As shown by cosmologists space is presumably a flat expanse. In his book the Fabric of the Cosmos, Brian Greene shows that one of the possible shapes of this flat universe is a Torus. My question is, are there other shapes that would fulfill the same criteria, if so, how?
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Anything homeomorphic to a torus would fulfill the criteria – Jim Apr 22 '15 at 15:47
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Can you give an example? – Jaywalker Apr 22 '15 at 15:53
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A coffee cup. Or at least the shape of one – Jim Apr 22 '15 at 16:00
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Does a coffee cup shape qualify as flat space? – Jaywalker Apr 22 '15 at 16:18
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@Jaywalker It can - it is continuously deformable into a torus. You could take a flat torus and continuously deform it into a coffee cup shape, without losing flatness - a flat coffee cup I guess. – Kyle Oman Apr 22 '15 at 16:28
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There is quite a nice description of this area in the Wikipedia article on the shape of the universe. As well as the torus possible shapes include the Poincaré dodecahedral space and the Picard horn.
Googling will find you lots of stuff about the duodecahedral shape, and despite its potential for causing sniggers there is quite a bit on Picard's horn as well.
John Rennie
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1The OP mentioned "flat" topologies, though. As far as I know, those two other examples, the Poincaré dodecahedral space and the Picard horn are actually not flat, but positively and negatively curved, respectively. A Klein bottle and its higher-dimensional cousins are flat, I believe. – John Oct 05 '18 at 17:45