Fifty Years of Sphere Eversions

105
50 years of sphere eversions Tony Phillips Stony Brook, 2016 Second edition, augmented

Transcript of Fifty Years of Sphere Eversions

Page 1: Fifty Years of Sphere Eversions

50 years of sphere eversions

Tony Phillips

Stony Brook, 2016

Second edition, augmented

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Stephen Smale - 1959

TAMS 90 281-290

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Stephen Smale - 1959

TAMS 90 281-290

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Arnold Shapiro - 1960

Shapiro’s eversion was only generally known much later. Hedied 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

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Nicolaas Kuiper -1961

Kuiper in 1961 – Photo by Paul Halmos

CMH 35 85-92

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May, 1966

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Marcel Froissart, Bernard Morin 1967

One of the clay models Morin used to communicate his mental pictures.

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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.

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“Turning a Sphere Inside-Out” - 1976

Nelson Max

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“Turning a Sphere Inside-Out” - 1976

Nelson Max

Stephen Smale

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“Turning a Sphere Inside-Out” - 1976

Nelson Max

Charles Pugh

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“Turning a Sphere Inside-Out” - 1976

Nelson Max

Nelson Max

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Interactive: John Hughes -1984

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George Francis: A Topological Picturebook - 1987

Francis observed that Morin’seversion 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.

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Thurston’s “Outside-In” - 1994

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The Optiverse - 1996

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Iain Aitchison - 2010

http://arxiv.org/abs/1008.0916The `Holiverse': holistic eversion of the 2-sphere in R3

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Arnaud Chéritat- 2014

http://arxiv.org/abs/1410.4417“Yet another sphere eversion”

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What happens to the rest of the surface?

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Greenhemisphere

Greenhemisphere

Yellowhemisphere

Yellowhemisphere

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Greenhemisphere

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Greenhemisphereattached here

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Greenhemisphereattached here

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Greenhemisphereattached here

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Greenhemisphereattached here

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Greenhemisphereattached here

Greenhemisphere

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Yellowhemisphere

Yellowhemisphere

Greenhemisphere

Greenhemisphere

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What happens to the hemispheres during the eversion?

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We trace the attaching surface ofone of the hemispheres.

Upper green hemisphereattached here.

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One of the fourhemisphere-attachingsurfaces, tracedthrough the eversion.

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Celestial sphere

Celestial sphere

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