Perhaps the most well-known higher-dimensional object is the hypercube,
the four-dimensional analog to the three-dimensional cube. Many were
introduced to it in the story {\it A Wrinkle in Time}, by Madeleine
L'Engle, or ``And He Built A Crooked House...'', by Robert Heinlein. Anyone
who wishes to understand four-dimensional objects would do well to study
the hypercube as a first attempt.
One way to understand four dimensions is to look closely at how we can
understand three dimensions using only images drawn in the plane. How
we use these two-dimensional images to reconstruct in our heads the
full three-dimensional objects can serve as a basis for trying to use
three-dimensional pictures to construct four-dimensional objects in
our heads in a similar way. A careful study of two and three
dimensions, and in particular, how we can move from one to the other,
is necessary for this to be successful.
To understand the four-dimensional hypercube, then, we should look at
the three-dimensional cube, and its analog in two dimensions, the
square. Indeed, we can go even further by looking also at the
corresponding one-dimensional object, the line segment, and even the
zero-dimensional version, the point. This family of objects --- the
point, segment, square, cube, and hypercube --- form a wonderfully
interconnected collection, whose relationships we now investigate.
The first thing to note is that the lower dimensional versions appear
as parts of the higher dimensional ones. For example, the cube has
faces that are squares, the square has edges that are segments, and
even the segment has ends that are points. This suggests that the
hypercube should have faces that are cubes, and indeed, this is the
case. Just as the cube has square faces meeting at segments (its
edges), and points (its corners), the hypercube has cubical faces that
attach to each other along their square faces, and which meet at
segments and points. But how many of these are there, and how do they
connect together? To understand the hypercube better, we will try to
count the number of pieces that make it up. In particular, we want to
know how many cubical faces it has.
We begin by looking back at the lower-dimensional analogs. To make
things easier, we will call all the objects ``cubes'', but also give
the dimension; in this way, the 3-cube is the standard cube, the
2-cube is the square, the 1-cube the segment, and the 0-cube the
point. The hypercube is then the 4-cube. This gives us the ability
to talk about an $n$-cube without having to specify which one we mean;
the goal is to be able to say things about the $n$-cube that are true
no matter {\it what\/} the $n$ is. These will be the statements that
help us understand the hypercube.
Our first observation is that the $n$-cube can be formed my moving the
$(n-1)$-cube in a direction perpendicular to itself. For example, if
we move a segment (a 1-cube) a unit distance in a direction
perpendicular to the segment, it will sweep out a square (a 2-cube) in
doing so. Similarly, if we move a square a unit distance in a
direction perpendicular to the square, it sweeps out a cube as it
moves. The same is true even for the point; moving it in any
direction a unit distance causes it to sweep out a line segment. The
sequence is clear: a point moves to form a line segment; the segment
moves to form a square; the square moves to form a cube. The natural
conclusion, then, is that if we move a cube in a direction
perpendicular to itself, the cube should sweep out a hypercube.
The difficulty is in finding this perpendicular direction. In our
three-space, there is no such direction. It is in imagining such a
direction that we take our first real step into the fourth dimension.
Even though we don't have a physical example to point to, we can still
try to conceptualize the properties such a collection of directions
would have: there would be four directions, each perpendicular to all
the others. In this space, we could construct a hypercube. Since we
don't really have such a place, our mental pictures of it always will
be incomplete or distorted, but what would be required for a
four-dimensional space should be fairly clear.
We can use the ``$(n-1)$-cube moving through time'' mechanism to help count
the parts that make up the $n$-cube. For example, suppose we want to
count the number of edges of the square; then we can first look at how
these are formed by the moving line segment as it sweeps out the square.
One edge (the bottom edge, say) is the initial segment just before it starts
moving. The opposite edge (the top) is the position of the segment when it stops
moving. The other two edges are formed by the endpoints of the segment as
it moves: each endpoint sweeps out a new segment (a 0-cube in motion sweeps
out a 1-cube). This gives two copies of the original segment (one at the
top and one at the bottom) and two new segments (one for each of the two
endpoints of the moving segment) for a total of four edges.
Lets do the same thing for the square faces of the cube (formed by a
square in motion). One of these (the bottom) is the original square
before it moves, and one (the top) is the square when it stops moving.
The remaining faces are formed by the {\it edges\/} of the moving
square, each edge sweeping out a new square face for the cube (a 1-cube
in motion sweeps out a 2-cube). This means the cube has two squares
(top and bottom) plus four more squares (one for each edge of the
original moving square), for a total of six.
The other parts of the $n$-cube also can be counted in this way. For
example, it is easy to count the number of corners in an $n$-cube by this
means. There are twice as many as in the $(n-1)$-cube: the corners are
the corners of the base $(n-1)$-cube before it moves plus the corners of
the $(n-1)$-cube when it is at the top. This means that the point has 1
corner (itself), the segment has 2, the square has 4, and the cube has 8.
In general, the $n$-cube has $2^n$ corners. This means that the 4-cube, or
hypercube, must have $2^4=16$ corners. This is the first concrete
information we have obtained about the hypercube!
In general, if we want to know the number of $m$-cubes in an $n$-cube,
we get twice the number of $m$-cubes in an $(n-1)$-cube (the ones that
are at the top and bottom), plus the number of $(m-1)$-cubes in the
$(n-1)$-cube (each sweeps out an $m$-cube as the $(n-1)$-cube moves
from the bottom to the top). This means that if we know the number of
parts in a cube of a particular dimension, we can determine the number
of parts in the cube that is one dimension higher. Carrying out this
procedure generates the following table:
$$\centerline{\vbox{\tabskip=2em plus 1fil\offinterlineskip
\halign{\hfil#\hskip 2em\vrule height 12pt depth 5pt&&\hfil$#$\cr
$n$& \hidewidth m=0\kern-4pt& 1& 2& 3& 4\cr
\omit \hfil\hskip 2em\vrule height 3pt
\vadjust{\hrule}\cr
\omit \hfil\hskip 2em\vrule height 3pt\cr
0& 1& 0& 0& 0& 0\cr
1& 2& 1& 0& 0& 0\cr
2& 4& 4& 1& 0& 0\cr
3& 8& 12& 6& 1& 0\cr
4& 16& 32& 24& 8& 1\cr
}}}$$
The bottom row represents the number of parts in the 4-cube. We
see that there are 16 corners, 32 edges, 24 squares and 8 cubes in the
hypercube! If we followed the sweeping cube carefully, we could even
tell which of these were connected to which other ones. This gives us
the combinatorial structure of the hypercube. (As an aside, we could
continue to extend this table, and count the parts of the 5-cube,
6-cube, or any higher-dimensional cube we want.)
Of course, we really want to {\it see\/} the hypercube; how can we do
that? Again, we move to the lower-dimensional case, and ask how could
we get a two-dimensional person to ``see'' a 3-cube? If we imagine a
world of two-dimensional people living in a plane, what methods could
they use to understand a cube using only their two-dimensional world
and things within it?
There are several approaches they could use, with our help. One
method would be for us to shine a light on the three-dimensional cube
and let it cast a shadow on their two-dimensional world. They could
view this two-dimensional shadow and try to recognize it as a distorted
or squashed view of the actual cube, and reconstruct the cube in their
minds. This is exactly what we do when we look at a photograph, or a
television screen. We see only a two-dimensional view of an object,
but we are adept at building three-dimensional pictures out of this in
our minds. Of course, we have three-dimensional experiences to help
us with this, but our friends in the two-dimensional world would have
a much harder time doing this.
Mathematically, this process is called {\it projection}, and is one of
the key methods of viewing higher-dimensional objects. Projection is
used throughout this exhibit; many of the images are actually
three-dimensional ``shadows'' of the four-dimensional objects from
which we must reconstruct the originals in hour minds.
Our minds are good at turning two-dimensional pictures into
three-dimensional mental images. This process is made even easier and
more concrete if we have a sequence of images that represent a series
of views of a three-dimensional object in motion. For example, the
television or a movie in a theater are quite effective in conveying a
three-dimensional impression. This same idea is used in the movie
sequences and some of the still images in this show to help us better
understand the four-dimensional objects whose shadows we see.
Projections, and sequences of projections, form one important method of
understanding higher dimensions. Returning to our friends in the
two-dimensional world, we can develop other mechanisms to help them
visualize three-dimensional objects. One would be to pass the object
through their two-dimensional world. If they could see the portion of
the object that intersects with their plane of existence, then they
would see a series of ``slices'' of the three-dimensional object.
This ``time sequence'' of shapes could be used to reconstruct the full
three-dimensional shape in their minds. This is very much like how a
doctor uses a CAT scan, which is a series of two-dimensional pictures,
to develop a three-dimensional image of the human body being scanned.
Similarly, we can be called on to reconstruct four-dimensional
objects in hour minds when we are presented with a sequence of
three-dimensional slices of the object. Several images in this
exhibit investigate the idea of slicing as a means of understanding a
higher-dimensional object.
In conclusion, one's ability to visualize four-dimensional objects
rests largely on a good understanding of two- and three-dimensions.
Not only can these dimensions provide valuable information about
higher-dimensional phenomena, but they also give us the power to
produce beautiful images of them.
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