Timeline for An intelligent ant living on a torus or sphere – Does it have a universal way to find out?
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Sep 16, 2020 at 11:47 | answer | added | Jérôme JEAN-CHARLES | timeline score: 1 | |
| Sep 12, 2020 at 18:31 | history | edited | Rodrigo de Azevedo | CC BY-SA 4.0 |
Fixed typo
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| Sep 11, 2020 at 6:57 | review | Close votes | |||
| Sep 16, 2020 at 3:09 | |||||
| Sep 8, 2020 at 11:15 | vote | accept | Claus | ||
| Sep 7, 2020 at 19:58 | answer | added | supercat | timeline score: 2 | |
| Sep 7, 2020 at 15:29 | comment | added | user44143 | @usul and stef, I would assume that any point has a simply connected neighborhood (eg the area immediately underneath the ant) which the ant can verify to be simply connected. If so the ant can triangulate the surface with verifiably simply connected triangles, and then the ant has enough information to decide whether the surface is a sphere or whether K5 can be drawn on it. | |
| Sep 7, 2020 at 15:07 | comment | added | Stef | "Draw the complete graph with five vertices K5. If it can be drawn without any edges crossing, then it must be the torus." Deciding whether "It can be drawn" is harder than "Try to draw it for a bit. Still haven't succeeded? Must be a sphere" especially if you don't know the size of the planet. | |
| Sep 7, 2020 at 14:49 | history | edited | Claus | CC BY-SA 4.0 |
adding references to make post more self-contained
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| Sep 7, 2020 at 10:31 | comment | added | Ilmari Karonen | There's a related question on Math.SE from 2014. | |
| Sep 7, 2020 at 4:46 | comment | added | usul | Interestingly, the answers highlight that a finite ant cannot prove she lives on a sphere, she can only disprove it (by finding a hole). (She can also prove there are no holes of a certain size/scale, given enough time.) I guess this is reflected in the $\Pi_1$ versus $\Sigma_1$ nature of your proposed tests (for example #4). | |
| Sep 6, 2020 at 18:38 | history | became hot network question | |||
| Sep 6, 2020 at 18:36 | comment | added | Asaf Karagila♦ | Develop space travel, obviously. | |
| Sep 6, 2020 at 18:05 | comment | added | Nate Eldredge | To see the difference between 1 and 2, you could step up one dimension and note that $S^3$ is simply connected (i.e. has trivial fundamental group, every loop can be shrunk to a point) but does admit nowhere vanishing tangent vector fields (so its hair can be combed). | |
| Sep 6, 2020 at 16:40 | comment | added | Kevin Walker | As Mike Shulman points out, 2 and 3 are closely related. Other than that relationship, I would say that all of the methods are distinct. | |
| Sep 6, 2020 at 14:36 | answer | added | Mirco A. Mannucci | timeline score: 19 | |
| Sep 6, 2020 at 14:13 | comment | added | Mike Shulman | I would say that 2 is more equivalent to 3, since the hairy ball theorem reduces to the Euler characteristic (being the sum of the indices of a vector field). The results of 4 and 5 sort of involve Euler characteristic too, but are more subtle, so I would not call them equivalent. | |
| Sep 6, 2020 at 12:13 | comment | added | user44143 | Once the ant triangulates the surface, it can apply the standard algorithms for classification of surfaces to identify it uniquely as a sphere, connected sum of $g$ tori or connected sum of $k$ projective planes. Everything else will follow from that. | |
| Sep 6, 2020 at 11:50 | answer | added | Carlo Beenakker | timeline score: 12 | |
| Sep 6, 2020 at 10:37 | history | asked | Claus | CC BY-SA 4.0 |