A sculpture of the Boy surface as a special immersion of the real projective plane in Euclidean 3-space was installed in front of the library of
the Mathematisches Forschungsinstitut Oberwolfach library building on January 28,
1991 (Mathematisches Forschungsinstitut Oberwolfach; Karcher and Pinkall 1997).
The Boy surface can be described using the general method for nonorientable surfaces, but this was not known until the analytic equations were found by Apéry
(1986). Based on the fact that it had been proven impossible to describe the surface
using quadratic polynomials, Hopf had conjectured that quartic polynomials were also
insufficient (Pinkall 1986). Apéry's immersion
proved this conjecture wrong, giving the equations explicitly in terms of the standard
form for a nonorientable surface,
(1)
(2)
(3)
Plugging in
(4)
(5)
(6)
and letting
and then gives the Boy surface,
three views of which are shown above.
The parameterization can also be written
as
(7)
(8)
(9)
for and .
Three views of the surface obtained using this parameterization are shown above.
R. Bryant devised the beautiful parametrization
(10)
(11)
(12)
where
(13)
and , giving the Cartesian coordinates
of a point on the surface as
(14)
(15)
(16)
In fact, a homotopy (smooth deformation) between the
Roman surface and Boy surface is given by the equations
(17)
(18)
(19)
as varies from 0 to 1, where corresponds to the Roman
surface and
to the Boy surface shown above.