Is there any space in which circles can be tiled without gaps?
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
966616
add a comment |
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
966616
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
966616
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
966616
asked Nov 19 at 3:57
zooby
966616
asked Nov 19 at 3:57
zooby
966616
asked Nov 19 at 3:57
zooby
966616
asked Nov 19 at 3:57
zooby
966616
966616
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 |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3004504%2fis-there-any-space-in-which-circles-can-be-tiled-without-gaps%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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