Editing Boy's surface (section)

{{short description|Self-intersecting compact surface, an immersion of the real projective plane}}
[[File:Boy Surface-animation-small.gif|thumb|An animation of Boy's surface]]

In [[geometry]], '''Boy's surface''' is an [[immersion (mathematics)|immersion]] of the [[real projective plane]] in [[three-dimensional space]]. It was discovered in 1901 by the German mathematician [[Werner Boy]], who had been tasked by his doctoral thesis advisor [[David Hilbert]] to prove that the projective plane ''could not'' be immersed in three-dimensional space. 

Boy's surface was first [[parametrized]] explicitly by [[Bernard Morin]] in 1978.<ref name="Morin 1978">{{cite journal
 |last1=Morin |first1=Bernard
 |date=13 November 1978
 |title=Équations du retournement de la sphère
 |trans-title=Equations of the eversion of the sphere
 |language=fr
 |journal=Comptes Rendus de l'Académie des Sciences
 |series=Série A
 |volume=287
 |pages=879–882
 |url=http://ayuba.fr/pdf/cras_geometry_1978c.pdf
 }}</ref> Another parametrization was discovered by Rob Kusner and [[Robert Bryant (mathematician)|Robert Bryant]].<ref name="Kusner 1987">{{cite journal |last1=Kusner|first1=Rob|title=Conformal geometry and complete minimal surfaces|url=https://www.ams.org/bull/1987-17-02/S0273-0979-1987-15564-9/S0273-0979-1987-15564-9.pdf|journal=Bulletin of the American Mathematical Society |series=New Series|volume=17|year=1987|issue=2|pages=291–295|doi=10.1090/S0273-0979-1987-15564-9|doi-access=free}}.</ref> Boy's surface is one of the two possible immersions of the real projective plane which have only a single triple point.<ref name="Goodman2009">{{cite journal|last=Goodman|first=Sue|author2=Marek Kossowski|year=2009|title=Immersions of the projective plane with one triple point|journal=Differential Geometry and Its Applications|volume=27|issue=4|pages=527–542|issn=0926-2245|doi=10.1016/j.difgeo.2009.01.011|url=https://cdr.lib.unc.edu/downloads/2r36v672h|doi-access=free}}</ref>

Unlike the [[Roman surface]] and the [[cross-cap]], it has no other [[singular point of an algebraic variety|singularities]] than [[self-intersection]]s (that is, it has no [[Pinch point (mathematics)|pinch-points]]).