Boy Surface
The Boy surface is a nonorientable surface that is one possible parametrization of the surface obtained by sewing a Möbius strip to the edge of a disk. Two other topologically equivalent parametrizations are the cross-cap and Roman surface. The Boy surface is a model of the projective plane without singularities and is a sextic surface.
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[(2x^2-y^2-z^2)(x^2+y^2+z^2)+2yz(y^2-z^2)+zx(x^2-z^2)+xy(y^2-x^2)]](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/BoySurface/Inline3.gif) |
(1)
|
 |  | ![1/2sqrt(3)[(y^2-z^2)(x^2+y^2+z^2)+zx(z^2-x^2)+xy(y^2-x^2)]](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/BoySurface/Inline6.gif) |
(2)
|
 |  | ![1/8(x+y+z)[(x+y+z)^3+4(y-x)(z-y)(x-z)].](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/BoySurface/Inline9.gif) |
(3)
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Plugging in
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
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and letting ![u in [0,pi]](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/BoySurface/Inline19.gif)
and ![v in [0,pi]](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/BoySurface/Inline20.gif)
then
gives the Boy surface, three views of which are shown above.
The 
parameterization can also be written as
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
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for ![u in [-pi/2,pi/2]](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/BoySurface/Inline31.gif)
and ![v in [0,pi]](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/BoySurface/Inline32.gif)
.
Three views of the surface obtained using this parameterization are shown above.
R. Bryant devised the beautiful parametrization
 |  | ![-3/2I[(z(1-z^4))/(z^6+sqrt(5)z^3-1)]](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/BoySurface/Inline35.gif) |
(10)
|
 |  | ![-3/2R[(z(1+z^4))/(z^6+sqrt(5)z^3-1)]](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/BoySurface/Inline38.gif) |
(11)
|
 |  | ![I[(1+z^6)/(z^6+sqrt(5)z^3-1)]-1/2,](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/BoySurface/Inline41.gif) |
(12)
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where
 |
(13)
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and 
, giving the Cartesian coordinates of a
point on the surface as
 |  |  |
(14)
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 |  |  |
(15)
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(16)
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In fact, a homotopy (smooth deformation) between the Roman surface and Boy surface is given by the equations
 |  |  |
(17)
|
 |  |  |
(18)
|
 |  |  |
(19)
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as 
varies from 0 to 1, where 
corresponds
to the Roman surface and 
to the Boy
surface shown above.
In 
, the parametric representation is
 |  | ![3[(u^2+v^2+w^2)(u^2+v^2)-sqrt(2)vw(3u^2-v^2)]](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/BoySurface/Inline67.gif) |
(20)
|
 |  |  |
(21)
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 |  |  |
(22)
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 |  |  |
(23)
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and the algebraic equation is
![64(x_0-x_3)^3x_3^3-48(x_0-x_3)^2x_3^2(3x_1^2+3x_2^2+2x_3^2)+12(x_0-x_3)x_3[27(x_1^2+x_2^2)^2-24x_3^2(x_1^2+x_2^2)+36sqrt(2)x_2x_3(x_2^2-3x_1^2)+x_3^4]+(9x_1^2+9x_2^2-2x_3^2)[-81(x_1^2+x_2^2)^2-72x_3^2(x_1^2+x_2^2)+108sqrt(2)x_1x_3(x_1^2-3x_2^2)+4x_3^4]=0](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/BoySurface/NumberedEquation2.gif) |
(24)
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(Apéry 1986). Letting
 |  |  |
(25)
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 |  |  |
(26)
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 |  |  |
(27)
|
 |  |  |
(28)
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gives another version of the surface in 
.
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Apéry, F. Models of the Real Projective Plane: Computer Graphics of Steiner and Boy Surfaces. Braunschweig, Germany: Vieweg, 1987.
Apéry, F. "An Algebraic Halfway Model for the Eversion of the Sphere." Tôhoku Math. J. 44, 103-150, 1992.
Apéry, F.; and Franzoni, G. "The Eversion of the Sphere: a Material Model of the Central Phase." Rendiconti Sem. Fac. Sc. Univ. Cagliari 69, 1-18, 1999.
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Brehm, U. "How to Build Minimal Polyhedral Models of the Boy Surface." Math. Intell. 12, 51-56, 1990.
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Mathematisches Forschungsinstitut Oberwolfach. "The Boy Surface at Oberwolfach." http://www.mfo.de/general/boy/.
Nordstrand, T. "Boy's Surface." http://jalape.no/math/boytxt.
Petit, J.-P. and Souriau, J. "Une représentation analytique de la surface de Boy." C. R. Acad. Sci. Paris Sér. 1 Math 293, 269-272, 1981.
Pinkall, U. "Regular Homotopy Classes of Immersed Surfaces." Topology 24, 421-434, 1985.
Pinkall, U. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 64-65, 1986.
Stewart, I. Game, Set and Math. New York: Viking Penguin, 1991.
Tardy, C. "La fameuse Surface de Boy." http://ctardy.free.fr/jadore/sciences/boy/.
Toth, G. Finite Möbius Groups, Minimal Immersion of Spheres, and Moduli. Berlin: Springer-Verlag, 2002.
Trott, M. The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, pp. 38-39, 2006. http://www.mathematicaguidebooks.org/.
Wang, P. "Renderings."
http://www.ugcs.caltech.edu/~peterw/portfolio/renderings/
Weisstein, Eric W. "Boy Surface." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BoySurface.html
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