Timeline for Is the universe flat?

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Apr 3, 2014 at 5:40	comment	added	user12262		Christoph: First I have to correct: The relations $AB = JK$ and $AJ = AK = BJ = BK$ are of course also satisfied by those four points being "corners of a square" (with "diagonals" $AB$ and $JK$) which is of course plane since precisely $AB = JK = \sqrt{2} \, AJ$. My point was then that on a torus surface (or even on a cylinder) four points can be easily found such that $AB = JK$ and $AJ = AK = BJ = BK$ and $AB < AJ$. "en.wikipedia.org/wiki/Torus#Flat_torus" -- Indeed, one more reason to be extremely wary to use coordinates in cosmology, or in physics overall.
Apr 2, 2014 at 21:17	comment	added	Christoph		@user12262: see en.wikipedia.org/wiki/Torus#Flat_torus
Apr 2, 2014 at 21:01	comment	added	user12262		Christoph: "eg a torus can be equipped with a flat connection" -- On any usual torus surface e.g. there are easily 4 points, $A$, $B$, $J$, $K$ such that $AB = JK$ and $AJ = AK = BJ = BK$. (In the sketch e.g. $A$ and $B$ on the red ring symmetrically above and below the outer black circumference; together with $J$, $K$ on the inner black ring left and right of the red ring.) Those are explicitly not plane to each other: their Cayley-Menger determinant doesn't vanish. (So: the universe is flat?? ...)
Apr 2, 2014 at 19:42	history	answered	Christoph	CC BY-SA 3.0