Editing Boy's surface

{{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]]).

==Parametrization==
[[File:BoysSurfaceKusnerBryant.svg|thumb|A view of the Kusner–Bryant parametrization of the Boy's surface]]
Boy's surface can be parametrized in several ways. One parametrization, discovered by Rob Kusner and [[Robert Bryant (mathematician)|Robert Bryant]],<ref name="Wells1988">{{cite book|contribution=Surfaces in conformal geometry (Robert Bryant)|author=Raymond O'Neil Wells|title=The Mathematical Heritage of Hermann Weyl (May 12–16, 1987, Duke University, Durham, North Carolina)|volume=48|url=https://books.google.com/books?id=e0MECAAAQBAJ&pg=PA227|year=1988|series=Proc. Sympos. Pure Math.|publisher=American Mathematical Soc.|isbn=978-0-8218-1482-6|pages=227–240|doi=10.1090/pspum/048/974338}}</ref> is the following: given a complex number ''w'' whose [[magnitude (mathematics)|magnitude]] is less than or equal to one (<math> \| w \| \le 1</math>), let
:<math>\begin{align}
  g_1 &= -{3 \over 2} \operatorname{Im} \left[ {w \left(1 - w^4\right) \over w^6 + \sqrt{5} w^3 - 1} \right]\\[4pt]
  g_2 &= -{3 \over 2} \operatorname{Re} \left[ {w \left(1 + w^4\right) \over w^6 + \sqrt{5} w^3 - 1} \right]\\[4pt]
  g_3 &= \operatorname{Im} \left[ {1 + w^6 \over w^6 + \sqrt{5} w^3 - 1}  \right] - {1 \over 2}\\
\end{align}</math>

and then set 
:<math>\begin{pmatrix}x\\ y\\ z\end{pmatrix} = \frac{1}{g_1^2 + g_2^2 + g_3^2} \begin{pmatrix}g_1\\ g_2\\ g_3\end{pmatrix}</math>

we then obtain the [[Cartesian coordinates]] ''x'', ''y'', and ''z''   of a point on the Boy's surface.

If one performs an inversion of this parametrization centered on the triple point, one obtains a complete [[minimal surface]] with three [[End (topology)|ends]] (that's how this parametrization was discovered naturally).  This implies that the Bryant–Kusner parametrization of Boy's surfaces is "optimal" in the sense that it is the "least bent" immersion of a [[projective plane]] into [[three-space]].

===Property of Bryant–Kusner parametrization===
{{Wikibooks|Famous Theorems of Mathematics|Boy's surface}}
If ''w'' is replaced by the negative reciprocal of its [[complex conjugate]], <math display="inline">-{1 \over w^\star},</math> then the functions ''g''<sub>1</sub>, ''g''<sub>2</sub>, and ''g''<sub>3</sub> of ''w'' are left unchanged.

By replacing {{math|''w''}} in terms of its real and imaginary parts {{math|1=''w'' = ''s'' + ''it''}}, and expanding resulting parameterization, one may obtain a parameterization of Boy's surface in terms of [[rational function]]s of {{math|''s''}} and {{math|''t''}}. This shows that Boy's surface is not only an [[algebraic surface]], but even a [[rational surface]]. The remark of the preceding paragraph shows that the [[generic fiber]] of this parameterization consists of two points (that is that almost every point of Boy's surface may be obtained by two parameters values).

===Relation to the real projective plane===
Let <math>P(w) = (x(w), y(w), z(w))</math> be the Bryant–Kusner parametrization of Boy's surface. Then
:<math> P(w) = P\left(-{1 \over w^\star} \right). </math>

This explains the condition <math>\left\| w \right\| \le 1</math> on the parameter: if <math>\left\| w \right\| < 1,</math> then <math display="inline"> \left\| - {1 \over w^\star} \right\| > 1 .</math> However, things are slightly more complicated for <math> \left\| w \right\| = 1.</math>  In this case, one has <math display="inline">-{1 \over w^\star} = -w .</math> This means that, if <math> \left \| w \right\| = 1, </math> the point of the Boy's surface is obtained from two parameter values: <math>P(w) = P(-w).</math> In other words, the Boy's surface has been parametrized by a disk such that pairs of diametrically opposite points on the [[perimeter]] of the disk are equivalent. This shows that the Boy's surface is the image of the [[real projective plane]], RP<sup>2</sup> by a [[smooth function|smooth map]]. That is, the parametrization of the Boy's surface is an [[immersion (mathematics)|immersion]] of the real projective plane into the [[Euclidean space]].

==Symmetries==
[[File:Surface_de_Boy.stl|thumb|[[STL (file format)|STL 3D model]] of Boy's surface]]
Boy's surface has 3-fold [[symmetry]].  This means that it has an axis of discrete rotational symmetry: any 120° turn about this axis will leave the surface looking exactly the same.  The Boy's surface can be cut into three mutually [[congruence (geometry)|congruent]] pieces.

==Applications==
Boy's surface can be used in [[sphere eversion]] as a [[half-way model]].  A half-way model is an immersion of the sphere with the property that a rotation interchanges inside and outside, and so can be employed to evert (turn inside-out) a sphere.  Boy's (the case p&nbsp;=&nbsp;3) and [[Morin surface|Morin's]] (the case p&nbsp;=&nbsp;2) surfaces begin a sequence of half-way models with higher symmetry first proposed by George Francis, indexed by the even integers 2p (for p odd, these immersions can be factored through a projective plane).  Kusner's parametrization yields all these.

==Models==
[[Image:Boyflaeche.JPG|thumb|Model of a Boy's surface in [[Oberwolfach Research Institute for Mathematics]]]]

=== Model at Oberwolfach ===
The [[Oberwolfach Research Institute for Mathematics]] has a large model of a Boy's surface outside the entrance, constructed and donated by [[Mercedes-Benz]] in January 1991. This model has 3-fold [[rotational symmetry]] and minimizes the [[Willmore energy]] of the surface. It consists of steel strips representing the image of a [[Polar coordinate system|polar coordinate grid]] under a parameterization given by Robert Bryant and Rob Kusner. The meridians (rays) become ordinary [[Möbius strip]]s, i.e. twisted by 180 degrees. All but one of the strips corresponding to circles of latitude (radial circles around the origin) are untwisted, while the one corresponding to the boundary of the unit circle is a Möbius strip twisted by three times 180 degrees &mdash; as is the emblem of the institute {{harv|Mathematisches Forschungsinstitut Oberwolfach|2011}}.

=== Model made for Clifford Stoll ===
A model was made in glass by glassblower Lucas Clarke, with the cooperation of [[Adam Savage]], for presentation to [[Clifford Stoll]]. It was featured on Adam Savage's [[YouTube]] channel, [[Tested.com|Tested]]. All three appeared in the video discussing it.<ref>{{cite web |last1=Adam |first1=Savage |title=This Object Should've Been Impossible to Make |url=https://www.youtube.com/watch?v=rMPrlvlUIMc |website=YouTube |date=21 June 2023 |access-date=22 June 2023}}</ref>

== References ==
=== Citations ===
{{Reflist}}

=== Sources ===
{{refbegin}}
* {{citation |last=Kirby|first=Rob|author-link=Robion Kirby|date=November 2007|title=What is Boy's surface?|url = https://www.ams.org/notices/200710/tx071001306p.pdf |journal=Notices of the AMS |volume =54  |issue = 10 |pages=1306–1307 }} This describes a piecewise linear model of Boy's surface.
** {{citation |last=Casselman|first=Bill|date=November 2007|title=Collapsing Boy's Umbrellas |url = https://www.ams.org/notices/200710/200710-about-the-cover.pdf |journal=Notices of the AMS |volume = 54 |issue = 10 |page = 1356 }} Article on the cover illustration that accompanies the Rob Kirby article.
* {{citation |author = Mathematisches Forschungsinstitut Oberwolfach |title=The Boy surface at Oberwolfach |url = https://www.mfo.de/about-the-institute/history/boy-surface/karcher_pinkallboy_surface.pdf |year=2011 }}.
* Sanderson, B. [http://www.maths.warwick.ac.uk/~bjs/proj.pdf ''Boy's will be Boy's''], (undated, 2006 or earlier).
* {{MathWorld|urlname=BoySurface|title=Boy's Surface}}
{{refend}}

== External links ==
{{Commons}}
*[https://mathcurve.com/surfaces.gb/boy/boy.shtml Boy's surface] at MathCurve; contains various visualizations, various equations, useful links and references
*[http://plus.maths.org/issue27/features/mathart/BoyUnfold.html A planar unfolding of the Boy's surface] – applet from ''Plus Magazine''.
* [http://www.maths.ed.ac.uk/~aar/surgery/notes.htm Boy's surface resources], including the [http://www.maths.ed.ac.uk/~aar/surgery/boy.pdf original article], and an [http://www.maths.ed.ac.uk/~aar/surgery/boyinboy.jpg embedding] of a topologist in the [https://web.archive.org/web/20160712171217/http://www.mfo.de/about-the-institute/history/Boy-Surface/the-boy-surface-at-oberwolfach Oberwolfach Boy's surface].
* [http://www.andrewlipson.com/boys.htm A LEGO Boy's surface]
* [https://web.archive.org/web/20160827051210/http://teach.hilderbuild.com/tiki-index.php?page=Paper%20Boy%27s A paper model of Boy's surface] – pattern and instructions
* [http://www.math.univ-toulouse.fr/~cheritat/boy-surface/index.html A model of Boy's surface] in [[Constructive Solid Geometry]] together with assembling instructions
* [https://www.youtube.com/watch?v=9gRx66xKXek ''Boy's surface''] visualization video from the Mathematical Institute of the Serbian Academy of the Arts and Sciences
* [https://www.youtube.com/watch?v=rMPrlvlUIMc ''This Object Should've Been Impossible to Make''] Adam Savage making a museum stand for a glass model of the surface
{{Compact topological surfaces}}

{{DEFAULTSORT:Boy's Surface}}
[[Category:Surfaces]]
[[Category:Geometric topology]]
[[Category:Eponyms in geometry]]