961 – Photo by Paul Halmos CMH 35 85-92
- 18. May, 1966
- 19. Marcel Froissart, Bernard Morin 1967 One of the clay models Morin used to communicate his mental pictures.
- 20. Marcel Froissart, Bernard Morin 1967 http://www.lutecium.fr/mirror/www.jp-petit.com/science/maths_f/Retournement_sphere/PLS_79.pdf. These pictures were published much later (1979) by Morin and Jean-Pierre Petit.
- 21. “Turning a Sphere Inside-Out” - 1976 Nelson Max
- 22. “Turning a Sphere Inside-Out” - 1976 Nelson Max Stephen Smale
- 23. “Turning a Sphere Inside-Out” - 1976 Nelson Max Charles Pugh
- 24. “Turning a Sphere Inside-Out” - 1976 Nelson Max Nelson Max
- 25. Interactive: John Hughes -1984
- 26. George Francis: A Topological Picturebook - 1987 Francis observed that Morin’s eversion is one of a family of “tobacco pouch” eversions. The central stage can have 2, 3, 4, 5, …-fold symmetry. 3: Boy’s Surface 4: Morin’s Surface from George Francis’ illustrations of Morin’s eversion.
- 27. Thurston’s “Outside-In” - 1994
- 28. The Optiverse - 1996
- 29. Iain Aitchison - 2010 http://arxiv.org/abs/1008.0916The `Holiverse': holistic eversion of the 2-sphere in R3
- 30. Arnaud Chéritat- 2014 http://arxiv.org/abs/1410.4417“Yet another sphere eversion”
- 76. What happens to the rest of the surface?
- 80. Green hemisphere
- 94. What happens to the hemispheres during the eversion?
- 95. We trace the attaching surface of one of the hemispheres. Upper green hemisphere attached here.
- 103. One of the four hemisphere-attaching surfaces, traced through the eversion.
Editor's Notes
- Slide the saddles apart
- 0 Going to rotate green saddle 180 degrees. Move strips out of the way.
- 1a
- 1 Now ready to turn green saddle
- 2 45 degrees
- 3 90 degrees
- 4 135 degrees
- y-eversion-5.fla (end of saddle rotations) Now slide strips to a more symmetric configuration.
- y-eversion-5a.fla (end of saddle rotations)
- 6
- 7
- 8new
- 8 Now ready to move the saddles together.
- 9 A little more adjusting.
- 10
- 11 Everywhere there are two yellow sides facing each other. This is a double covering of the central part of Kuiper’s immersion of the projective plane.
- 12 This is the only place where there’s any change in the z-direction.
- 13=12R Now we run the process backwards with the switched layers.
- 14=11R
- 10R=15
- 9R=16
- 8newR
- 8R=17
- 7aR
- 7R
- 6R
- 5aR
- 5R
- 4R
- 3R
- 2R
- 1R
- 1aR
- 0R
- 0aR Now you may wonder, what happened to the hemispheres?
- Watch the circle where the upper green hemisphere is attached.
- Watch the circle where the upper green hemisphere is attached.
- Y-eversion-0aaX.fla
- Y-eversion-0aX.fla
- y-eversion-0.fla
- y-eversion-5X.fla
- y-eversion-11X
- Y-eversion-11RX
- y-eversion-5R, y-eversion-5RX
- Y-eversion-0RX
- The green hemisphere has migrated over to the right. y-eversion-0aRX
- 2cRXa
- 5Rbb