THE HIGHER WORLD
It is indeed strange, the manner in which we must begin to think about
the higher world.
Those simplest objects analogous to those which are about us on every
side in our daily experience such as a door, a table, a wheel are
remote and uncognisable in the world of four dimensions, while the
abstract ideas of rotation, stress and strain, elasticity into which
analysis resolves the familiar elements of our daily experience are
transferable and applicable with no difficulty whatever. Thus we are
in the unwonted position of being obliged to construct the daily and
habitual experience of a four-dimensional being, from a knowledge of
the abstract theories of the space, the matter, the motion of it;
instead of, as in our case, passing to the abstract theories from the
richness of sensible things.
What would a wheel be in four dimensions? What the shafting for the
transmission of power which a four-dimensional being would use.
The four-dimensional wheel, and the four-dimensional shafting are
what will occupy us for these few pages. And it is no futile or
insignificant enquiry. For in the attempt to penetrate into the nature
of the higher, to grasp within our ken that which transcends all
analogies, because what we know are merely partial views of it, the
purely material and physical path affords a means of approach pursuing
which we are in less likelihood of error than if we use the more
frequently trodden path of framing conceptions which in their elevation
and beauty seem to us ideally perfect.
For where we are concerned with our own thoughts, the development of
our own ideals, we are as it were on a curve, moving at any moment
in a direction of tangency. Whither we go, what we set up and exalt
as perfect, represents not the true trend of the curve, but our own
direction at the present—a tendency conditioned by the past, and by
a vital energy of motion essential but only true when perpetually
modified. That eternal corrector of our aspirations and ideals, the
material universe draws sublimely away from the simplest things we can
touch or handle to the infinite depths of starry space, in one and
all uninfluenced by what we think or feel, presenting unmoved fact
to which, think it good or think it evil, we can but conform, yet
out of all that impassivity with a reference to something beyond our
individual hopes and fears supporting us and giving us our being.
And to this great being we come with the question: “You, too, what is
your higher?”
Or to put it in a form which will leave our conclusions in the shape
of no barren formula, and attacking the problem on its most assailable
side: “What is the wheel and the shafting of the four-dimensional
mechanic?”
In entering on this enquiry we must make a plan of procedure. The
method which I shall adopt is to trace out the steps of reasoning by
which a being confined to movement in a two-dimensional world could
arrive at a conception of our turning and rotation, and then to apply
an analogous process to the consideration of the higher movements. The
plane being must be imagined as no abstract figure, but as a real body
possessing all three dimensions. His limitation to a plane must be the
result of physical conditions.
We will therefore think of him as of a figure cut out of paper placed
on a smooth plane. Sliding over this plane, and coming into contact
with other figures equally thin as he in the third dimension, he will
apprehend them only by their edges. To him they will be completely
bounded by lines. A “solid” body will be to him a two-dimensional
extent, the interior of which can only be reached by penetrating
through the bounding lines.
Now such a plane being can think of our three-dimensional existence in
two ways.
First, he can think of it as a series of sections, each like the solid
he knows of extending in a direction unknown to him, which stretches
transverse to his tangible universe, which lies in a direction at right
angles to every motion which he made.
Secondly, relinquishing the attempt to think of the three-dimensional
solid body in its entirety he can regard it as consisting of a
number of plane sections, each of them in itself exactly like
the two-dimensional bodies he knows, but extending away from his
two-dimensional space.
A square lying in his space he regards as a solid bounded by four
lines, each of which lies in his space.
A square standing at right angles to his plane appears to him as simply
a line in his plane, for all of it except the line stretches in the
third dimension.
He can think of a three-dimensional body as consisting of a number of
such sections, each of which starts from a line in his space.
Now, since in his world he can make any drawing or model which involves
only two dimensions, he can represent each such upright section as it
actually is, and can represent a turning from a known into the unknown
dimension as a turning from one to another of his known dimensions.
To see the whole he must relinquish part of that which he has, and take
the whole portion by portion.
Consider now a plane being in front of a square, fig. 34. The square
can turn about any point in the plane—say the point A. But it cannot
turn about a line, as AB. For, in order to turn about the line AB,
the square must leave the plane and move in the third dimension. This
motion is out of his range of observation, and is therefore, except for
a process of reasoning, inconceivable to him.
[Illustration: Fig. 34.]
Rotation will therefore be to him rotation about a point. Rotation
about a line will be inconceivable to him.
The result of rotation about a line he can apprehend. He can see the
first and last positions occupied in a half-revolution about the line
AC. The result of such a half revolution is to place the square ABCD
on the left hand instead of on the right hand of the line AC. It would
correspond to a pulling of the whole body ABCD through the line AC,
or to the production of a solid body which was the exact reflection
of it in the line AC. It would be as if the square ABCD turned into
its image, the line AB acting as a mirror. Such a reversal of the
positions of the parts of the square would be impossible in his space.
The occurrence of it would be a proof of the existence of a higher
dimensionality.
Let him now, adopting the conception of a three-dimensional body as
a series of sections lying, each removed a little farther than the
preceding one, in direction at right angles to his plane, regard a
cube, fig. 36, as a series of sections, each like the square which
forms its base, all rigidly connected together.
[Illustration: Fig. 35.]
If now he turns the square about the point A in the plane of _xy_,
each parallel section turns with the square he moves. In each of the
sections there is a point at rest, that vertically over A. Hence he
would conclude that in the turning of a three-dimensional body there
is one line which is at rest. That is a three-dimensional turning in a
turning about a line.
* * * * *
In a similar way let us regard ourselves as limited to a
three-dimensional world by a physical condition. Let us imagine that
there is a direction at right angles to every direction in which we can
move, and that we are prevented from passing in this direction by a
vast solid, that against which in every movement we make we slip as the
plane being slips against his plane sheet.
We can then consider a four-dimensional body as consisting of a series
of sections, each parallel to our space, and each a little farther off
than the preceding on the unknown dimension.
Take the simplest four-dimensional body—one which begins as a cube,
fig. 36, in our space, and consists of sections, each a cube like fig.
36, lying away from our space. If we turn the cube which is its base in
our space about a line, if, _e.g._, in fig. 36 we turn the cube about
the line AB, not only it but each of the parallel cubes moves about a
line. The cube we see moves about the line AB, the cube beyond it about
a line parallel to AB and so on. Hence the whole four-dimensional body
moves about a plane, for the assemblage of these lines is our way of
thinking about the plane which, starting from the line AB in our space,
runs off in the unknown direction.
[Illustration: Fig. 36.]
In this case all that we see of the plane about which the turning takes
place is the line AB.
But it is obvious that the axis plane may lie in our space. A point
near the plane determines with it a three-dimensional space. When it
begins to rotate round the plane it does not move anywhere in this
three-dimensional space, but moves out of it. A point can no more
rotate round a plane in three-dimensional space than a point can move
round a line in two-dimensional space.
We will now apply the second of the modes of representation to this
case of turning about a plane, building up our analogy step by step
from the turning in a plane about a point and that in space about a
line, and so on.
In order to reduce our considerations to those of the greatest
simplicity possible, let us realise how the plane being would think of
the motion by which a square is turned round a line.
Let, fig. 34, ABCD be a square on his plane, and represent the two
dimensions of his space by the axes A_x_ A_y_.
Now the motion by which the square is turned over about the line AC
involves the third dimension.
He cannot represent the motion of the whole square in its turning,
but he can represent the motions of parts of it. Let the third axis
perpendicular to the plane of the paper be called the axis of _z_. Of
the three axes _x_, _y_, _z_, the plane being can represent any two in
his space. Let him then draw, in fig. 35, two axes, _x_ and _z_. Here
he has in his plane a representation of what exists in the plane which
goes off perpendicularly to his space.
In this representation the square would not be shown, for in the plane
of _xz_ simply the line AB of the square is contained.
The plane being then would have before him, in fig. 35, the
representation of one line AB of his square and two axes, _x_ and _z_,
at right angles. Now it would be obvious to him that, by a turning
such as he knows, by a rotation about a point, the line AB can turn
round A, and occupying all the intermediate positions, such as AB_{1},
come after half a revolution to lie as A_x_ produced through A.
Again, just as he can represent the vertical plane through AB, so he
can represent the vertical plane through A´B´, fig. 34, and in a like
manner can see that the line A´B´ can turn about the point A´ till it
lies in the opposite direction from that which it ran in at first.
Now these two turnings are not inconsistent. In his plane, if AB
turned about A, and A´B´ about A´, the consistency of the square would
be destroyed, it would be an impossible motion for a rigid body to
perform. But in the turning which he studies portion by portion there
is nothing inconsistent. Each line in the square can turn in this way,
hence he would realise the turning of the whole square as the sum of
a number of turnings of isolated parts. Such turnings, if they took
place in his plane, would be inconsistent, but by virtue of a third
dimension they are consistent, and the result of them all is that the
square turns about the line AC and lies in a position in which it is
the mirror image of what it was in its first position. Thus he can
realise a turning about a line by relinquishing one of his axes, and
representing his body part by part.
Let us apply this method to the turning of a cube so as to become the
mirror image of itself. In our space we can construct three independent
axes, _x_, _y_, _z_, shown in fig. 36. Suppose that there is a fourth
axis, _w_, at right angles to each and every one of them. We cannot,
keeping all three axes, _x_, _y_, _z_, represent _w_ in our space; but
if we relinquish one of our three axes we can let the fourth axis take
its place, and we can represent what lies in the space, determined by
the two axes we retain and the fourth axis.
[Illustration: Fig. 37.]
Let us suppose that we let the _y_ axis drop, and that we represent
the _w_ axis as occupying its direction. We have in fig. 37 a drawing
of what we should then see of the cube. The square ABCD, remains
unchanged, for that is in the plane of _xz_, and we still have that
plane. But from this plane the cube stretches out in the direction of
the _y_ axis. Now the _y_ axis is gone, and so we have no more of the
cube than the face ABCD. Considering now this face ABCD, we see that
it is free to turn about the line AB. It can rotate in the _x_ to _w_
direction about this line. In fig. 38 it is shown on its way, and it
can evidently continue this rotation till it lies on the other side of
the _z_ axis in the plane of _xz_.
We can also take a section parallel to the face ABCD, and then letting
drop all of our space except the plane of that section, introduce
the _w_ axis, running in the old _y_ direction. This section can be
represented by the same drawing, fig. 38, and we see that it can rotate
about the line on its left until it swings half way round and runs in
the opposite direction to that which it ran in before. These turnings
of the different sections are not inconsistent, and taken all together
they will bring the cube from the position shown in fig. 36 to that
shown in fig. 41.
[Illustration: Fig. 38.]
Since we have three axes at our disposal in our space, we are not
obliged to represent the _w_ axis by any particular one. We may let any
axis we like disappear, and let the fourth axis take its place.
[Illustration: Fig. 39.]
[Illustration: Fig. 40.]
[Illustration: Fig. 41.]
In fig. 36 suppose the _z_ axis to go. We have then simply the plane of
_xy_ and the square base of the cube ACEG, fig. 39, is all that could
be seen of it. Let now the _w_ axis take the place of the _z_ axis and
we have, in fig. 39 again, a representation of the space of _xyw_, in
which all that exists of the cube is its square base. Now, by a turning
of _x_ to _w_, this base can rotate around the line AE, it is shown
on its way in fig. 40, and finally it will, after half a revolution,
lie on the other side of the _y_ axis. In a similar way we may rotate
sections parallel to the base of the _xw_ rotation, and each of them
comes to run in the opposite direction from that which they occupied at
first.
Thus again the cube comes from the position of fig. 36. to that of
fig. 41. In this _x_ to _w_ turning, we see that it takes place by
the rotations of sections parallel to the front face about lines
parallel to AB, or else we may consider it as consisting of the
rotation of sections parallel to the base about lines parallel to AE.
It is a rotation of the whole cube about the plane ABEF. Two separate
sections could not rotate about two separate lines in our space without
conflicting, but their motion is consistent when we consider another
dimension. Just, then, as a plane being can think of rotation about
a line as a rotation about a number of points, these rotations not
interfering as they would if they took place in his two-dimensional
space, so we can think of a rotation about a plane as the rotation
of a number of sections of a body about a number of lines in a plane,
these rotations not being inconsistent in a four-dimensional space as
they are in three-dimensional space.
We are not limited to any particular direction for the lines in the
plane about which we suppose the rotation of the particular sections to
take place. Let us draw the section of the cube, fig. 36, through A,
F, C, H, forming a sloping plane. Now since the fourth dimension is at
right angles to every line in our space it is at right angles to this
section also. We can represent our space by drawing an axis at right
angles to the plane ACEG, our space is then determined by the plane
ACEG, and the perpendicular axis. If we let this axis drop and suppose
the fourth axis, _w_, to take its place, we have a representation of
the space which runs off in the fourth dimension from the plane ACEG.
In this space we shall see simply the section ACEG of the cube, and
nothing else, for one cube does not extend to any distance in the
fourth dimension.
If, keeping this plane, we bring in the fourth dimension, we shall have
a space in which simply this section of the cube exists and nothing
else. The section can turn about the line AF, and parallel sections can
turn about parallel lines. Thus in considering the rotation about a
plane we can draw any lines we like and consider the rotation as taking
place in sections about them.
[Illustration: Fig. 42.]
To bring out this point more clearly let us take two parallel lines,
A and B, in the space of _xyz_, and let CD and EF be two rods running
above and below the plane of _xy_, from these lines. If we turn these
rods in our space about the lines A and B, as the upper end of one,
F, is going down, the lower end of the other, C, will be coming up.
They will meet and conflict. But it is quite possible for these two
rods each of them to turn about the two lines without altering their
relative distances.
To see this suppose the _y_ axis to go, and let the _w_ axis take its
place. We shall see the lines A and B no longer, for they run in the
_y_ direction from the points G and H.
[Illustration: Fig. 43.]
Fig. 43 is a picture of the two rods seen in the space of _xzw_. If
they rotate in the direction shown by the arrows—in the _z_ to _w_
direction—they move parallel to one another, keeping their relative
distances. Each will rotate about its own line, but their rotation will
not be inconsistent with their forming part of a rigid body.
Now we have but to suppose a central plane with rods crossing it
at every point, like CD and EF cross the plane of _xy_, to have an
image of a mass of matter extending equal distances on each side of a
diametral plane. As two of these rods can rotate round, so can all, and
the whole mass of matter can rotate round its diametral plane.
This rotation round a plane corresponds, in four dimensions, to the
rotation round an axis in three dimensions. Rotation of a body round a
plane is the analogue of rotation of a rod round an axis.
In a plane we have rotation round a point, in three-space rotation
round an axis line, in four-space rotation round an axis plane.
The four-dimensional being’s shaft by which he transmits power is a
disk rotating round its central plane—the whole contour corresponds
to the ends of an axis of rotation in our space. He can impart the
rotation at any point and take it off at any other point on the
contour, just as rotation round a line can in three-space be imparted
at one end of a rod and taken off at the other end.
A four-dimensional wheel can easily be described from the analogy of
the representation which a plane being would form for himself of one of
our wheels.
Suppose a wheel to move transverse to a plane, so that the whole disk,
which I will consider to be solid and without spokes, came at the same
time into contact with the plane. It would appear as a circular portion
of plane matter completely enclosing another and smaller portion—the
axle.
This appearance would last, supposing the motion of the wheel to
continue until it had traversed the plane by the extent of its
thickness, when there would remain in the plane only the small disk
which is the section of the axle. There would be no means obvious in
the plane at first by which the axle could be reached, except by going
through the substance of the wheel. But the possibility of reaching it
without destroying the substance of the wheel would be shown by the
continued existence of the axle section after that of the wheel had
disappeared.
In a similar way a four-dimensional wheel moving transverse to our
space would appear first as a solid sphere, completely surrounding
a smaller solid sphere. The outer sphere would represent the wheel,
and would last until the wheel has traversed our space by a distance
equal to its thickness. Then the small sphere alone would remain,
representing the section of the axle. The large sphere could move
round the small one quite freely. Any line in space could be taken as
an axis, and round this line the outer sphere could rotate, while the
inner sphere remained still. But in all these directions of revolution
there would be in reality one line which remained unaltered, that is
the line which stretches away in the fourth direction, forming the
axis of the axle. The four-dimensional wheel can rotate in any number
of planes, but all these planes are such that there is a line at right
angles to them all unaffected by rotation in them.
An objection is sometimes experienced as to this mode of reasoning from
a plane world to a higher dimensionality. How artificial, it is argued,
this conception of a plane world is. If any real existence confined to
a superficies could be shown to exist, there would be an argument for
one relative to which our three-dimensional existence is superficial.
But, both on the one side and the other of the space we are familiar
with, spaces either with less or more than three dimensions are merely
arbitrary conceptions.
In reply to this I would remark that a plane being having one less
dimension than our three would have one-third of our possibilities of
motion, while we have only one-fourth less than those of the higher
space. It may very well be that there may be a certain amount of
freedom of motion which is demanded as a condition of an organised
existence, and that no material existence is possible with a more
limited dimensionality than ours. This is well seen if we try to
construct the mechanics of a two-dimensional world. No tube could
exist, for unless joined together completely at one end two parallel
lines would be completely separate. The possibility of an organic
structure, subject to conditions such as this, is highly problematical;
yet, possibly in the convolutions of the brain there may be a mode of
existence to be described as two-dimensional.
We have but to suppose the increase in surface and the diminution in
mass carried on to a certain extent to find a region which, though
without mobility of the constituents, would have to be described as
two-dimensional.
But, however artificial the conception of a plane being may be, it is
none the less to be used in passing to the conception of a greater
dimensionality than ours, and hence the validity of the first part of
this objection altogether disappears directly we find evidence for such
a state of being.
The second part of the objection has more weight. How is it possible
to conceive that in a four-dimensional space any creatures should be
confined to a three-dimensional existence?
In reply I would say that we know as a matter of fact that life is
essentially a phenomenon of surface. The amplitude of the movements
which we can make is much greater along the surface of the earth than
it is up or down.
Now we have but to conceive the extent of a solid surface increased,
while the motions possible tranverse to it are diminished in the
same proportion, to obtain the image of a three-dimensional world in
four-dimensional space.
And as our habitat is the meeting of air and earth on the world, so
we must think of the meeting place of two as affording the condition
for our universe. The meeting of what two? What can that vastness be
in the higher space which stretches in such a perfect level that our
astronomical observations fail to detect the slightest curvature?
The perfection of the level suggests a liquid—a lake amidst what vast
scenery!—whereon the matter of the universe floats speck-like.
But this aspect of the problem is like what are called in mathematics
boundary conditions.
We can trace out all the consequences of four-dimensional movements
down to their last detail. Then, knowing the mode of action which
would be characteristic of the minutest particles, if they were
free, we can draw conclusions from what they actually do of what the
constraint on them is. Of the two things, the material conditions and
the motion, one is known, and the other can be inferred. If the place
of this universe is a meeting of two, there would be a one-sideness
to space. If it lies so that what stretches away in one direction in
the unknown is unlike what stretches away in the other, then, as far
as the movements which participate in that dimension are concerned,
there would be a difference as to which way the motion took place. This
would be shown in the dissimilarity of phenomena, which, so far as
all three-space movements are concerned, were perfectly symmetrical.
To take an instance, merely, for the sake of precising our ideas,
not for any inherent probability in it; if it could be shown that
the electric current in the positive direction were exactly like the
electric current in the negative direction, except for a reversal of
the components of the motion in three-dimensional space, then the
dissimilarity of the discharge from the positive and negative poles
would be an indication of a one-sideness to our space. The only cause
of difference in the two discharges would be due to a component in
the fourth dimension, which directed in one direction transverse to
our space, met with a different resistance to that which it met when
directed in the opposite direction.