Geometeer
Boy surface



A ruled Boy surface

in plastic rod.


A differently shaped hole around ∞,

to be seen with red-blue stereo glasses: or for a different depth cue, this view.
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lets you turn and move it.


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The Klein bottle is a half-twisted version of the Pac-Maniverse, with the just the sides of our map flipped.  If every point corresponds to the diametrically opposite point, and -Man crosses the map edge this way
we get the projective plane, so called because its topology is the topology of projective geometry.  (Each direction gives a different 'point at ∞' for parallel lines to meet in: but for this purpose opposite directions coincide, so opposite points at ∞ coincide: just like opposite points on the map edge, if we disdain distances.)  Squeezing the usual and half-twisted versions into 3D is easy, though the Klein bottle has to cross itself, but the difficulty of this one is on the front and back of
a wonderful geometry book:
       
David Hilbert found it so hard that he gave his graduate student Werner Boy the thesis topic of proving it impossible.  Instead, Boy found a way to do it!  Up to mirror images and deformations, Boy's way is the only way.
Bernard Morin taught me to mould a finite Boy surface in my fingers.
I have also carved it, in several materials, but not found a way to make this form math-photogenic (i. e., easy to see the structure, in a picture).
Morin memories:    We met after a 1975 conference in Cargèse (where people were so like Asterix en Corse you'd think they studied it), when I was living in Geneva.  Morin did not attend, but other mathematicians from Strasbourg did, and told him about the carvings I had shown.  (His are superb: if only there was a web site for them all!)  One sleepy going-home time he called me, and told me over the phone how to turn a sphere inside out.  No intricate notation, no visible diagrams, just vivid words painting pictures on my brain.  I bought up the local stock of transparent soap and started carving a sequence of self-intersecting surfaces, in my frequent "point at ∞" style…  Two weeks later I took them to Strasbourg, and learned I had done almost exactly what he described.  (There were a couple of unnecessary moves.)
Sadly, soap does not last a century/3, and I have no adequate photos.

Morin told me his habit of asking people for an algebraic formula for the Boy surface, with all answering "Surely it's in Geometry and the Imagination?"  It's not.  I went and looked, and found an explicit mention of finding a formula, as an unsolved problem.  I told Morin this — he remembered only the absence, and going back to a book is complicated for the blind — and he went "Aah, this makes it worth some effort!"  (Solving a problem in Hilbert and Cohn-Vossen is not exactly solving a Hilbert problem, but solving anything Hilbert looked at is nice.)  He solved it, and the work led to other good things.

Being Morin's blackboard hand while he gave a talk on sphere eversion at the University of Geneva was exciting, but even more was a walk down town, as his seeing eye.  It takes skill and concentration, if you lack practice, to steer somebody in heavy pedestrian traffic; but I was also trying to look for a street name and his target pipe-and-tobacco shop, while not darting off course to search; and I was also hearing how to transform the Boy surface into the Roman surface, that shape into the cross-cap, and back to the Boy.  I needed to stand dead still with my eyes shut, and just listen and imagine.
I did make a first set of models for that, in my fingers with mouldable wax.  They lasted until eaten by ants in Singapore.
To see how a surface meets itself, you need to see through it.  Carved clear soap doesn't last, and glass-blown crossings tend to looky lumpy.  (This spoils a lot of glass Klein bottles.)  You can see through a surface of straight lines — rods or wires — and they can pass cleanly between themselves by careful planning.

All straight lines go through ∞, so this forces a point of the surface to go there, which I like anyway, as in the like the Möbius band with the hole around it,
or around ∞, as we see more of the shape from various directions.  So I looked for a way to do that, and found the pattern shown here, with various shapes for the curved 'hole' edge that supports the straights.  I made a 50cm maquette of one that dreams of being several storeys high (replacing the lines with parabolas, this could become a fountain), but the rest are still virtual. 

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The shape in the plastic rods, interactively displayed.







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