What surface is created by a continuous rotation in n-dimension?
Given a random rotation in n-dimensions, what surface is created?
In 2 dimensions this is obviously the circle. The same is true in 3 dimensions. (Any continuous rotation in 3 dimensions of a random point around the origin will produce a circle or a degenerate point if the rotation axis is through the point).
But in 4-dimensions once could have rotation which is a combination of a rotation in the $xy$ axis combined with a rotation in the $zw$ axis, and if these rotations are not rational ratios of each other, a point a 4D space will never return to its starting point. So this can't be a circle. (What shape is it? A torus? Will it go through all points in space?)
Similarly what shapes are produced in n-dimensions by a continuous n-dimensional rotation of a point around the origin? Is there a general theory behind this?
algebraic-topology differential-topology
asked 2 days ago
zoobyzooby
983616
add a comment |
Given a random rotation in n-dimensions, what surface is created?
In 2 dimensions this is obviously the circle. The same is true in 3 dimensions. (Any continuous rotation in 3 dimensions of a random point around the origin will produce a circle or a degenerate point if the rotation axis is through the point).
But in 4-dimensions once could have rotation which is a combination of a rotation in the $xy$ axis combined with a rotation in the $zw$ axis, and if these rotations are not rational ratios of each other, a point a 4D space will never return to its starting point. So this can't be a circle. (What shape is it? A torus? Will it go through all points in space?)
Similarly what shapes are produced in n-dimensions by a continuous n-dimensional rotation of a point around the origin? Is there a general theory behind this?
algebraic-topology differential-topology
asked 2 days ago
zoobyzooby
983616
What object are you rotating? A point?
– Randall
2 days ago
@Matt Hmm, OK I'll use that definition of orthogonal transformation. I mean it just depends on your definition of what a "rotation" is in 4 dimensions.
– zooby
2 days ago
@Randall. Yes an arbitrary point (not the origin).
– zooby
2 days ago
add a comment |
Given a random rotation in n-dimensions, what surface is created?
In 2 dimensions this is obviously the circle. The same is true in 3 dimensions. (Any continuous rotation in 3 dimensions of a random point around the origin will produce a circle or a degenerate point if the rotation axis is through the point).
But in 4-dimensions once could have rotation which is a combination of a rotation in the $xy$ axis combined with a rotation in the $zw$ axis, and if these rotations are not rational ratios of each other, a point a 4D space will never return to its starting point. So this can't be a circle. (What shape is it? A torus? Will it go through all points in space?)
Similarly what shapes are produced in n-dimensions by a continuous n-dimensional rotation of a point around the origin? Is there a general theory behind this?
algebraic-topology differential-topology
asked 2 days ago
zoobyzooby
983616
Given a random rotation in n-dimensions, what surface is created?
In 2 dimensions this is obviously the circle. The same is true in 3 dimensions. (Any continuous rotation in 3 dimensions of a random point around the origin will produce a circle or a degenerate point if the rotation axis is through the point).
But in 4-dimensions once could have rotation which is a combination of a rotation in the $xy$ axis combined with a rotation in the $zw$ axis, and if these rotations are not rational ratios of each other, a point a 4D space will never return to its starting point. So this can't be a circle. (What shape is it? A torus? Will it go through all points in space?)
Similarly what shapes are produced in n-dimensions by a continuous n-dimensional rotation of a point around the origin? Is there a general theory behind this?
algebraic-topology differential-topology
algebraic-topology differential-topology
asked 2 days ago
zoobyzooby
983616
asked 2 days ago
zoobyzooby
983616
asked 2 days ago
zoobyzooby
983616
asked 2 days ago
zoobyzooby
983616
asked 2 days ago
zoobyzooby
983616
983616
What object are you rotating? A point?
– Randall
2 days ago
@Matt Hmm, OK I'll use that definition of orthogonal transformation. I mean it just depends on your definition of what a "rotation" is in 4 dimensions.
– zooby
2 days ago
@Randall. Yes an arbitrary point (not the origin).
– zooby
2 days ago
add a comment |
What object are you rotating? A point?
– Randall
2 days ago
@Matt Hmm, OK I'll use that definition of orthogonal transformation. I mean it just depends on your definition of what a "rotation" is in 4 dimensions.
– zooby
2 days ago
@Randall. Yes an arbitrary point (not the origin).
– zooby
2 days ago
What object are you rotating? A point?
– Randall
2 days ago
What object are you rotating? A point?
– Randall
2 days ago
@Matt Hmm, OK I'll use that definition of orthogonal transformation. I mean it just depends on your definition of what a "rotation" is in 4 dimensions.
– zooby
2 days ago
@Matt Hmm, OK I'll use that definition of orthogonal transformation. I mean it just depends on your definition of what a "rotation" is in 4 dimensions.
– zooby
2 days ago
@Randall. Yes an arbitrary point (not the origin).
– zooby
2 days ago
@Randall. Yes an arbitrary point (not the origin).
– zooby
2 days ago
add a comment |
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What object are you rotating? A point?
– Randall
2 days ago
@Matt Hmm, OK I'll use that definition of orthogonal transformation. I mean it just depends on your definition of what a "rotation" is in 4 dimensions.
– zooby
2 days ago
@Randall. Yes an arbitrary point (not the origin).
– zooby
2 days ago