The Logic of Cube 2: Hypercube

In an attempt to better understand this film, here are two posts from an IMDB message board that attempts to explain what exactly is occurring.

The first is by Nonym and gives a mathematic perspective on the cube structure:

It's actually 6-dimensional: it has 4 dimensions of space and 2 of time. But before I explain why it is 6-dimensional, I'll explain why it isn't 4-dimensional.

Most people would consider any normal old cube to be 3-dimensional, but it is in fact 4-dimensional, since it occupies the 3 dimensions of space and the one dimension of time. If it didn't occupy the 4 dimensions, the "normal" cube would literally only exist for an instant. (just as a 2-dimensional square has infinitessimal depth and only exists at a single point in the third dimension)

Now, imagine we build a perfectly normal cube with a side-length of 5m at 06:00:00 and dismantle it at 06:06:59. So, this cube has dimensions of 5mx5mx5mx419s. Now put someone in that cube. If that person exits the cube, they just exit into whatever place the cube was built in (analagous to exiting a house into your neighborhood, or exiting your car into the parking lot or driveway). If you were to attach other cubes to the sides of the first one to create a network of cubes, it would be no different from the cubes in the first movie (Note, however, that it is implied that the entire hypercube consits of only a single room at different points in time). Also, if you were to just wait until 06:06:59, when the cube is supposed to be dismantled, the cube is simply dismantled around you and you are left in whatever place the cube was built in the first place (like having your house being dismantled around you to be left in your yard, or your car dismantled around you to be left in your driveway). Obviously, such a cube is no different from the boring old cube from the first movie.

So, how many dimensions do you need to get a cube like the one in the movie? The answer is 6, and the cube isn't really a "hypercube" either... it's closer to a "hypertoroid". I'll explain why: In the movie, as I have said, it is implied that the entire cube is only one room. So, when a person leaves through a side of the cube, they simply appear on the other side (like we saw near the end of the movie, though earlier in the movie, they would actually appear back in the cube at a different point in time). Instead of starting with a cube and try to figure out how we would accomplish this, let's start out with a line. We want to deform the line so that by going in one direction, we'll eventually end up back where we started. How do we do this? By turning the line into a circle. So now we have a 2-d object, which locally appears to be 1-dimensional (since one can still only move back and forth along the "line".) Now, let's say we want to do the same thing with a square, so that if one moves off one edge, he reappears on the opposite side, or if he walks in one direction, he'll end up back where he started. We do this by mapping the square onto the 3-dimensional shape called a torus (a doughnut or bagel shape). The reason we use a torus instead of a sphere is the sphere would only truly wrap in one direction (for example, on earth, if you were to just keep going north, you wouldn't suddenly appear at the south pole after reaching the north pole, instead you start coming back down on the other side of the earth.) Also note that while the torus is 3-dimensional, it still locally appears to be 2-dimensional, since we are only walking on its surface. Now all that we do is apply the same mapping of a cube onto a 4-dimensional hypertoroid, and we get a shape which is locally 3-dimensional, but wraps in all directions - just like the hypercube in the movie.

So we now have a shape which has 4 spatial dimensions and 1 time dimension, for a total of 5 dimensions. But, with only one dimension of time, it still flows perfectly normally. This is simply solved by adding another dimension of time, but which is orthogonal to the other time dimension and the 4 spatial ones at inside of the cube, but not at the edges. (orthogonal means "at a 90 degree angle". If 2 axes or dimensions are orthogonal, then motion in one does not correspond to motion in the other, ie if I move forward, then I haven't also moved left or right. If they are not orthogonal, then motion in one *does* correspond to motion in the other.) In this way, we can have the time in one axis pass normally, while time in the other axis passes as we move through the doors in the cube. This even allows it to seem as if passing through a door moves you into a completely different room, since now you're in a different point in time in one of the axes, and your timeline doesn't intersect with the original timeline (until the cube collapses at the end, that is). Another effect of this (though it would take too long to explain why) is that people moving in the same direction through time would experience time passing at the same rate, but 2 people moving in different directions would both appear to one-another to move more slowly (seemingly [but not] paradoxically, and also contrary to the movie's portrayal of one appearing to move more slowly while the other appears to move faster. Thus, many effects in the movie can be explained by adding this second time dimension.

So, the hypercube isn't 4-dimensional: a 4-dimensional cube is nothing special. In fact, it is a 6-dimensional toroid.
The second post is by brand_nu_sim and points out further inconsistencies:

Despite that extended ending, the ending still does not make sense, for numerous reasons:

1) a) If Kate's mission was to kill Sasha, it makes no sense that she didn't kill her, was shocked when she died, and in fact made a concerted effort specifically to PROTECT her from the spinning razor cube. You may argue that Kate didn't know Sasha was the uberhacker at that point, but if that's true, why was she not more frantically concerned with finding and learning more about the other characters, when she got separated from them, in order to complete her mission? I'm not sure I even buy that it's true, because it seemed from her actions at the end that she recognized the "device" around Sasha's neck immediately, and if part of her mission was to recover that device it should have been a dead giveaway that Sasha was the hacker.

1) b) It also doesn't make sense that if Kate was sent in to kill the hacker and/or recover the device, she wasn't given any instructions on how to navigate the hypercube, or any information on anyone else who had been placed in the hypercube (e.g., knowing about the violent sociopath might have helped).

2) a) It doesn't make sense that the uberhacker could have "given them a hypercube", and known where it was, and known what it was for, and yet be stupid enough to intentionally go inside of it and/or be stupid enough not to know how to get around safely or escape. (As the world's greatest hacker - and nothing else that we know of - she should have known that it had no internal controls, no hackable interfaces whatsoever.)

2) b) The movie tried to one-up its predecessor by giving the characters real reasons for being thrown into the cube, but almost none of them make sense. (Note: It also does not make sense that the lack of scientists or panic in the hangar when the hypercube collapses indicates that they expected it to fail. In that case, they would have just assassinated anyone they wanted dead, rather than go through the completely useless trouble of transporting them to the hangar and into the hypercube. Also, there's a ridiculous lack of monitoring devices in the hangar. Think about it: the government would scientifically monitor the hell out of an enormous, expensive, high-tech creation.)

Examples of characters without reasons:
i) If we assume they wanted Becky Young dead for knowing something she shouldn't have, there was no reason to throw Simon Grady in there when he didn't yet know a single thing about the reason for her disappearance (besides, if she disappeared at the same time as everyone else, how is Simon even on the case yet?).
ii) There's no reason to execute a senile, alzheimer's-suffering theoretical mathematician who simply knows what a (non-lethal, theoretical) hypercube is.
iii) There's no reason to execute a hacker/programmer who has no idea you stole his concept of rooms with differing time speeds (why did they even steal that?).
iv) There's no reason to execute your lawyer - quite the contrary! You might really need her some day!

3) a) The movie operated on a self-contradictory notion of what the hypercube was. It was stated in the film that the fourth dimension was NOT time, and yet inside the hypercube, parallel realities existed AND many of them were time-shifted - neither of these have anything to do with a fourth dimension if it is not time (nor does the shifting of gravity, or sped-up and slowed-down zones). Basically instead of choosing a definition of what the fourth dimension was, the movie experiments with many possibilities, thereby negating all plausibility.

3) b) Moreover, the parallel realities intermingled, which makes no sense - specifically when the two Mrs. Paleys touched hands, two realities existed in the exact same space, which undermines any definition of what a reality is. (If the hypercube contains its own, encompassing reality - and indeed it SHOULD - there's no reason for multiple realities as we know them to exist within it, that is, to be the definition of its reality, because this too is NOT what the fourth dimension is.) Besides, it's incredibly "convenient" that the acting number of realities inside the hypercube was very very small (much smaller than the suggested number of rooms!), and that despite the "instability" of the hypercube no two realities happened to coexist such that two or more people occupied the same space at the same time, and that none of the realities we witnessed were lucky enough to see one of the characters being brought into the hypercube in the first place and thereby see the guards using the exit (or likewise to see Kate making her final escape and jump in after her)!

3) c) Most importantly, it doesn't actually make sense that although multiple time-shifted realities co-existed within the hypercube, the cube could progressively (as in over time) collapse. Some of those realities would have to exist in a time when the cube was already in the process of collapsing, thus negating said realities because they were outside the (shrinking) bounds of the hypercube - and since we've already established that multiple realities are allowed to exist in the same space and time, most or all of the characters, for most or all of the film, should have been simultaneously in a state of being and a state of non-being, because a reality wherein they were outside the (smaller) bounds of the hypercube existed at the exact same space and time. The most simplistic solution to this would be that ALL the characters are alive in the water in the hangar at the end of the film, because non-being in the hypercube is being outside of the hypercube, which for simplicity's sake ought to be being alive and well in the hangar. Maybe they were all shot as they appeared in the hangar, dozens of copies of the same people, over and over, during the course of the film, and we just didn't see it...?

3) d) Kate's final escape does not make sense. The film implies that the hypercube is collapsing and what it leaves behind is total nonexistence, and there is no good reason why that collapse should be coming from every direction except the floor. Besides, who's to say that particular room had gravity directed the way it normally is, where down is down? Actually, the hypercube's collapse should have come from all four dimensions, one of which Kate would have been unable to see. So she should have been rendered nonexistent. OR since the final minutes implied all realities and all rooms were merging together, ALL living characters (which would have been a lot, because of the multiple realities - more than one of each character) should have survived. BUT then they would all be merged into the same space as the collapse continued, thus negating them all. So yeah, nobody should have escaped the cube at all, unless that particular floor panel was THE exit, but then because of the realities being merged together in the collapse, it would simultaneously be NOT the exit - so nobody should have escaped (or everyone, existing in a single space and time, should have simultaneously escaped and not escaped, but on their escape they'd re-enter our normal 3 dimensions and cohabit the same space, exploding into an atomic pulp).

3) e) Also, the collapsing of all the realities and rooms together would bring the room with the spinning razor cube and the rooms with the deadly crystal shafts all together as well, thus killing everybody.

4) a) Did I mention time doesn't make sense in the movie either? The realities are time-shifted, meaning different iterations (copies) of the same people enter the cube, do certain things, and die at different times. Plus, time moves slower or faster in certain rooms. Hence there is no way that all the different watches from Jerry would display the same time, as they were acquired in different rooms from different, time-shifted Jerrys. Also, the room in which Max and Julia are decomposing corpses making out seems to imply (since we never see them again) that those are the ones from the reality we know - which makes sense, sort of, since they were in a fast-time room. However, the time in that room was not THAT fast compared to the previous rooms (besides, if it were, they'd be decomposing so fast before Kate's eyes that we'd see the progress). This would also lead us to assume that when Kate and Sasha are in a room with bodies in a similar state of decomposition, that is likely a similarly fast-timed room, however the next time we see Simon (who could not have been in an equally fast-timed room for equally long, since he has been moving), his age is no different, and he is moving no faster or slower in the room he comes from.

4) b) Another huge problem is at the very end when the general says Kate has been gone for 6 minutes and 59 seconds. This leads us to believe that all the characters were thrown into the hypercube at 6 pm (as it is now, we're lead to believe, 6:06:59). However, just before leaving the hypercube Kate looks at a watch that says 6:06:59, and that watch, whether it has been in the exact same room speeds as Kate or not, should have on average been through the same amount of relative time as Kate (note I already pointed out that the watches displaying the identical time makes no sense, but let's assume they all averaged out perfectly somehow). This means either that all the characters were actually all thrown in there ninety-whatever minutes ago (however long the movie is up to that point), OR that the average room speed in the hypercube is roughly 13 times faster than time as we know it, which is arbitrary and ridiculous. Why build an inescapable killing device AND make the people inside die 13 times faster? What do you care, since they're stuck there anyway, and the inevitable collapse will kill whoever is left at the same time regardless?

5) So Kate was sent into the hypercube with a mission. She knows the way in (so does whoever put the victims in there), and let's assume the ending proved she knows the way out (so does whoever put the victims in there). She did her job (so did whoever put the victims in there). Why kill her, and not whoever put the victims in there? Or if they were killed too, why not kill them all by putting them into this, that, or the other cube, since that seems to be the preferred way of killing anyone with any knowledge of the cube? But then, doesn't that become a vicious cycle? Whoever Kate works for is exactly the same as whoever the person operating the cube works for, and that operator seems to have the exact same reasons to be killed as Kate, and then whoever kills him has the exact same reasons (same forbidden knowledge) to be killed, et cetera. So everyone with any knowledge of this project should be killed until there are only two people left, and then they'd get stuck in a war of "You go first, I insist" until they shot each other or destroyed the cube. Basically, if we're to make any sense of the obsession with secrecy that made them kill Kate, we need to accept that the cube is an utterly doomed experiment, and take comfort in the fact that the killers will also be killed, because rules are rules.

6) Finally, this isn't a major point, but visually, the "boundaries" of rooms, in terms of where their different speeds or gravities kick in, is inconsistent. OK, that had nothing to do with the ending. Sorry.

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