Geometeer
Hanging on the wall










These versions of the Klein bottle all go through the point at ∞.  A hole around that point makes the shapes clearer, as well as saving wax and casting-metal.


Here, the curve of self-crossing

goes through ∞ also.
A large piece of the surface would look like two planes at right angles, with a disturbance near the origin.




In this form

the point at ∞ is 'opposite' the self-crossing curve; the surface flattens out toward it, and looks like one plane with a disturbance near the origin. 



A more compact version

of the shape above.



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Where does this unsmooth Klein meet itself?  Two Möbius bands are sharing the usual one edge.
  A mathematician named Klein
  Felt the Möbius band is divine:
      As he said, "If you glue
      The edges of two
  You get a nice bottle, like mine."

I suspect the Klein bottle started out as the dignified Kleinsche Fläche (Klein surface), but some English translator read it as Kleinsche Flasche (bottle), and now even the Germans call it that.  In most images it does look like a whisky bottle sucking itself.  I have a nice knitted one like that (a gift from Ti.Zero), but I haven't made one. 

The spun and twisted figure-8 (this one
carved in jeweller's wax) is rarer: Once you see how these are 'different immersions of the same thing', you are half way to being a topologist.  That 'same thing' is a twisted version of Pac-Man's universe.  His usual one,
like ours, is finite but unbounded: unlike ours, it is a torus, not a 3-sphere or some stange folded version of one.  When he reaches the edge of our map of it, he is also at the opposite side (like ±180° being the same point on a circle), and heads back 'in'.  Opposites meet and become each other, like Yin and Yang, or Communist and Catholic.

This version of 'opposite' glues together the top and bottom of our map, forming a tube, then glues the tube ends to give a donut surface.  Pac-Man lives in a flat space, but if we do that gluing in 3D we see it as curved: only in 4D can we give an un-bent, un-cut copy of his universe.  Our spacetime is curved by gravity, but it is not 'bent around' some higher flat dimension any more than his is 'flat around' four dimensions.  Each just has the internal metrical properties that it has.

A different, more diametrical idea of 'opposite' glues side points this way,
giving the Klein bottle, with tube ends glued against the torus matching.  Pushing this, un-cut, into 3D space not only bends it out of shape, but forces it to meet itself.  (Like his usual space, it sits comfortably in 4D.)

The spun-8 and the self-absorbing whiskey forms are very different ways to put the 'abstract' Klein bottle (the 2D twisted Pac-Maniverse) into 3D space: the curves along which they meet themselves go differently around them, and if you try to turn one into the other you get crumpled points.  (All ways to put the sphere into 3D are the same in this sense — any one can turn into any other with crossings but without crumpling — is what 'turning the sphere inside out' is all about.  Mathematicians care more that they are all the same than that two in particular deform into each other.)

The problem with the old joke that a topologist is a mathematician who cannot tell the difference between a donut [
note the US spelling: a doughnut is topologically different] and a cup of coffee is not that she can.  Purely topologically, they are 'the same'.  The problem is that once you think of soft cups and donuts, this particular equivalence is obvious.  An explicit visualisation is heavy-handed overkill. So, do topologists spend their days in trivial blindness? 

The objects here show that deciding when two things are topologically the same, and proving the answer, can be a real challenge.  Are the shapes at left the same, give or take a point at ∞?

Showing that the topological viewpoint is useful is not the task of a sculpture page!  However, the fuller version of the tensegrities section will show off the engineering force of at least some topological ideas.



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ruled and unruly