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Mathematics Goes to the Movies

by Burkard Polster and Marty Ross

**Cube 2: Hypercube (2002)**

Cube 2 is the sequel to The Cube. Just like in The Cube, a couple of people wake up in a strange place that is made up of interconnected cubical rooms. Important for us are only the following among these people: Psychologist Kate Filmore (Kari Matchett), private investigator Simon Grady (Geraint Wyn Davies), the blind girl Sasha (Grace Lynn Kung) who turns out to be a mysterious super hacker, computer game programmer Max Reisler (Matthew Ferguson), the engineer Jerry Whitehall (Neil Crone), the senile mathematician Mrs. Paley (Barbara Gordon), the lawyer Julia (Lindsey Connell), and the famous physicist Dr Rozenzweig (Andrew Scorer).

Each of the cubical rooms has six doors, or portals, one in each wall, ceiling and floor. On the other side of each of these portals is another cubical room. As the group stumbles from room to room strange things happen: rooms loop in on themselves. Rooms that are adjacent one moment are no longer adjacent one second later, time moves differently in different rooms, the direction in which gravity acts is different in different rooms, some rooms show alternate realities, deadly traps appear out of nowhere, etc.

As in the Cube, the whole movie is about this new group of people interacting with each other as they are trying to find a way out of this place.

18:20

JERRY: I’ve been trying to get a handle on the configuration of these
rooms. All I can say is…

SIMON: They don’t make any sense.

JERRY: That’s right. They sure don’t.

MAX: It’s as if the rooms are moving around very quickly.

JERRY: There’s got to be some kind of logic to it. These rooms just seem
to repeat. You go in one direction the room just loops back on itself.

It turns out that Jerry designed the door handles for the doors. However, when
pressed he admits that he has heard some rumours relating to …

25:20

JERRY: Quantum teleportation.

MAX: Pardon?

JERRY: They were just rumours.

25:50

MRS. PALEY: It’s a tesseract.

JERRY: Christ, she’s losing it.

MRS. PALEY: Isn’t it beautiful?

KATE: Isn’t what beautiful, Mrs. Paley?

MRS. Paley points at the diagram on one of the panels on the floor.

JERRY: Holy Shit. If you look at it from just the right angle… What did
you call it again Mrs. Paley?

MRS. PALEY: It’s a tesseract, sweetheart.

JERRY: Tesseract?

……

JERRY: Tesseract. It’s a tesseract.

Now comes Jerry’s explanation of what a tesseract is interrupted by more
or less meaningless comments by some of the other people. Let’s skip these
and concentrate on the explanation, which is actually very accurate.

JERRY: A tesseract, it’s another name for a hypercube. .. a four-dimensional
cube. All the elements are there. I mean, rooms repeating, rooms folding in
on themselves. Teleportation. It could all very well add up. Look. Here. See?
Let’s call one dimension length and represent that with a simple line.
Now, two dimensions are length and width which is represented by a simple square.
He first draws a line and then a square

If we extend this by one more dimension we get a cube, which has three dimensions:
length, width and depth. ….

Here he extends the square into a cube.

Here’s the really funky part. If you take this cube and we extend we extend
it one more dimension, we get

KATE: A tesseract.

MAX: I thought time was considered to be the fourth dimension.

JERRY: Sure that is one idea. But what if you have a fourth spatial dimension.

MAX: There is no such thing.

……

KATE: Let’s just say we are in this hyper…

JERRY: Cube.

KATE: Whatever. Does this diagram show us how to get out?

JERRY: Well, uh… no. A hypercube isn’t supposed to be real. It’s
just a theoretical construct.

KATE: Oh, well that makes me feel better.

SIMON: Is there a theory on how we might get out of this theoretical construct?

JERRY: I don’t know.

MRS. PALEY: Don’t worry, dear. It’s just a matter of time.

SIMON: Oh, that’s so much clearer. Thank you.

As we already mentioned, this explanation is actually a very accurate description
of how a four-dimensional cube, that is, a cube with four spatial dimensions,
can be constructed abstractly. It shows why mathematician also refer to a point
as a 0-dimensional cube, a line segment as a 1-dimensional cube, and a square
as a 2-dimensional cube. Mathematically, there is no problem to continue like
this to first construct a 4-dimensional cube, then a 5-dimensional cube, and
so forth. Jerry could have gone on and explained a little bit more as follows:

1. (The number of rooms.) A 1-dimensional cube is bounded by two 0-dimensional
cubes, a 2-dimensional cube is bounded by four 1-dimensional cubes and a 3-dimensional
cube by six 2-dimensional cubes…. 2, 4, 6, … This suggests that
a 4-dimensional cube should be bounded by 8 regular cubes, which is actually
the case. In turn, this would suggest that the world these people are stumbling
around in consists of only eight rooms.

2. (No way out.) Again by analogy it is possible to figure out why things seem
to loop in on themselves when you move in one direction. For a moment, let’s
assume the group consisted of 1-dimensional beings that are caught in the boundary
of a 2-dimensional cube, i.e., the periphery of a square. Since this periphery
is a loop consisting of four 1-dimensional cubes, moving in one direction, the
group would always end up back in the same room they started from after traversing
three other rooms. The same stays true in three and four dimensions. That means
that if we are really dealing with periphery of a hypercube, moving in one direction
(in through one door, out through the opposite door), will get you back to the
same room after traversing three other rooms. Furthermore, since the boundaries
of all cubes are closed, there cannot really be any “way out.”

3. (The diagram) When you project a wire frame model of an ordinary 3-dimensional
cube onto a screen, that is, a 2-dimensional world, you get a shadow in which
still many of the features of the original cube are visible. You usually see
all the vertices, all the edges, and even all the faces of the cube. Of course,
the faces are somewhat distorted. Here is one particularly nice shadow of a
cube, which is called a Schlegel projection of the cube. (insert a diagram of
this). We count 8 vertices, 12 edges and 5 faces. Actually, the one missing
face corresponds to the outside of this diagram. So, there are really 6 faces.
Mathematically it is not a problem to cast a shadow of a wire frame of a hypercube
onto a 3-dimensional screen such as our 3-dimensional world. This shadow will
be a 3-dimensional wire frame. A particularly nice shadow of the hypercube is
the analogue of the above 2-dimensional diagram (insert a diagram of this).
A drawing of this shadow is what Mrs. Palay discovers on the panels. If you
look closely, you count 7 distorted cubical rooms + one that corresponds to
the outside of the diagram = 8 cubical rooms, as predicted above.

All this would explain some of the strange things that are happening in the
world that the group is trapped in. However, all this math only gets developed
very incompletely in the movie and all kinds of other bits and pieces that are
usually associated with the fourth dimension get mixed into the story. For example,
time is sometimes also considered as a fourth dimension, as Max notes at some
point. However, this does not have anything to do with a mathematical hypercube,
which is really a spatial object in which all dimension a spatial dimensions
and not three spatial and one temporal dimension. Nevertheless, in the story
strange things happen in that the time seems to pass at different speeds in
different rooms, etc. A lot of the pseudoscience that was supposed explain what
else was going on in the Hypercube is described in the Behind the Scences feature
“Making of Cube 2: Hypercube” on the DVD. This part of the DVD is
one of the most interesting features of the DVD and is highly recommended.

29:15

Leaving out nonessential bits again.

SIMON: I think we should make a map.

JERRY: I don’t think a map is really going to help us. Look at that?

KATE: That wasn’t there a minute ago.

They find a number on the ceiling.

…..

JERRY: 60659 rooms, Christ.

KATE: This place must be huge.

MRS. PALEY: Oh yes, yes … in a hypercube, there could be 60 million rooms.

JERRY: She could be right.

Oh well, ….

36:15

MAX: Where is here?

SIMON: I don’t know. Your guess is as good as mine.

JERRY: They could have built a hypercube structure anywhere. And if this thing
really does fold space, we could be literately anywhere.

38:10

They find a dead body a man with all kinds of numbers scribbled all over him.

Grab another screen shot. Somebody remarks that the numbers on the body are
upside down (which they are not) and that therefore it seems likely that they
were written by the man himself. Furthermore the numbers are written in the
same handwriting as those numbers that they keep finding in the rooms. Mrs.
Paley recognizes the body and knows the name of this man, Dr Rozenzweig. Max
clarifies the man is a leading theoretical physicist in the field of quantum
chaos. Jerry read his book and fills in some more details. All more or less
pointless.

39:50

They find more numbers on the walls of the room they found the body in. Looks
like he tried to calculate his way out and there is a solution in the form of
the number 60659 that we saw before.

Again all these numbers and formulas don’t seem to make any sense and
were probably just put in by the film makers for effect. 41:50

SIMON: Mrs. Paley. What do you do for a living?

MRS. PALEY: Oh, nothing. I’m retired.

…..

SIMON: Before you retired, what did you do?

MRS. PALEY: Nothing very exciting. I was a theoretical mathematician.

45:00

After Mrs. Paley just held hands through one of the doors with a copy of herself.

JERRY: I have an idea. … I know what just happened there was a little
shocking, but it actually makes total sense if we’re in a really multi-dimensional
quantum environment.

….One fundamental idea of a quantum universe, is that parallel realities
can actually exist simultaneously. …. I read it in Rozenzweig’s
book, it was a big part of his theory. What if whoever designed this stinking
thing somehow managed to create a place where parallel realities can crossover
like that.

KATE: So what you are saying is that what we just saw was Simon and Mrs. Paley
in a parallel universe.

53:50

They hit on the room with the Razor Cube. The way this grows is in some way
the nicest piece of maths in the movie. From a square, to an octahedron, to
a stellated cube to many nested cubes (fill in details).

KATE: It’s a square. It’s just floating.

MAX: What is it?

MRS. PALEY: It’s beautiful.

SIMON: You recognize this too, Mrs. Paley?

MRS. PALEY: Well, not exactly-- …….

Up to here everything is clear. Now comes the interesting transition from octahedron
to stellated cube which boils down to the following. First note that by bisecting
the octahedron along one of the three squares we get two roofs. This gives a
total of six such roofs. Consider the octahedron as being the superposition
of these six roofs (this means that every edge is the superposition of three
edges coming from different roofs). Now, simultaneously, opposite roofs start
floating apart and the individual roofs start turning in the clockwise direction
(viewed from the respective tip of each roof). This is done in a completely
symmetric fashion such that after each roof has turned 45 degrees, all roofs
lock into place in the following configuration (a stellated cube).

During the next transition the tips of the roofs move simultaneously towards
the center of the structure.

until all the roofs are pointing inside, at which point more cubes appear. Together
these cubes form a nested structure and any two of these cubes taken together
form

a projection of a hypercube as described above.

KATE: Maybe that’s what a four-dimensional object really looks like.

MRS. PALEY: It’s stunning. The math of it. It’s a perfect quadrangular
oscillation.

Here the cubes are start to “wave” from the centre out. Following
this they start spinning all in different directions, bouncing around as a sphere
made up of rotating cubes (the razor cube) and growing until they fill the room
(chopping everything and everybody that gets in their way into tiny little pieces.)

1:04:30

Find some more writings on a wall 2FAO8E 60659

SASHA: It could mean anything.

KATE: Right. Maybe it is a coordinate of some type but in four dimensions.

1:20:58

Everything collapses onto one room all the markings in all the rooms appear
in one room. Here things become interesting again mathematically, because all
of a sudden we are dealing with another mathematical world consisting of just
one room in which opposite sides are identified. This means that if you leave
the room through one door, you re-enter it through the opposite door. This is
exactly what Kate does. To escape from the raving mad Simon she flees through
one door and then she surprises him by attacking him from behind.

6:06:59 is the time when things implode which explains the number 60659 which
they have been stumbling across everywhere.

Opening sequence

During the opening titles the camera moves across the blueprint to one of the
cubical rooms as it is being drawn. Along the way it comes across some seemingly
random bits of mathematics.

First we have a formula which is labelled “binomial theorem”

(x+2)^(-n)=\sum _{k=0}{\infty} {-n \choose k} x^k 2^{-n-k}

then an inequality

1/3 n^{1/2} \leq \Delta (n) \leq (1/6n)^{1/2}\sqrt {1/3[1+2(1+3/(5n)]^{1/2}}

Then Delta(N) = …..

Next comes a nice way of growing the title Hypercube, first in one, two, three,
and then (stylised) four spatial dimensions. The one-dimensional version is
just nine strokes of equal length. These get “extruded” into the
2nd dimension to create the 2-dimensional (readable) title. This 2-dimensional
title gets extruded into the 3rd dimension, and then finally into the fourth
dimension. This captures nicely the way

a 4-dimensional cube can be constructed as described by Jerry. In fact, even
in terms of the blueprint we move across various dimensions. We start with a
single blue point (the 0th dimension), which turns into a 2-dimensional blueprint,
which turns into a 3-dimensional blueprint, and finally hints at something beyond
the 3rd dimension. Again, quite a few interesting remarks about creating this
opening sequence are contained in the Behind the Scenes feature “Making
of Cube 2: Hypercube”.