331:. A little more explanation: if you have a collection of cuboids in Euclidean space, meeting face-to-face, you can form a collection of 2-manifolds by drawing three equatorial quadrilaterals in each cuboid and connecting pairs of quadrilaterals from cuboids that share faces. Sometimes (in the meshing literature) this collection of surfaces is called the "spatial twist continuum". This paper finds a collection of cuboids from which this construction forms a Boy's surface. It can also be interpreted as a similar construction on the surface of a four-polytope with cuboid faces. —
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533:', hoping that it would help those who (like myself) weren't aware of who 'Hilbert' referred to. I am not, however, certain that it is indeed in reference to David Hilbert, but his biographical information (lifetime, profession, nationality, etc.) strongly suggests that this is true. Please correct my edit if I am mistaken. Thank you.
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pi - phi. This last part is pretty easy, you can chose a subset of the spherical harmonics that satisfy it. This is how John Hughes found his parametrization of Boy's surface -- he projected a 7-vertex polygon model on the spherical harmonics to solve for the three functions, also taking use of the
1028:
The first external link of the article contains various parameterizations and references to the articles where they are defined. It explains also (in French) why Boy's surface differs from other realizations of projective plane (all singular points are simple crossing points — no cusps). I guess that
918:
By the way, beside a lacking correct parametrization, other fundamental properties are lacking: I believe that Boy's surface is algebraic. Therefore, an implicit equation must be provided, or, at least, its degree must be given (from the picture, I guess 6). Also, it must been said if Boy's surface
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It looks like the parameterization used for the statue is from Kusner, which is (as far as I can tell) different from the parameterization currently given in the article which is attributed to Bryant. That parameterization with the credit to Bryant also appears in Apery's book. If there are no
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Upon even more in depth digging it turns out that this class of parameterizations is due to a collaboration between Bryant and Kusner, and they are commonly refered to as "Bryant-Kusner parameterizations". So a compromise it is, and unless no one objects I'll make the change.
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attributes the parametrization formula to Rob Kusner instead of Robert Bryant. Instead it credits Bryant with a general result about rational immersions of some kind of minimal surfaces, of which this parametrization is a special case apparently. regards,
1190:- From the set of three ovals, each enclosing the next, that gives you three roughly J shaped sloping and slanting arches, with their feet forming a regular hexagon, each arch going over the next and linking opposite vertices of the hexagon.
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this afternoon. If you think that looks right feel free to insert it as you see fit (I assume you have done most of the organizing of this article.) Let me know if you would like a smaller verision too, it does seem rather big now. —
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I'd like to propose moving the section on the
Structure of Boy's Surface in front of the section on parameterization. Knowledge is addressed to a general audiance, who are more likely to relate to the pictures than the equations.
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to represent the complex variable of the definition. Doing so would avoid confusing the non-mathematicians in our audience (and also promote clarity and rigour among the mathematicians!) I suggest using the letter
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which are themselves irrational functions of the real and imaginary parts of a complex number, named by the same letter as the third coordinate of the points of the surface. The problem is that, the expression of
1193:- Looking at the arches two at a time lets you see that each strip bridging opposite edges of the hexagon can be set up to fit with the others, though the creasing still needs a straightforward rounding off.
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IMHO the easiest way to derive a parametrization is to start with the unit circle, and write x(theta, phi), y(theta, phi), z(theta, phi) making sure that each coordinate function is periodic in phi -: -->
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I think the images could use more transparency and smoothness. Drawing them opaque and polygonized like this makes it harder to see what's happening. There's a sketchy idea of what I mean on my page
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Here's a model of Boy's that one can move around, for those of us who like to play with things. If anyone else thinks that it's worth it feel free to link it, but I'll refrain since I made it. —
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but maybe a raytrace would look better. I'd also like to see a more prominent treatment of the locus of self-intersection points (a single triple point connecting three loops of double points). —
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In fact, the argument presented in the "Relating the Boy's surface..." section is bogus: what's said there does not imply that the parametrization is injective in the interior of the disk.
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I wish I had the skill to derive a parametrization based on trig functions and the stitching illustrated here. Has no one done so? Would it be a double or a quadruple cover? —
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admits a rational parametrization. I guess that it is true, because, otherwise, Boy's surface would not be isomorphic (that is birationally equivalent) to the projective plane.
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for the complex variable, as is often done. This would obviously require a thorough edit of the article, at least the
Parametrization section, to replace all instances of
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The given parametrization of Boy's surface is either inconsistent or some essential information is lacking: the coordinates of a point are expressed rationally in terms of
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The page says that the Boy surface is homeomorphic to RP^2, but that is just impossible, as there are a compact non orientable surface cannot be embedded in R^3...
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Given the three-fold symmetry of the usual embeddings, wouldn't it make more sense to start with a hexagon with opposite edges glued, instead of a square? —
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Is this merely a visual aid that at least helped me, or is there a deeper association? And is it worth it/acceptable to put any of this in the article?
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This is explained in section "Property of Bryant-Kusner parametrization": this is a two fold parameterization, and the condition that the magnitude of
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Pretty! And a lot more understandable than what's there now. I'll work on getting it in and making the other images a little less space-consuming. —
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1181:- First, set up or imagine the rings as symmetrical ovals, e.g. ellipses, each placed symmetrically in a plane orthogonal to the others.
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I have added an external link to a page that answers to my preceding questions. It remains to edit the article to include these answers.
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the answer to your question is in this link and its references. By the way, it would be useful to expand the article, using this link.
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That wouldn't be you in the references would it ;-)? I think I get the idea. After a bit of smoothing it does make for a nice model
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I like it! It's much more helpful IMHO to have something one can rotate freely instead of seeing it animated. So I'll add it. --
1184:- Then look at them along the body diagonal of a cube symmetrically containing them, to see the threefold rotational symmetry.
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I find the "Construction" section pretty obscure. Some diagrams seem pretty necessary. Does anyone know where that's from? --
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objections, we can change the mention in the "statue" section. Thanks, High on a tree! Sorry this is rather late.
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are conventionally used as the three spatial coordinates in
Cartesian 3-space, it would be wise to choose some letter
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U. Brehm, How to build minimal polyhedral models of the Boy surface. Math. Intelligencer 12(4):51-56 (1990).
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Something should be said about left and right-handed Boy's surfaces, which are not regularly homotopic. --
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is not greater than 1 is here only for having a bijective parameterization (outside the unit sphere).
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Doing some google digging finds which talks about a discreet Boy's surface, with a reference
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Thus this complicate parametrization is simply a strange parametrization of the unit sphere.
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It should be noted that the article by
Hermann Karcher and Ulrich Pinkall which can be found
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That's irrelevant. The problem is with the word homeomorphic, which is evidently wrong.
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1187:- Then bisect them at a plane orthogonal to that diagonal and discard one part.
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348:. The tight immersions stuff seems quite interesting space for a new article? --
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discussed the imposibility of tight imersions of the real projective plane.
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Eliminating confusion caused by a poor choice of variable names
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There being no objection, I went ahead and made the change. --
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seems to have some interesting stuff, but its in french? --
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http://torus.math.uiuc.edu/jms/Papers/isama/color/opt2.htm
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As long as you're finding things in Google, there's also
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Seems like Boys surface is used as a halfway stage in a
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http://www.jp-petit.com/science/maths_f/maths_f.htm
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752:{\displaystyle x^{2}+y^{2}+z^{2}=1.}
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491:It's not embedded, it's immersed. —
174:by Knowledge editors, which is now
38:It is of interest to the following
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607:{\displaystyle g_{1},g_{2},g_{3},}
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1247:Mid-priority mathematics articles
1127:I agree. Moreover, it is written
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115:Knowledge:WikiProject Mathematics
1135:) in the article. I'll do that.
190:Old comments on parameterization
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402:02:38, 20 September 2007 (UTC)
353:22:06, 14 September 2006 (UTC)
336:20:16, 14 September 2006 (UTC)
329:Another discrete Boy's surface
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275:22:45, 13 September 2006 (UTC)
221:21:56, 13 September 2006 (UTC)
1:
1170:Connection to Borromean rings
1049:of the real projective plane
109:and see a list of open tasks.
1252:Old requests for peer review
1242:C-Class mathematics articles
1228:20:20, 3 November 2023 (UTC)
1145:14:47, 29 October 2015 (UTC)
1122:13:22, 29 October 2015 (UTC)
520:02:47, 3 December 2008 (UTC)
377:21:05, 6 December 2006 (UTC)
289:other interesting links are
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208:18:01, 21 October 2005 (UTC)
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988:two-period parametrization
445:Left and right handed boys
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700:implies immediately that
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293:A New Polyhedral Surface
199:15:47, 6 Jun 2005 (UTC)
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469:—Preceding
462:False claim
172:peer review
170:received a
112:Mathematics
103:mathematics
59:Mathematics
1236:Categories
1220:Coolwanglu
1198:PMLawrence
350:Salix alba
311:Salix alba
267:Salix alba
1211:New Video
1051:was found
1137:D.Lazard
1031:D.Lazard
975:D.Lazard
958:D.Lazard
921:D.Lazard
483:contribs
471:unsigned
177:archived
1001:Tamfang
940:Tamfang
905:Tamfang
525:Hilbert
512:Mariano
139:on the
30:C-class
1055:a13ean
545:A13ean
454:(Talk)
436:(Talk)
421:A13ean
399:A13ean
385:A13ean
232:A13ean
36:scale.
1093:than
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1224:talk
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903:. —
451:C S
433:C S
205:agr
197:agr
131:Mid
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1095:z
1087:z
1083:y
1079:x
1057:(
1033:(
1018:(
1003:(
977:(
960:(
954:z
942:(
923:(
907:(
887:2
882:3
878:g
874:+
869:2
864:2
860:g
856:+
851:2
846:1
842:g
817:2
812:3
808:g
804:+
799:2
794:2
790:g
786:+
781:2
776:1
772:g
744:=
739:2
735:z
731:+
726:2
722:y
718:+
713:2
709:x
686:3
682:g
678:,
673:2
669:g
665:,
660:1
656:g
635:z
632:,
629:y
626:,
623:x
602:,
597:3
593:g
589:,
584:2
580:g
576:,
571:1
567:g
514:(
477:(
423:(
313:(
269:(
238:)
234:(
143:.
42::
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