A sphere eversion is a regular homotopy that turns a sphere inside out in Euclidean
space. It connects the standard embedding
of the 2-sphere to an embedding
with the opposite orientation. Every stage must be an immersion,
so self-intersections are allowed but sharp creases are not. Smale (1959, p. 281)
proved that any two immersions of the 2-sphere
in
are regularly homotopic, establishing the existence of a sphere eversion.
In 1961, Arnold Shapiro devised an explicit eversion but did not publicize it. Phillips (1966) heard of the result and, in trying to reproduce it, actually devised an independent method of his own. Yet another eversion was devised by Morin, which became the basis for the movie by Max (1977). Morin (1978) gave explicit algebraic equations describing the process. The original method of Shapiro was subsequently published by Francis and Morin (1980).
One way to construct a symmetric sphere eversion is to deform the round sphere into a halfway model, whose symmetry reverses the orientation
of the immersed sphere. The second half of the eversion
is obtained by applying that symmetry to the first half
in reverse order (Apéry 1992, pp. 108-109). The double
cover of the real projective plane by
the sphere, followed by the Boy
surfaceimmersion, gives another type of halfway
model (Sullivan 1999, pp. 265-266).
where ,, and . The parameter is expressed in terms of the
latitude on the sphere. The north
pole and south pole both map to the origin
as . The denominator
satisfies , so the formulas are
defined for every finite value of .Rotation
of the image through about the vertical axis corresponds to , which reverses the
orientation of the parameter sphere. The extended map
is , but not , at the poles (Apéry
1992, pp. 121-123). Trott (2006) uses this model as an example of implicitization
by a Gröbner basis.
The Season 1 episode "Sniper Zero" (2005) of the television crime drama NUMB3RS
mentions sphere eversion.
. It connects the standard embedding
of the 2-sphere to an embedding
with the opposite orientation. Every stage must be an immersion,
so self-intersections are allowed but sharp creases are not. Smale (1959, p. 281)
proved that any two 
immersions of the 2-sphere
in 
are regularly homotopic, establishing the existence of a sphere eversion.
![D=1+tau^2-tau[sin(2theta)-cos(2theta)]](/images/equations/SphereEversion/Inline13.svg)
, 
, and 
. The parameter 
is expressed in terms of the
latitude 
on the sphere. The north
pole and south pole both map to the origin
as 
. The denominator
satisfies 
, so the formulas are
defined for every finite value of 
. Rotation
of the image through 
about the vertical axis corresponds to 
, which reverses the
orientation of the parameter sphere. The extended map
is 
, but not 
, at the poles (Apéry
1992, pp. 121-123). Trott (2006) uses this model as an example of implicitization
by a Gröbner basis.
