THE EVIDENCES FOR A FOURTH DIMENSION
The method necessarily to be employed in the search for the evidences
of a fourth dimension, consists primarily in the formation of the
conceptions of four-dimensional shapes and motions. When we are in
possession of these it is possible to call in the aid of observation,
without them we may have been all our lives in the familiar presence of
a four-dimensional phenomenon without ever recognising its nature.
To take one of the conceptions we have already formed, the turning of a
real thing into its mirror image would be an occurrence which it would
be hard to explain, except on the assumption of a fourth dimension.
We know of no such turning. But there exist a multitude of forms which
show a certain relation to a plane, a relation of symmetry, which
indicates more than an accidental juxtaposition of parts. In organic
life the universal type is of right- and left-handed symmetry, there
is a plane on each side of which the parts correspond. Now we have
seen that in four dimensions a plane takes the place of a line in
three dimensions. In our space, rotation about an axis is the type of
rotation, and the origin of bodies symmetrical about a line as the
earth is symmetrical about an axis can easily be explained. But where
there is symmetry about a plane no simple physical motion, such as we
are accustomed to, suffices to explain it. In our space a symmetrical
object must be built up by equal additions on each side of a central
plane. Such additions about such a plane are as little likely as any
other increments. The probability against the existence of symmetrical
form in inorganic nature is overwhelming in our space, and in organic
forms they would be as difficult of production as any other variety
of configuration. To illustrate this point we may take the child’s
amusement of making from dots of ink on a piece of paper a lifelike
representation of an insect by simply folding the paper over. The
dots spread out on a symmetrical line, and give the impression of a
segmented form with antennæ and legs.
Now seeing a number of such figures we should naturally infer a folding
over. Can, then, a folding over in four-dimensional space account for
the symmetry of organic forms? The folding cannot of course be of the
bodies we see, but it may be of those minute constituents, the ultimate
elements of living matter which, turned in one way or the other, become
right- or left-handed, and so produce a corresponding structure.
There is something in life not included in our conceptions of
mechanical movement. Is this something a four-dimensional movement?
If we look at it from the broadest point of view, there is something
striking in the fact that where life comes in there arises an entirely
different set of phenomena to those of the inorganic world.
The interest and values of life as we know it in ourselves, as we
know it existing around us in subordinate forms, is entirely and
completely different to anything which inorganic nature shows. And in
living beings we have a kind of form, a disposition of matter which
is entirely different from that shown in inorganic matter. Right-
and left-handed symmetry does not occur in the configurations of dead
matter. We have instances of symmetry about an axis, but not about
a plane. It can be argued that the occurrence of symmetry in two
dimensions involves the existence of a three-dimensional process, as
when a stone falls into water and makes rings of ripples, or as when
a mass of soft material rotates about an axis. It can be argued that
symmetry in any number of dimensions is the evidence of an action in
a higher dimensionality. Thus considering living beings, there is an
evidence both in their structure, and their different mode of activity,
of a something coming in from without into the inorganic world.
And the objections which will readily occur, such as those derived from
the forms of twin crystals and the theoretical structure of chemical
molecules, do not invalidate the argument; for in these forms too the
presumable seat of the activity producing them lies in that very minute
region in which we necessarily place the seat of a four-dimensional
mobility.
In another respect also the existence of symmetrical forms is
noteworthy. It is puzzling to conceive how two shapes exactly equal can
exist which are not superposible. Such a pair of symmetrical figures
as the two hands, right and left, show either a limitation in our
power of movement, by which we cannot superpose the one on the other,
or a definite influence and compulsion of space on matter, inflicting
limitations which are additional to those of the proportions of the
parts.
We will, however, put aside the arguments to be drawn from the
consideration of symmetry as inconclusive, retaining one valuable
indication which they afford. If it is in virtue of a four-dimensional
motion that symmetry exists, it is only in the very minute particles
of bodies that that motion is to be found, for there is no such thing
as a bending over in four dimensions of any object of a size which we
can observe. The region of the extremely minute is the one, then, which
we shall have to investigate. We must look for some phenomenon which,
occasioning movements of the kind we know, still is itself inexplicable
as any form of motion which we know.
Now in the theories of the actions of the minute particles of bodies
on one another, and in the motions of the ether, mathematicians
have tacitly assumed that the mechanical principles are the same as
those which prevail in the case of bodies which can be observed, it
has been assumed without proof that the conception of motion being
three-dimensional, holds beyond the region from observations in which
it was formed.
Hence it is not from any phenomenon explained by mathematics that we
can derive a proof of four dimensions. Every phenomenon that has been
explained is explained as three-dimensional. And, moreover, since in
the region of the very minute we do not find rigid bodies acting on
each other at a distance, but elastic substances and continuous fluids
such as ether, we shall have a double task.
We must form the conceptions of the possible movements of elastic and
liquid four-dimensional matter, before we can begin to observe. Let
us, therefore, take the four-dimensional rotation about a plane, and
enquire what it becomes in the case of extensible fluid substances. If
four-dimensional movements exist, this kind of rotation must exist, and
the finer portions of matter must exhibit it.
Consider for a moment a rod of flexible and extensible material. It can
turn about an axis, even if not straight; a ring of india rubber can
turn inside out.
What would this be in the case of four dimensions?
Let us consider a sphere of our three-dimensional matter having a
definite thickness. To represent this thickness let us suppose that
from every point of the sphere in fig. 44 rods project both ways, in
and out, like D and F. We can only see the external portion, because
the internal parts are hidden by the sphere.
[Illustration: Fig. 44.
_Axis of x running towards the observer._]
In this sphere the axis of _x_ is supposed to come towards the
observer, the axis of _z_ to run up, the axis of _y_ to go to the right.
[Illustration: Fig. 45.]
Now take the section determined by the _zy_ plane. This will be a
circle as shown in fig. 45. If we let drop the _x_ axis, this circle
is all we have of the sphere. Letting the _w_ axis now run in the
place of the old _x_ axis we have the space _yzw_, and in this space
all that we have of the sphere is the circle. Fig. 45 then represents
all that there is of the sphere in the space of _yzw_. In this space
it is evident that the rods CD and EF can turn round the circumference
as an axis. If the matter of the spherical shell is sufficiently
extensible to allow the particles C and E to become as widely separated
as they would be in the positions D and F, then the strip of matter
represented by CD and EF and a multitude of rods like them can turn
round the circular circumference.
Thus this particular section of the sphere can turn inside out, and
what holds for any one section holds for all. Hence in four dimensions
the whole sphere can, if extensible turn inside out. Moreover, any part
of it—a bowl-shaped portion, for instance—can turn inside out, and so
on round and round.
This is really no more than we had before in the rotation about a
plane, except that we see that the plane can, in the case of extensible
matter, be curved, and still play the part of an axis.
If we suppose the spherical shell to be of four-dimensional matter, our
representation will be a little different. Let us suppose there to be a
small thickness to the matter in the fourth dimension. This would make
no difference in fig. 44, for that merely shows the view in the _xyz_
space. But when the _x_ axis is let drop, and the _w_ axis comes in,
then the rods CD and EF which represent the matter of the shell, will
have a certain thickness perpendicular to the plane of the paper on
which they are drawn. If they have a thickness in the fourth dimension
they will show this thickness when looked at from the direction of the
_w_ axis.
Supposing these rods, then, to be small slabs strung on the
circumference of the circle in fig. 45, we see that there will not
be in this case either any obstacle to their turning round the
circumference. We can have a shell of extensible material or of fluid
material turning inside out in four dimensions.
And we must remember that in four dimensions there is no such thing as
rotation round an axis. If we want to investigate the motion of fluids
in four dimensions we must take a movement about an axis in our space,
and find the corresponding movement about a plane in four space.
Now, of all the movements which take place in fluids, the most
important from a physical point of view is vortex motion.
A vortex is a whirl or eddy—it is shown in the gyrating wreaths of
dust seen on a summer day; it is exhibited on a larger scale in the
destructive march of a cyclone.
A wheel whirling round will throw off the water on it. But when
this circling motion takes place in a liquid itself it is strangely
persistent. There is, of course, a certain cohesion between the
particles of water by which they mutually impede their motions. But
in a liquid devoid of friction, such that every particle is free from
lateral cohesion on its path of motion, it can be shown that a vortex
or eddy separates from the mass of the fluid a certain portion, which
always remain in that vortex.
The shape of the vortex may alter, but it always consists of the same
particles of the fluid.
Now, a very remarkable fact about such a vortex is that the ends of the
vortex cannot remain suspended and isolated in the fluid. They must
always run to the boundary of the fluid. An eddy in water that remains
half way down without coming to the top is impossible.
The ends of a vortex must reach the boundary of a fluid—the boundary
may be external or internal—a vortex may exist between two objects
in the fluid, terminating one end on each object, the objects being
internal boundaries of the fluid. Again, a vortex may have its ends
linked together, so that it forms a ring. Circular vortex rings of
this description are often seen in puffs of smoke, and that the smoke
travels on in the ring is a proof that the vortex always consists of
the same particles of air.
Let us now enquire what a vortex would be in a four-dimensional fluid.
We must replace the line axis by a plane axis. We should have therefore
a portion of fluid rotating round a plane.
We have seen that the contour of this plane corresponds with the ends
of the axis line. Hence such a four-dimensional vortex must have its
rim on a boundary of the fluid. There would be a region of vorticity
with a contour. If such a rotation were started at one part of a
circular boundary, its edges would run round the boundary in both
directions till the whole interior region was filled with the vortex
sheet.
A vortex in a three-dimensional liquid may consist of a number of
vortex filaments lying together producing a tube, or rod of vorticity.
In the same way we can have in four dimensions a number of vortex
sheets alongside each other, each of which can be thought of as a
bowl-shaped portion of a spherical shell turning inside out. The
rotation takes place at any point not in the space occupied by the
shell, but from that space to the fourth dimension and round back again.
Is there anything analogous to this within the range of our observation?
An electric current answers this description in every respect.
Electricity does not flow through a wire. Its effect travels both ways
from the starting point along the wire. The spark which shows its
passing midway in its circuit is later than that which occurs at points
near its starting point on either side of it.
Moreover, it is known that the action of the current is not in the
wire. It is in the region enclosed by the wire, this is the field of
force, the locus of the exhibition of the effects of the current.
And the necessity of a conducting circuit for a current is exactly
that which we should expect if it were a four-dimensional vortex.
According to Maxwell every current forms a closed circuit, and this,
from the four-dimensional point of view, is the same as saying a vortex
must have its ends on a boundary of the fluid.
Thus, on the hypothesis of a fourth dimension, the rotation of the
fluid ether would give the phenomenon of an electric current. We must
suppose the ether to be full of movement, for the more we examine into
the conditions which prevail in the obscurity of the minute, the more
we find that an unceasing and perpetual motion reigns. Thus we may say
that the conception of the fourth dimension means that there must be a
phenomenon which presents the characteristics of electricity.
We know now that light is an electro-magnetic action, and that so far
from being a special and isolated phenomenon this electric action is
universal in the realm of the minute. Hence, may we not conclude that,
so far from the fourth dimension being remote and far away, being a
thing of symbolic import, a term for the explanation of dubious facts
by a more obscure theory, it is really the most important fact within
our knowledge. Our three-dimensional world is superficial. These
processes, which really lie at the basis of all phenomena of matter,
escape our observation by their minuteness, but reveal to our intellect
an amplitude of motion surpassing any that we can see. In such shapes
and motions there is a realm of the utmost intellectual beauty, and one
to which our symbolic methods apply with a better grace than they do to
those of three dimensions.