Geometeer
On the edge


Both sides of the 1-half-twist

Möbius band.
Right-click stl and "Save As…" a geometry file, that you can see at any angle or lighting in an STL viewer (there are plenty for free download), or print as a solid object with a 3D printer (sadly not free, but getting cheaper).




The 3-half-twists Möbius band

stl   a central slot stresses the unity of the outer edge.
Virtual, in wax for casting:

without the slot.





stl   This circle-edge Möbius band has circular symmetry in four dimensions, distorted when I project to three.

stl   Same, with a central slot.



stl   The edge on its way to circularity.

The Möbius band is famous for having just one side.

One side, that goes around twice.

It is a little less famous for — unlike a cylinder — having just one edge,
that also goes around twice.  This is why cutting it down the middle
stl  
does not separate it: It is still joined along the original edge.
            A mathematician confided
            The Möbius band is one-sided
                You get quite a laugh
                If you cut it in half
            For it stays in one piece, though divided.
If this Möbius band was transparent, the main edge would look like
    very different from the edge at left,

which is knotted.  (So, if you look carefully, is the edge of the slot.)

Knot theory formalises this by saying that can deform into without self-collisions, while cannot.  Mathematicians have a lot of proofs that things are impossible, and can prove that can never deform into a flat circle, but the interest here is that can.  Moreover, 'deform' has the special meaning of ambient isotopy, which does not just move the 3D curve: It drags the surrounding 3D space smoothly, and invertibly, along with it.  For the edge of the 1-half-twist Möbius band, this deformation drags the band along with it, into a smooth, non-self-intersecting surface… that still has only one side.  The usual surface with a circular edge is a disc, but a disc has two sides, so it cannot be that.    It looks like… what?
       
This olive-wood surface is in two parts, for easier carving 'inside' and to make the circular edge viewable from all directions.  Put together (right), they make a surface with only that edge, like the version at left. 

The intricate part has an elliptical rim, easily plugged with the dome part, which to a topologist is a bent disc: in contrast, any disc fitted to the circular edge finds the wood in the way.  These two edges are deformed views of the edges in this,
where the small edge easily glues to a disc, the big one less so.  Raising the eliptical hole-edge to share a centre with the circle produces the shape at the top of the page, which is thus a Möbius band with a hole around it.  (This shape can also be made as a soap-film surface, but that version is hard to photograph.)

I showed one I had carved in wax to the owner of the wonderful Tiepolo antique gallery, to ask if he knew a skilful place to cast it in silver.  He didn't, but he liked the shape so much I gave it him.  Next time he came back from Beijing, he gave me

this copy, carved in jade.


Back when OpenGL was just GL (Silicon Graphics proprietary 'graphics language'), I made a 3D animation of the Möbius band with a hole in it transforming to a Möbius band with a hole around it, by way of shapes like the one at left.  A student is refactoring that for the web, in HTML5.




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This applet needs Java3D on your browser:
a left-button drag turns, right-button ↑↓↔.





This circle-edged Möbius band has a square-edged hole around it.