A short story about math and physics grad students who, while drinking together at a bar, stumble upon the ability to draw a superposition of different polygons:
(quoted from The Pexagon)
Eric looked both scared and excited. “I was going to make a bad physics joke. You know, something can exist in multiple states until it’s observed—like Schrödinger’s cat. So I just emptied my head and decided to draw a pentagon and a hexagon at the same time. The joke was supposed to be that it would have 5.5 sides until we looked at it. But I don’t understand, once a wave function collapses, it’s supposed to stay collapsed. Not pop back into superposition. Honestly, I thought the whole von Neumann–Wigner interpretation of quantum mechanics was just handwaving.” As we talked, we kept looking at the figure on the napkin that randomly became a pentagon or hexagon whenever we all looked away or blinked.

I suppose the author's idea was to explore what would happen to math if it was subject to the same counterintuitive rules as quantum mechanics. Of course, quantum physics is always described in mathematical terms (operators on Hilbert spaces for QM, connections on manifolds for QFT, etc.). Moreover, the field of Quantum Logic specifically seeks to extend those sorts of ideas to mathematics beyond mathematical physics. In that sense mathematics already does include those ideas.
By my reckoning, this is more about physics than about math. (The drawing on the napkin is a physical object which is in a superposition of states due to the laws of quantum physics. The story does not really investigate whether there is a new mathematical object corresponding to it.) So, in my opinion, this only barely counts as an example of "mathematical fiction", but I am adding it to the database on the recommendation of frequent site contributor Allan Goldberg.
It was "published" as a podcast on 600 Second Saga and is also available in textual form here.
