July 15, 2011 12:08 AM Subscribe

Consider two rings that are intertwined. (These lie in 3-dimensional space, but when you draw it on paper it looks like a venn diagram.) In 3-dimensional space they cannot be unlinked. The questions is, if you had one more dimension, can you unlink them?

posted by gzimmer to Grab Bag (14 answers total) 8 users marked this as a favorite

posted by gzimmer to Grab Bag (14 answers total) 8 users marked this as a favorite

Trivially so, in fact - or rather, in four-dimensional space it'd be hard to claim they were linked in the first place. Here's an easy analogue with one- and two-dimensional space. In one dimension, a "circle" (all points equidistant from a center point) is two points. Here's two one-dimensional circles, A and B:

posted by NMcCoy at 12:51 AM on July 15, 2011 [2 favorites]

A B A BYou can't pull them apart because they're linked - the right A point blocks the left B point from sliding off to the right. In two dimensions, of course, you can just pull the left B around.

posted by NMcCoy at 12:51 AM on July 15, 2011 [2 favorites]

Using the fourth dimension to solve puzzles in three dimensions is the whole idea of a game currently in development: Miegakure

Indeed, here's an example of linking three-dimensional rings from the game.

posted by firesine at 3:19 AM on July 15, 2011 [2 favorites]

Indeed, here's an example of linking three-dimensional rings from the game.

posted by firesine at 3:19 AM on July 15, 2011 [2 favorites]

Contrary opinion: if the "ring" actually has four dimensions, it isn't a ring any more. The same way a circle isn't a circle anymore when it has three dimensions: it must either be a plate, sphere or a ring.

posted by gjc at 4:31 AM on July 15, 2011

posted by gjc at 4:31 AM on July 15, 2011

Hmm...I'm not so sure.

I really like the example of the link in the first answer with the coin trapped by the picture frame. If the coin is only manipulated in a two-dimensional manner, it is trapped. It still has a third dimension, though, and it is still a coin. Only by utilizing this third dimension can it break free. By analogy, if this phenomenon concerning 2d - 3d can happen, it seems reasonable it can happen 3d - 4d. So yes, it is imaginable that you could unlink the rings if you had a 4th dimension to work with and knew what you were doing.

posted by 3FLryan at 8:04 AM on July 15, 2011 [1 favorite]

It seems like the 3-dimensional thing that looks like two linked rings could be a projection of several different 4-dimentional objects, in the same way that a Venn diagram is a 2-D representation of both linked rings and unlinked but overlapping rings.

posted by aimedwander at 8:10 AM on July 15, 2011 [1 favorite]

posted by aimedwander at 8:10 AM on July 15, 2011 [1 favorite]

Mathematician here.

*Contrary opinion: if the "ring" actually has four dimensions, it isn't a ring any more. The same way a circle isn't a circle anymore when it has three dimensions: it must either be a plate, sphere or a ring.*

It's not the rings that have four dimensions, it is the ambient space that the rings are living in. For instance if I draw a line on a piece of paper, that is a one-dimensional subset of a two-dimensional ambient space. But if I think of the piece of paper as sitting in three-dimensional space (i.e. our world) then that same line is a one-dimensional subset of a three-dimensional ambient space.

Thinking about four spatial dimensions is hard. Our brains are wired for dealing with objects in three dimensions (for obvious reasons) so we have to employ some tricks to understand more dimensions.

Adding another dimension means that you have another degree of freedom, another way in which things can differ. To move from three to four dimensions, we can think about adding another descriptor to three-dimensional objects: let's call that color. So objects would have four coordinates (x,y,z,c), where x,y,z describe location, and c describes the color. Two objects are touching if and only if all the spatial coordinates are the same,*and* the color is the same.

So if we apply that to unlinking the two linked circles in four dimensions, it's easy. Color one circle red and the other one green. Since their colors are different, we can move them spatially (just using the x,y,z coordinates) and they won't ever touch each other since the c coordinates are always different.

posted by number9dream at 9:52 AM on July 15, 2011 [5 favorites]

It's not the rings that have four dimensions, it is the ambient space that the rings are living in. For instance if I draw a line on a piece of paper, that is a one-dimensional subset of a two-dimensional ambient space. But if I think of the piece of paper as sitting in three-dimensional space (i.e. our world) then that same line is a one-dimensional subset of a three-dimensional ambient space.

Thinking about four spatial dimensions is hard. Our brains are wired for dealing with objects in three dimensions (for obvious reasons) so we have to employ some tricks to understand more dimensions.

Adding another dimension means that you have another degree of freedom, another way in which things can differ. To move from three to four dimensions, we can think about adding another descriptor to three-dimensional objects: let's call that color. So objects would have four coordinates (x,y,z,c), where x,y,z describe location, and c describes the color. Two objects are touching if and only if all the spatial coordinates are the same,

So if we apply that to unlinking the two linked circles in four dimensions, it's easy. Color one circle red and the other one green. Since their colors are different, we can move them spatially (just using the x,y,z coordinates) and they won't ever touch each other since the c coordinates are always different.

posted by number9dream at 9:52 AM on July 15, 2011 [5 favorites]

Read the short novel "The Boy Who Reversed Himself" by William Sleator for an interesting viewpoint on this whole realm of thought. It's a YA Sci-Fi book, but it does a good job discussing what each dimension would look like from a higher dimension.

posted by tacodave at 1:15 PM on July 15, 2011

posted by tacodave at 1:15 PM on July 15, 2011

Lines that intersect in 2 dimensions also intersect in three dimensions. The original point of intersection doesn't change. Rings are three dimensional objects, they cannot be expressed with a fourth dimension added. No matter what color they are.

Adding a fourth dimension, like time, doesn't change the relationship of the rings to each other. Take two interlocked rings, set them on the table, wait for some time, and they don't magically unlink.

Think of it this way: how much does a circle weigh? Nothing. But what about a circle in three dimensions? OK, now you can manipulate the circle on edge. But it is still a circle. It has no mass. It is just all the points equidistant from the center on a plane.

If you add a fourth dimension to the rings, as opposed to just the universe they exist in, I believe they are still constrained by the original three dimensional linked-ness. (The same way two interlocking circles would remain interlocked if a dimension is added and they are morphed into rings.)

posted by gjc at 4:27 PM on July 15, 2011

Adding a fourth dimension, like time, doesn't change the relationship of the rings to each other. Take two interlocked rings, set them on the table, wait for some time, and they don't magically unlink.

Think of it this way: how much does a circle weigh? Nothing. But what about a circle in three dimensions? OK, now you can manipulate the circle on edge. But it is still a circle. It has no mass. It is just all the points equidistant from the center on a plane.

If you add a fourth dimension to the rings, as opposed to just the universe they exist in, I believe they are still constrained by the original three dimensional linked-ness. (The same way two interlocking circles would remain interlocked if a dimension is added and they are morphed into rings.)

posted by gjc at 4:27 PM on July 15, 2011

Obvious Flatland Shoutout for more (literary, fictional, metaphorical) wanderings in n-space.

posted by lalochezia at 4:52 PM on July 15, 2011

posted by lalochezia at 4:52 PM on July 15, 2011

They don't magically unlink because they still have the same time coordinate. If there were a way to keep the first three dimensions the same, but make the time coordinates different, they would unlink.

posted by 3FLryan at 5:36 PM on July 15, 2011

(much like the coin remains trapped in the picture frame until you manipulate the third dimension with relation to the coin but not the picture frame)

posted by 3FLryan at 5:37 PM on July 15, 2011

posted by 3FLryan at 5:37 PM on July 15, 2011

But along the time dimension, there exists two locations, one in the past, and one in the future, where the rings are not linked, whereas in between there are no spatial locations in which the rings are unlinked. Therefore, translation along the temporal dimension will allow you to unlink them.

posted by HiroProtagonist at 8:15 PM on July 17, 2011

This may be off-topic, but this same principle, applied to 2D-to-3D transforms is a key element in Super Paper Mario (for the Wii) problem solving.

posted by bookdragoness at 12:57 PM on July 29, 2011

posted by bookdragoness at 12:57 PM on July 29, 2011

This thread is closed to new comments.

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/four_dimensions/index.html#knotty

posted by Chionophilia at 12:37 AM on July 15, 2011 [7 favorites]