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1 [made Community Wiki]

Something about the "Quaternion Demonstrator" (the belt trick demonstrating that $\mathbf{SO}\left(3\right)$ has a double cover). An exhibit could centre on three distinct, accessible and interesting to the general public (like myself) mathematical topics:

1) As a model for spin 1/2 particles - I recall being enthralled as a young teen by the idea that some objects might not come back to their same state after a $2 \pi$ rotation. At that age, on seeing the belt trick, I recall one of my first reactions was "Interesting, but might not we build something with a fancier arrangement of ribbons and strings that would need, say $6 \pi$ rotations to bring it back to the "same state"?" I think it would be interesting to say in the exhibit that there is sound mathematics behind the assertion that, no, there is no such fancier arrangement, so that, unless the topology of our Universe is radically different from what we can imagine, there is very strong evidence, grounded on mathematics alone, that half-integer spin is the only possibility - and we don't need billion dollar particle accelerators to know this.

2) The quaternions and the idea of number systems beyond "everyday" rational and real numbers. That only restricted systems can be built if one wants to preserve "real world" properties like continuity of the "multiplication" - that mathematics isn't just postulating arbitrary axiom systems and playing games with them. History of complex numbers could be included, maybe even a feel for Hatcher's Algebraic Topology proof thereof (something like the YouTube clip http://www.youtube.com/watch?v=nRO_4IYOdq8).

3) Thinking about the belt trick itself (the physical thingie, rather than the mathemetics of the $\mathbf{SO}\left(3\right)$ double cover) for me stridently raises the question of what the distinction between mathematics and physics really is, or even whether there is one. The belt trick is compelling to even small children - I showed it to my five year old recently and was astonished to find that she seemed to understand many of the ideas of symmetry involved and played around with different numbers of twists and untangling them for quite some time. Of course, most serious mathematicians will say that the belt trick is not a proof, but when you begin to look at it hard, and see that the ribbon is directly encoding a "history" of rotations of a Frenet-Serret frame, you realise that the contraption is a pretty spot on analogue of the mathematical construction of a universal cover - so much so that you begin to wonder whether the mathematical construction isn't part of the subconscious visual processing and understanding of the physical contraption in almost anyone - mathematician or layperson.