Movement: local and global, in two to four dimensions
Current move-to-Berlin plan: I leave about a week into October, and stay at one of the Max Planck gueshouses till the end of that month. Uwe comes with me for a week of apartment hunting and then returns to Milwaukee to finish packing, sorting out boa-constrictor-importing and cyclocross racing season.
I went out to Mad Planet last night! I am sorely out of dancing shape. But it was fantastic. I need to remember how much I love dancing. It felt like dormant parts of me were waking up in anticipation all Friday, dancing around my office and in the bathroom. I have especially been enjoying doing physical things since submitting my dissertation. Paddling around on windsurfing boards at Wasa, spending weekend afternoons around Milwaukee at beaches splashing around or (ineptly) playing beach volleyball. I actually have a tan! I only manage to get one about one summer in four, I think.
I spent a fun hour-ish yesterday afternoon figuring out how to explain why one cannot keep a string knotted in four-dimensional space. I made little diagrams! Oooh I should make computer figures put them here:
So we have a knot: A loop of one-dimensional string, tangled up in itself in three dimensions (has to be at least three, so it can pass by itself). And, in those three dimensions, we can (by snipping the string, threading it through itself, then re-joining the snip) make a tangled loop that cannot be untangled without snipping it: what we usually consider a knot, as opposed to an untangled loop. But what if there were four spatial dimensions? That extra dimension would give us enough space so that the loop can always be untangled.
This is one way to see this fairly clearly: Any untangling can be reduced to the problem of having two lengths of the string which are laying one on top of the other, and shifting the order of the lengths without affecting any of the other parts of the string.
On the top left is the three dimensional view that's the "normal reference". At the top, the strings are lying mostly parallel to axis 1, connected to arbitrary stuff at the top and bottom. The red string is laying on top of the blue string, further along axis 3 and closer to us. There's a bit of the red string that lays parallel to axis 2.
Now imagine there is a fourth spatial dimension. Each point in the first picture has an extra possible direction attached to it. Draw the same picture considering the 2-3-4 axes, on the top right. This 2-3-4 subspace is a slice in the four-dimensional space through the zero-point of axis 1. We see the bit of the red string laying parallel to axis 2, and only a point of the blue string. In fact, since the direction coming out of the figure is 4, and the blue string extends in dimension 1, this really is only a point in this sub-space. So now we can picture lifting the bit of the red string into dimension 4, and moving it past the point marking the blue string until it is at a smaller coordinate of axis 3. Everything stays fixed in dimensions 1 and 2, so we don't tangle any string ends doing this.
The configuration after moving is shown in the bottom two pictures, the blue string is now on top. And now, any knot you make in three spatial dimensions can be untangled if you have access to a fourth.
(Time is a 4th dimension, but we can't use it to untangle knots because 1) we need it to do the "movement" bit in and 2) the 1-spatial-dimension loops are sweeping out cylinder-y things in time, unless the loop only flashes into existence for an instant)
And lastly, this meme was kinda fun:
Hmm, doesn't seem to show. That PersonalDNA one. I'm a benevolent inventor!
I went out to Mad Planet last night! I am sorely out of dancing shape. But it was fantastic. I need to remember how much I love dancing. It felt like dormant parts of me were waking up in anticipation all Friday, dancing around my office and in the bathroom. I have especially been enjoying doing physical things since submitting my dissertation. Paddling around on windsurfing boards at Wasa, spending weekend afternoons around Milwaukee at beaches splashing around or (ineptly) playing beach volleyball. I actually have a tan! I only manage to get one about one summer in four, I think.
I spent a fun hour-ish yesterday afternoon figuring out how to explain why one cannot keep a string knotted in four-dimensional space. I made little diagrams! Oooh I should make computer figures put them here:
So we have a knot: A loop of one-dimensional string, tangled up in itself in three dimensions (has to be at least three, so it can pass by itself). And, in those three dimensions, we can (by snipping the string, threading it through itself, then re-joining the snip) make a tangled loop that cannot be untangled without snipping it: what we usually consider a knot, as opposed to an untangled loop. But what if there were four spatial dimensions? That extra dimension would give us enough space so that the loop can always be untangled.
This is one way to see this fairly clearly: Any untangling can be reduced to the problem of having two lengths of the string which are laying one on top of the other, and shifting the order of the lengths without affecting any of the other parts of the string.
On the top left is the three dimensional view that's the "normal reference". At the top, the strings are lying mostly parallel to axis 1, connected to arbitrary stuff at the top and bottom. The red string is laying on top of the blue string, further along axis 3 and closer to us. There's a bit of the red string that lays parallel to axis 2.
Now imagine there is a fourth spatial dimension. Each point in the first picture has an extra possible direction attached to it. Draw the same picture considering the 2-3-4 axes, on the top right. This 2-3-4 subspace is a slice in the four-dimensional space through the zero-point of axis 1. We see the bit of the red string laying parallel to axis 2, and only a point of the blue string. In fact, since the direction coming out of the figure is 4, and the blue string extends in dimension 1, this really is only a point in this sub-space. So now we can picture lifting the bit of the red string into dimension 4, and moving it past the point marking the blue string until it is at a smaller coordinate of axis 3. Everything stays fixed in dimensions 1 and 2, so we don't tangle any string ends doing this.
The configuration after moving is shown in the bottom two pictures, the blue string is now on top. And now, any knot you make in three spatial dimensions can be untangled if you have access to a fourth.
(Time is a 4th dimension, but we can't use it to untangle knots because 1) we need it to do the "movement" bit in and 2) the 1-spatial-dimension loops are sweeping out cylinder-y things in time, unless the loop only flashes into existence for an instant)
And lastly, this meme was kinda fun:
Hmm, doesn't seem to show. That PersonalDNA one. I'm a benevolent inventor!