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Octagons can't be tiled in flat space but they can in hyperbolic space. Likewise pentagons can be tiled on a sphere.

Imagine you had some flat circles then you glued them by their edges to create a honey cone structure. You'd have to bend the circles a bit.

Is there some kind of hypothetical 2D surface on which circles can be tiled without gaps?

(Apart from the obvious 2 circles making halves of a sphere.)

It sounds like a crazy idea. Maybe it is.

zooby
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    Would you count $\Bbb{R}^2$ with the sup norm? If you define a circle to be there set of points at distance one from some center point, then circles in that space are what we usually call squares. – Sort of Damocles Nov 19 '18 at 04:02
  • Another obvious example is the elliptic plane made from 1 circle. As dbx says, you have to precisely say what kinds of spaces are allowed (closed Riemannian manifolds, I suppose)? – Zeno Rogue Nov 19 '18 at 17:34
  • I believe you cannot have three (or more) circles meeting in a single point in a Riemannian manifold (because then at least one would have an angle < 180 degrees there, and a circle cannot have such a sharp corner), so that would restrict the (connected) examples to the two we already have. – Zeno Rogue Nov 19 '18 at 17:40
  • Of course we could also make say, an ellipsoid out of two circles, or another simliar surface of revolution -- I meant two kinds of tilings. – Zeno Rogue Nov 19 '18 at 22:50

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