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Boy Surface

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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.

Boy surface sculpture at Oberwolfach

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,

f_1(x,y,z)=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)]
(1)
f_2(x,y,z)=1/2sqrt(3)[(y^2-z^2)(x^2+y^2+z^2)+zx(z^2-x^2)+xy(y^2-x^2)]
(2)
f_3(x,y,z)=1/8(x+y+z)[(x+y+z)^3+4(y-x)(z-y)(x-z)].
(3)
BoySurface

Plugging in

x=cosusinv
(4)
y=sinusinv
(5)
z=cosv
(6)

and letting u in [0,pi] and v in [0,pi] then gives the Boy surface, three views of which are shown above.

The R^3 parameterization can also be written as

x=(sqrt(2)cos^2vcos(2u)+cosusin(2v))/(2-sqrt(2)sin(3u)sin(2v))
(7)
y=(sqrt(2)cos^2vsin(2u)-sinusin(2v))/(2-sqrt(2)sin(3u)sin(2v))
(8)
z=(3cos^2v)/(2-sqrt(2)sin(3u)sin(2v))
(9)

for u in [-pi/2,pi/2] and v in [0,pi].

BoySurface2

Three views of the surface obtained using this parameterization are shown above.

BoySurfaceBryant

R. Bryant devised the beautiful parametrization

g_1=-3/2I[(z(1-z^4))/(z^6+sqrt(5)z^3-1)]
(10)
g_2=-3/2R[(z(1+z^4))/(z^6+sqrt(5)z^3-1)]
(11)
g_3=I[(1+z^6)/(z^6+sqrt(5)z^3-1)]-1/2,
(12)

where

g=g_1^2+g_2^2+g_3^2
(13)

and |z|<=1, giving the Cartesian coordinates of a point on the surface as

X=(g_1)/g
(14)
Y=(g_2)/g
(15)
Z=(g_3)/g.
(16)
RomanBoy

In fact, a homotopy (smooth deformation) between the Roman surface and Boy surface is given by the equations

x(u,v)=(sqrt(2)cos(2u)cos^2v+cosusin(2v))/(2-alphasqrt(2)sin(3u)sin(2v))
(17)
y(u,v)=(sqrt(2)sin(2u)cos^2v-sinusin(2v))/(2-alphasqrt(2)sin(3u)sin(2v))
(18)
z(u,v)=(3cos^2v)/(2-alphasqrt(2)sin(3u)sin(2v))
(19)

as alpha varies from 0 to 1, where alpha=0 corresponds to the Roman surface and alpha=1 to the Boy surface shown above.

In R^4, the parametric representation is

x_0=3[(u^2+v^2+w^2)(u^2+v^2)-sqrt(2)vw(3u^2-v^2)]
(20)
x_1=sqrt(2)(u^2+v^2)(u^2-v^2+sqrt(2)uw)
(21)
x_2=sqrt(2)(u^2+v^2)(2uv-sqrt(2)vw)
(22)
x_3=3(u^2+v^2)^2,
(23)

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
(24)

(Apéry 1986). Letting

x_0=1
(25)
x_1=x
(26)
x_2=y
(27)
x_3=z
(28)

gives another version of the surface in R^3.

See also

Cross-Cap, Immersion, Möbius Strip, Nonorientable Surface, Real Projective Plane, Roman Surface, Sextic Surface

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References

Apéry, F. "The Boy Surface." Adv. Math. 61, 185-266, 1986.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.Boy, W. "Über die Curvatura integra und die Topologie geschlossener Flächen." Math. Ann 57, 151-184, 1903.Brehm, U. "How to Build Minimal Polyhedral Models of the Boy Surface." Math. Intell. 12, 51-56, 1990.Carter, J. S. "On Generalizing Boy Surface--Constructing a Generator of the 3rd Stable Stem." Trans. Amer. Math. Soc. 298, 103-122, 1986.Fischer, G. (Ed.). Plates 115-120 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, pp. 110-115, 1986.Geometry Center. "Boy's Surface." http://www.geom.umn.edu/zoo/toptype/pplane/boy/.Hilbert, D. and Cohn-Vossen, S. §46-47 in Geometry and the Imagination. New York: Chelsea, 1999.Karcher, H. and Pinkall, U. "Die Boysche Fläche in Oberwolfach." Mitteilungen der DMV, issue 1, 45-47, 1997.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/.Update a linkWang, P. "Renderings." http://www.ugcs.caltech.edu/~peterw/portfolio/renderings/

Cite this as:

Weisstein, Eric W. "Boy Surface." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BoySurface.html

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