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[[File:Boy Surface-animation-small.gif|thumb|An animation of Boy's surface]]
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In [[geometry]], '''Boy's surface''' is an [[immersion (mathematics)|immersion]] of the [[real projective plane]] in 3-dimensional space found by [[Werner Boy]] in 1901 (he discovered it on assignment from [[David Hilbert]] to prove that the projective plane ''could not'' be immersed in [[Three-dimensional space|3-space]]). Unlike the [[Roman surface]] and the [[cross-cap]], it has no singularities (i.e., [[Pinch point (mathematics)|pinch-points]]), but it does self-intersect.
 
Boy's surface is discussed (and illustrated) in [[Jean-Pierre Petit]]'s ''Le Topologicon''.<ref>[http://www.savoir-sans-frontieres.com/JPP/telechargeables/English/topo_the_world_eng.htm Jean-Pierre Petit, ''Le Topologicon'']</ref>
 
Boy's surface was first parametrized explicitly by [[Bernard Morin]] in 1978. See below for another parametrization, discovered by Rob Kusner and Robert Bryant. 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|coauthors=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=09262245|doi=10.1016/j.difgeo.2009.01.011|url=http://www.sciencedirect.com/science/article/pii/S0926224509000242/pdfft?md5=6f38c1568e1c7976069612c1131ab879&pid=1-s2.0-S0926224509000242-main.pdf|accessdate=Dec 27, 2013}}</ref>
 
==Construction==
 
To make a Boy's surface:
#Start with a sphere. Remove a cap.
#Attach one end of each of three strips to alternate sixths of the edge left by removing the cap.
#Bend each strip and attach the other end of each strip to the sixth opposite the first end, so that the inside of the sphere at one end is connected to the outside at the other. Make the strips skirt the middle rather than go through it.
#Join the loose edges of the strips. The joins intersect the strips.
 
==Symmetry of the 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.
 
==Model at Oberwolfach==
[[Image:Boyflaeche.JPG|thumb|Model of a Boy's surface in Oberwolfach]]
The [[Mathematical Research Institute of Oberwolfach]] 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 which represent 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|2008}}.
 
==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's (the case p&nbsp;=&nbsp;2) surfaces are beginning of 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 factor through a projective plane).  Kusner's parametrization yields all these.
 
==Parametrization of Boy's surface==
[[Image:BoysSurfaceTopView.PNG|thumb|A view of the parametrization described here]]
Boy's surface can be parametrized in several ways. One parametrization,<ref>{{citation|last=Bryant|first=Robert|contribution=Surfaces in conformal geometry|title=The mathematical heritage of Hermann Weyl (May 12–16, 1987, Duke University, Durham, North Carolina)|series=Proc. Sympos. Pure Math.|year=1988|pages=227–240|volume=48|publisher=Amer. Math. Soc.|publication-place=Providence, RI}}</ref> discovered by Rob Kusner and [[Robert Bryant (mathematician)|Robert Bryant]], is the following: given a complex number ''z'' whose [[magnitude (mathematics)|magnitude]] is less than or equal to one (<math>\scriptstyle \| z \| \;\le\; 1</math>), let
:<math>\begin{align}
  g_1 &= -{3 \over 2} \mathrm{Im} \left[ {z (1 - z^4) \over z^6 + \sqrt{5} z^3 - 1} \right]\\
  g_2 &= -{3 \over 2} \mathrm{Re} \left[ {z (1 + z^4) \over z^6 + \sqrt{5} z^3 - 1} \right]\\
  g_3 &= \mathrm{Im} \left[ {1 + z^6 \over z^6 + \sqrt{5} z^3 - 1}  \right] - {1 \over 2}\\
\end{align}</math>
 
so that
:<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>
 
where ''x'', ''y'', and ''z'' are the desired [[Cartesian coordinates]] 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===
{{further information|[[b:Famous Theorems of Mathematics/Boy's_surface|Properties of the Bryant-Kusner parametrization]]}}
If ''z'' is replaced by the negative reciprocal of its [[complex conjugate]], <math>\scriptstyle -{1 \over z^\star},</math> then the functions ''g<sub>1</sub>'', ''g<sub>2</sub>'', and ''g<sub>3</sub>'' of ''z'' are left unchanged.
 
===Relating the Boy's surface to the real projective plane===
Let <math>\scriptstyle P(z) \;=\; (x(z),\, y(z),\, z(z))</math>, where <math>\scriptstyle \| z \| \;\le\; 1,</math> denote a [[point (geometry)|point]] on Boy's surface. Then
:<math> P(z) = P\left( -{1 \over z^\star} \right) </math>
 
but only if <math>\scriptstyle \| z \| \;=\; \sqrt{z z^\star} \;=\; 1.</math>  What if <math>\scriptstyle \| z \| \;<\; 1 ?</math>  Then
:<math> \left\| - {1 \over z^\star} \right\| > 1 </math>
 
because
:<math> - {1 \over z^\star} = {- z \over z^\star z} = {-z \over \| z \|^2} </math>
 
whose magnitude is
:<math> {\| z \| \over \| z \|^2} = {1 \over \| z \|},</math>
 
but <math>\scriptstyle \| z \| \;<\; 1,</math> so that
:<math> {1 \over \| z \|} > 1. </math>
 
Since ''P(z)'' belongs to the Boy's surface only when <math>\scriptstyle \|z\| \;\le\; 1,</math>, this means that <math>\scriptstyle P\left( -{1 \over z^\star} \right) </math> belongs to Boy's surface only if <math>\scriptstyle \| z \| \;=\; 1.</math>  Thus <math>\scriptstyle P(z) \;=\; P(-z) </math> if <math>\scriptstyle \| z \| \;=\; 1,</math> but all other points on the Boy's surface are unique. The set of complex values <math>\scriptstyle \|z\| \;\le\; 1</math> is the [[unit disk]]. Thus 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. Therefore the Boy's surface is [[homeomorphism|homeomorphic]] to the [[real projective plane]], ''RP<sup>2</sup>''.
 
==References==
{{commons}}
{{Reflist}}
*{{citation|last=Kirby|first=Rob|authorlink=Robion Kirby|date=November 2007|title=What is Boy's surface?|url=http://www.ams.org/notices/200710/tx071001306p.pdf|journal=Notices of the AMS |volume =54  |issue =10|pages=1306&ndash;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=http://www.ams.org/notices/200710/200710-about-the-cover.pdf|journal=Notices of the AMS |volume =54  |issue =10|pages=1356 }} Article on the cover illustration that accompanies the Rob Kirby article.
* {{citation|last=Kusner|first=Rob|title=Conformal geometry and complete minimal surfaces|url=http://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 (New series)|volume=17|year=1987|issue=2|pages=291&ndash;295|doi=10.1090/S0273-0979-1987-15564-9}}.
* {{citation|last=Mathematisches Forschungsinstitut Oberwolfach|title=The Boy surface at Oberwolfach|url=http://www.mfo.de/about-the-institute/history/Boy-Surface/the-boy-surface-at-oberwolfach/|year=2011}}.
* {{citation|last=Morin|first=Bernard|authorlink=Bernard Morin|title=Equations du retournement de la sphère|journal=C. R. Acad. Sci. Paris|year=1978|volume=287|issue=13|pages=A879–A882}}
* 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}}
 
==External links==
*[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 [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]
* [http://teach.hilderbuild.com/tiki-index.php?page=Paper%20Boy%27s A paper model of Boy's surface] - pattern and instructions
* [http://surfaces.gotfork.net/boy_jv.html Java-based model that can be freely rotated]
* [http://vis.cs.brown.edu/docs/pdf/Demiralp-2009-CLF.pdf Line field coloring using Boy's surface]
* [http://www.youtube.com/watch?v=9gRx66xKXek ''Boy's surface''] visualization video from the Mathematical Institute of the Serbian Academy of the Arts and Sciences
* [http://www.youtube.com/watch?v=dekzCBOlNAc Make your Boy surface] A movie and an article on a paper-craft of making the Boy surface by using a pair of scissors, a piece of paper, and a strip of scotch tape. [http://arxiv.org/abs/1303.6448 The plan] The plan of the paper-craft in the movie. (The website of it is written below the movie and is linked with the movie.)
{{DEFAULTSORT:Boy's Surface}}
[[Category:Surfaces]]
[[Category:Geometric topology]]

Latest revision as of 15:03, 12 January 2015

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