Here's What I Think (The Whole Picture)
This is sort of a mult-response to many of the topics here, for anyone still reading these boards.. I just rewatched it and feel the need to analyze a bit.
Be warned... SPOILERS be lurking below, mateys.
First issue... what is the fourth dimension?
My answer (note: I am definitely NOT a scientist, so bare with me on these opinions): Time. It's a dimension that cannot be depicted on a diagram because it requires motion to be displayed.. the dimension of time looks at a 3D object and all of its components over an infinite span of time. One could assume that the hypercube is one single cube with six doors. Each door leads to the same cube at a different point in time. In this way, the hypercube seems to stretch on into infinity, and it quite frankly does. Because by adding the element of infinite time, alternate univeres are developed.... see, not only does the 4th dimension look at a 3D object at arbitrary points in time.. it also looks at all POSSIBLE perspectives from all POSSIBLE points in time. So every choice and movement made by the cube's components (the people in the cube) creates millions of cubes... how each "snapshot" cube is organized seems to be mostly random in the multiverse of the hypercube.
Because of its complexity, the hypercube is extremely unstable. In the movie, the hypercube is obviously a work in progress, which might explain why IZON is vehement on tracking down Alex Trusk.. even if it means going into the hostile environment of the cube... because she is carrying some sort of vital information device.
Meanwhile, the cube can only stay "alive" for a short period of time. As the physicist is able to figure out, when the watch he happens upon (Jerry's) reached 6:06:59, the cube will collapse on itself. It seems the only way out, then, is to be in the right cube (probably the original 3D cube) by 60659. How do they find this cube out of the billions of cubes? They are led there by the collapse of the outermost "snapshot" cubes as the hypercube is imploding (you know, the ones that kill people).
As for getting into the cube, it seems that a portal must be created that allows one to physically enter the 4D environment. The liquid that Kate wakes up in looks like mercury or a similar chemical, so it must be the "portal" into the cube.. therefore, there could be more than one portal into the cube, as long as there is only one to get out (ie they would have to destroy other portals before people escape from the cube..otherwise, the hypercube's dimenstion will interfere with our real world physics, which could cause a catastrophic result for the rest of the earth.
We have to assume that the movie takes place far enough into the future for this quantum transportational system to be developed (after all, if you can't suspend your disbelief somewhat, you'll never enjoy any sort of scifi movie or book)
Issue 2: Why Them??
Why were these specific people chosen to go into the hypercube? Simple.
Recall the first Cube movie. As Holloway exclaims angrily told that she was chosen because she's a doctor.. "why not the 10 million other doctors in this country?" (or something like that).. .there is an amount of randomness to the selection process... but, they would most likely want to choose people close to home, because they would have greater control hiding their disappearances. In the end, they are just looking for guinea pigs to test out their new technological toy. Let's look at all the unwilling participants, and the two willings.
Dr. Phil Rosenzweig: This famous physicist was most obviously selected to test the hypercube. Would this genius be able to figure his way out? The answer is no. He is found dead (though he must have been alive elsewhere in the cube, do to the time factor.) I believe that the watch they find in the cube was originally left by Jerry and then picked up by Rosenzweig. The doctor was able to do his math based on the arbitrary time set up on the watch. In one despairing reality (obviously a "true" reality because he did not make it out of the cube [as far as we get to know]), the doctor is able to decipher the time when the cube will implode, but not how to get out. He writes the time stamp down in various places (60659) and somehow dies.. starvation, exhaustion, stroke.. doesn't really matter in the end.
A quick note on how he finds Jerry's watch: The main characters in the film that are "just waking up" have most likely already woken up in other realities... in one of them, Jerry suggests leaving things behind, and he reluctantly leaves his watch behind. This is integral to Rosenzweig's purpose as a character, because he would need the time reference to do his calculations. In the current reality, the one where Kate escapes, Jerry and co. find the watch, so Jerry remarks that he was going to suggest that they leave things behind, but he doesn't do to the disturbing find.
Colonel Maguire: This sad fellow was obviously aware of the original cube. He looks around for the numbers to help guide him out... a couple things here... why the briefcase? why him? Well, the viewer could make up pretty much any reason here, but one explanation could be that he crossed someone that he shouldn't have and is being punished. He has knowledge of the regular cube, so he is the test to see if that knowledge will help him escape a hypercube.. he doesn't even try, finding that suicide is his only choice. The briefcase is most likely a taunt. It gives him hope when he wakes up and finds it with him, then he realizes taht it is empty and therefore is useless.
Becky Young: Becky is an IZON employee, though we don't know what level. I would like to romanticize things a bit and say that she started snooping around above her clearance level and is being punished for it, but no real explanation is given. She certainly seems out of her league in the cube, so I doubt she is put in there to test her expertise in any way. Her main purpose for the movie is to create the character of Simon Grady.
Simon Grady: Simon is contracted by Becky's parents after she is abducted, after all, we don't know how long each person (except for Kate on the DVD) was in the cube or how long they were kept "plastic wrapped" before going in the cube. Simon starts investigating Becky and IZON takes notice. To hide their tracks regarding Becky, they abduct Simon Grady as well. Simon of course goes crazy and eats Becky multiple times (living in the cube for many "years" in one reality), so he probably wasn't the best PI to contract.
Julia: Julia seems to be another random grab. She represents IZON as a lawyer, is very wealthy, and rubs shoulders with the Hollywood crowd. Other than that, she seems pretty expendable. Perhaps she was bad at her job, or perhaps she had a big mouth about IZON's cases... either way, someone wanted her dead, so they used her as a guinea pig like Becky.
Max Reisler: Max's reason is easy to guess... he essentially designed parts of the cube; these ideas were stolen by Game Thrill/IZON and when he prepared to take legal action against Game Thrill, IZON takes him out and puts him in his own game.
Jerry Whitehall: Jerry, my least favorite character, though the actor did pretty well, is one of the engineers that designed the phyiscal aspect of the original 3D cube that was put into the 4th dimention. Like Worth before him, he proves that its best to research your projects before taking on random contracts to design strange doors or large outershells. I suspect that IZON was worried he might try to sell his designs elsewhere, which could eventually lead to a leakage of IZON being the first to contract these doors. So they keep him quiet by putting him in the cube.
Sasha: Sasha/Alexandra Trusk says she goes into the hypercube to escape IZON. Her necklace apparently holds an information device that is deemed vital by the IZON people. How does she get in the hypercube? Well, as she is a genius, one could stretch and say taht she was able to construct her own "portal" into the cube.. then when IZON tracks her down to her house and finds this portal, they destroy it and send an agent to go after her.
Kate Filmore: Kate is an agent of IZON sent into the cube on a suicide mission to destroy Alex Trusk and hopefully retrieve the information device. It takes Kate a while to discover Alex's otehr identity--which would make me assume that she wasn't given much information--there probably wasn't much info. to give, Alex seems to be very mysterious after all. When Kate does discover Alex, I think she has gained an attachment--it isn't uncommon for unlikely friendships to form as a result of a life-or-death situation. So, Kate might have hoped to take ALex out of the cube alive hoping that IZON might let her go... or, maybe she just wanted her captured alive to be better rewarded by IZON--either way, Kate was not happen when Sasha was killed by Simon. Kate does make it out of the cube, but why is Tracton ordered to kill her? Simple. She knows too much. She has fulfilled her task BETTER than expected (they thought she was a goner) but they had no room for someone with more information about the cube than they did. The information device may have been recording info while they were in the cube, which would explain why there was no need of a debriefing/interrogation about what happened.
That's all I'll cover for now... leave questions or comments and I'll try to address them... like I said, this is just for fun, I'm not a genius or a scientist so don't feel special if you discover a flaw in my ideas--I'm sure they are there. I'm jsut in the mood to discuss a movie taht I thoroughly enjoyed (though I liked the original cube better for its commentary on human nature.)