To further advance your topological explorations, get a Klein Bottle cut right down the middle. You can then trace the built-in Möbius loop with your fingertip or crayon. In the photo, you can see the edge of the glass forms a nice Möbius loop.
In this all-too-asymetrical universe, Acme's topological technicians strive to preserve parity. By slicing an Acme Klein Bottle, they generate one left handed Möbius loop and a matching right-handed one. Unlike matter/antimatter couplets, these can be placed next to each other without fear of total annihilation. Indeed, they fit so well that you can glue 'em together to recreate the original.
These are great classroom demos - students immediately see (and feel) how two Möbius loops combine to make a Klein Bottle.
Put one of these on an overhead projector, and shine it at the chalkboard. A student can then use chalk to trace the edge -- showing that the edge does come back on itself (as a Möbius loop should!).
We leave the edge semi-rough, like emery paper or the surface of Ramanujan's chalkboard.
Thanks to an advanced diamond band-saw, Acme's Slicer-meister can now cleave baby, medium, or big classical Klein bottles. (See how I do it - scroll to the bottom of this page)
Each half of a Baby Klein Bottle is about 110mm (4.5") tall; the matched set of two sides will set you back $100.
A sliced Medium Klein Bottle is around 150mm
(6") tall; both mirror-image halves cost $150.
The sliced Big Klein Bottle is a smidgen over
200mm (8") tall; the two matched halves cost $200.
Prices slightly higher north of the North Pole.
That's a Diamond Band Saw
Closeup of a baby Klein Bottle being cut. The bottle advances about 2mm per minute, and a water stream keeps the blade cool.
A medium Klein Bottle sliced
A kludge? You bet! You can't do physics without pleny of C-Clamps, gaffer's tape, and paperclips.
Not shown is the honking 6 Amp regulated power supply and the Basic Stamp controller. But you can see the rotation sensor - a white reed-switch a few mm above the chuck of the drill. This senses each rotation so the computer will drive the drill at a constant (very slow) speed.