Is there any space in which circles can be tiled without gaps?
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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.
noneuclidean-geometry
asked Nov 19 at 3:57
zooby
956616
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1
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favorite
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.
noneuclidean-geometry
asked Nov 19 at 3:57
zooby
956616
2
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.
– dbx
Nov 19 at 4:02
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 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 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 at 22:50
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
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.
noneuclidean-geometry
asked Nov 19 at 3:57
zooby
956616
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.
noneuclidean-geometry
noneuclidean-geometry
asked Nov 19 at 3:57
zooby
956616
asked Nov 19 at 3:57
zooby
956616
asked Nov 19 at 3:57
zooby
956616
asked Nov 19 at 3:57
zooby
956616
asked Nov 19 at 3:57
zooby
956616
956616
2
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.
– dbx
Nov 19 at 4:02
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 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 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 at 22:50
add a comment |
2
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.
– dbx
Nov 19 at 4:02
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 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 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 at 22:50
2
2
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.
– dbx
Nov 19 at 4:02
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.
– dbx
Nov 19 at 4:02
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 at 17:34
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 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 at 17:40
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 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 at 22:50
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 at 22:50
add a comment |
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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.
– dbx
Nov 19 at 4:02
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 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 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 at 22:50