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 be used as a basis for trying to use
three-dimensional pictures to construct the concept 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.
In order 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 interrelationships we now investigate.
The first thing to note is that the lower dimensional versions appear as
pieces 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 (it's 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 to the connect together? To understand the
hypercube better, we will try to count the number of pieces of various
dimensions that make it up. In particular, we want to know how many
cubical faces it has.
To do this, we look back at the lower-dimensional objects. 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, then it sweeps out a cube as it moves. The
same is even true 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 what properties such a collection of directions would have:
there would be four directions, each perpendicular to all the others. In
such a space, we could construct a hypercube. Since we don't really have
such a place, our mental pictures of it will always be somewhat 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.
Where do the other faces come from? They are formed by the {\it edges\/}
of the moving square, and each edge sweeps 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. (Try this out with the square and
the cube to see that it really works). 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!
As a final example, we can count the number of edges in the cube by noting
that four of them come from the square when it is at the bottom, four more
from the square at the top, and the remaining edges are formed as the
corners of the square sweep out new vertical edges of the cube. There were
four corners in the square, so four new edges are formed in the cube.
This gives a total of 12 edges in the cube.
Can we determine a formula for this, as we did for the number of corners
above? Indeed, we can, because the process was always the same: if we
wanted to know how many $m$-cubes there are in the $n$-cube, we counted
them by noting that the $n$-cube has an $(n-1)$-cube at the top and the
bottom, so we got twice the number of $m$-cubes in the $(n-1)$-cube (the
number at the bottom plus the number at the top). To this number we added
the number of $m$-cubes generated as the $(n-1)$-cube sweeps out the $n$-cube.
These are formed by the $(m-1)$-cubes in the $(n-1)$-cube (e.g., the
vertical edges, or 1-cubes, of the 3-cube were formed by the motion of
the corners, or 0-cubes, of the moving square, or 2-cube). If we let
$C(n,m)$ represent the number of $m$-cubes in the $n$-cube, then this
means that $C(n,m) = 2C(n-1,m) + C(n-1,m-1)$.
This formula means that if we know the number of parts in a
lower-dimensional cube, we can find the number of parts in a
higher-dimensional one. In particular, we can carry out this procedure
in order to find the number of parts the hypercube. To do so, we
build a table of values for $C(m,n)$ for various values of $m$ and
$n$:
$$\centerline{\vbox{\tabskip=2em plus 1fil\offinterlineskip
\halign{\hfil#\hskip 2em\vrule height 12pt depth 5pt&&\hfil$#$\cr
$n$& \hidewidth m=0\kern-4pt& 1& 2& 3& 4\cr
\omit \hfil\hskip 2em\vrule height 3pt
\vadjust{\hrule}\cr
\omit \hfil\hskip 2em\vrule height 3pt\cr
0& 1& 0& 0& 0& 0\cr
1& 2& 1& 0& 0& 0\cr
2& 4& 4& 1& 0& 0\cr
3& 8& 12& 6& 1& 0\cr
4& 16& 32& 24& 8& 1\cr
}}}$$
Here, 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 use 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 in both {\it A Rotation of Cubes\/} and {\it Iced Cubes\/} in
fundamental ways. The images of the hypercube that appear in both
pictures are actually three-dimensional ``shadows'' of the
four-dimensional hypercube from which we must reconstruct the
hypercube in hour minds. Just as a shadow of a cube is a distorted,
flattened view of the cube, so these images distort the hypercube.
It can be instructive to try to locate the various parts that we
counted above in one of these images. For example, in {\it A Rotation
of Cubes}, you should be able to count the 16 vertices of the
hypercube in any one of the five projections. Also, the 32 edges
should be fairly clear. Can you find the 8 cubical faces? Two of
them are colored orange and yellow, but there are six more. Because
the projection distorts the shape, the cubes will not seem to have
right angles, but remember that the shadow of a cube exhibits the same
behavior. Indeed, below each projection of the hypercube, there is a
corresponding projection into a plane of a three-dimensional cube. In
your mind, you can ``see'' the cube of which these are two-dimensional
shadows; in the same way, you must try to recognize how the
three-dimensional image above it is the shadow of a hypercube.
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
brought into play in {\it A Rotation of Cubes}. As it's name
implies, this image depicts a sequence of projections of a hypercube
as it rotates in four-space. Having not just one view, but several
related views of the hypercube, can help us to recognize its
four-dimensional nature even further.
Start by looking at the sequence of projections of the
three-dimensional cube that appear in the planes below the five
hypercubes. Follow the orange and yellow faces of the cube through
the sequence, and note how their relative sizes and distortions help
you to reconstruct their positions within the original cube of which
they are shadows. How do you tell which is in front, and which is
closer and which is farther away? How do you see a square face when
the shadow is a trapezoid? Once you have studied these images, move
to the corresponding views of the hypercube, and try to perform the
same analysis. Which parts are closer and which farther away? Are
the images you see really images of cubical faces of a hypercube?
Remember that the three-dimensional image is really just a ``flat''
shadow of a four-dimensional object; there is a dimension that has
been lost in performing the projection, just as a dimension is lost
when flattening the cube onto its shadow in the plane.
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 additional methods. If we
consider the cube to be hollow, and consisting only of its flat faces
attached to each other, we recognize that this means the cube is made
from six squares that are appropriately glues together. Squares,
however, are two-dimensional, and can be readily understood by our
two-dimensional friends. So another way of helping them understand
the cube would be to cut the edges of the cube and ``unfold'' it until
it lies flat in a plane. We could show this flat pattern to the
two-dimensional people, and could indicate which edges need to be
attached to form the cube. They could not make these attachments in
their own two-dimensional world, but could imaging them being made
and could get some understanding of the cube. In fact, we showed
them shadows of the cube being folded together, they could see even
better how the parts fit together to form the whole.
We could do the same with the hypercube and its eight cubical faces.
In fact, movies of the these cubes folding together to form a
hypercube are available in the electronic supplements to {\it A
Rotation of Cubes}, and are well worth a look.
One final method of giving our flat community a look at a
three-dimensional object would be to pass the object through their
two-dimensional world. If they could see the portion of the object
that intersected 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 images,
to develop a three-dimensional image of the human body being scanned.
The image {\it Iced Cubes\/} investigates the hypercube using this
technique. Here, we show five slices from a sequence of slices
through the hypercube by parallel three-dimensional hyperplanes, all
perpendicular to the long diagonal of the hypercube. The goal is to
try to reconstruct in hour heads the four-dimensional object from the
three-dimensional slices.
To do this, we first look very carefully at how the process lets us
understand a three-dimensional. The corresponding two-dimensional
slices of a cube are shown in the planes below the hypercube slices.
Of course, a cube can be sliced in many different ways, but the most
interesting views come when we slice it by planes perpendicular to its
long diagonal. That is, we start at one corner, and cut off more and more
of the cube until we reach the opposite corner.
As we first start to cut off the corner, our slice is an equilateral
triangle (one vertex along each edge emanating from the corner, and
one edge for each of the three faces meeting at the corner). As we
slice further, the triangle grows in size. Eventually, the slice
reaches the three corners of the cube that are adjacent to the
original corner. As we slice further, the corners of this triangle
become truncated (as our slicing plane begins to cut into the remaining
three faces of the cube). As we progress further, the triangle
becomes more truncated, and at half way through the cube we arrive at
a slice that is a perfectly regular hexagon. At this point, all six
faces of the cube are sliced in exactly the same way. Moving on from
here, the sequence progresses in reverse, with three of the edges of
the hexagon getting longer and three shorter (but the opposite ones
from before). The short edges disappear as our slice passes three
more of the vertices of the cube, leaving us with a triangular slice.
This triangle shrinks down to a point as we finally reach the corner
of the cube opposite to the one where we started.
This sequence of slices is illustrated in {\it Iced Cubes\/} by the flat images
appearing in the planes below the three-dimensional objects. Here we see
the largest triangle when the slices passes the first set of three
vertices, then the truncated triangle, and in the center the regular
hexagon. The sequence is reversed as we move off to the right. The cube
itself is represented by the dark blue lines. These form a view of a cube
when the viewpoint is directly toward a corner of the cube (see the
electronic materials associated with the exhibit for more details on
this).
Having understood the slices of a cube, we can try to use that
understanding to help imagine the analogous slices of a hypercube
formed by hyperplanes perpendicular to its long diagonal. In this
case, the slice first hits one corner and cuts it off. Since there
are four edges emanating from each corner, the hyperplane cuts each of
these edges, producing four vertices in our slice. Four cubical faces
of the hypercube meet at the corner that is being cut off, so each
is sliced by the hyperplane. From each of these cubes, one
corner is being cut off, so the slice forms a triangle in each cube
in exactly the same way as we saw above. Thus each cube contributes
an equilateral triangle to the slice of the hypercube, and we see that
the slice must form a tetrahedron in the slicing hyperplane, a regular
solid known to Plato.
As we cut further into the hypercube, this tetrahedron grows larger until
it comes to the four corners adjacent to the one being cut off. This
situation is shown in {\it A Rotation of Cubes\/} on the far left. Moving
past these four vertices, the corners of the tetrahedron become truncated,
just as the triangle was truncated in the cube slicing sequence; indeed,
at this point, the the four cubical faces are being sliced in exactly this
way, and as the slice moves further, we arrive at the point where they are
sliced as perfect hexagons. The other four cubes of the hypercube are now
beginning to be sliced, and each has a corner removed, so the slice is a
triangle. Together, the four hexagons and four triangles form a
semi-regular figure called the truncated tetrahedron, an Archimedean
solid. This can be seen in the second slice shown in the picture.
Moving past this point, the hexagons move toward truncated triangles
again, and the other four triangles grow. Eventually, the truncated
triangles turn back into triangles as the slice reaches another set of
four vertices of the hypercube. Now all eight cubes are being sliced
in exactly the same way, and each contributes an equilateral triangle
to the slice. These eight triangles form a figure known as an
octahedron, another Platonic solid. It is a perfectly regular object,
half way through the slicing sequence for the hypercube, as shown in
the central image in {\it A Rotation of Cubes}. This slice has six
of the vertices of the hypercube as its corners.
As the hyperplane continues to pass through the hypercube, the
sequence reverses itself. Moving past these six vertices, four of the
faces of the octahedron begin to form hexagons, but the opposite ones
from before. Again, we see the truncated tetrahedron (the fourth
slice in the series shown). Eventually the hyperplane passes four
more of the hypercube's vertices, and the slice becomes a tetrahedron
again (the right-most image in the series). From here, the slice
remains a tetrahedron which shrinks down to and disappears at the
corner of the hypercube opposite the original one where we started.
In the projection of the hypercube that we are using, both the
starting and ending corner are at the center of the figure; see the
electronic materials for a more complete picture of this sequence, and
also several other slicing sequences.
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