A RECAPITULATION AND EXTENSION OF THE PHYSICAL ARGUMENT
There are two directions of inquiry in which the research for the
physical reality of a fourth dimension can be prosecuted. One is the
investigation of the infinitely great, the other is the investigation
of the infinitely small.
[6] The contents of this chapter are taken from a paper read before
the Philosophical Society of Washington. The mathematical portion
of the paper has appeared in part in the Proceedings of the Royal
Irish Academy under the title, “Cayley’s formulæ of orthogonal
transformation,” Nov. 29th, 1903.
By the measurement of the angles of vast triangles, whose sides are the
distances between the stars, astronomers have sought to determine if
there is any deviation from the values given by geometrical deduction.
If the angles of a celestial triangle do not together equal two right
angles, there would be an evidence for the physical reality of a fourth
dimension.
This conclusion deserves a word of explanation. If space is really
four-dimensional, certain conclusions follow which must be brought
clearly into evidence if we are to frame the questions definitely which
we put to Nature. To account for our limitation let us assume a solid
material sheet against which we move. This sheet must stretch alongside
every object in every direction in which it visibly moves. Every
material body must slip or slide along this sheet, not deviating from
contact with it in any motion which we can observe.
The necessity for this assumption is clearly apparent, if we consider
the analogous case of a suppositionary plane world. If there were
any creatures whose experiences were confined to a plane, we must
account for their limitation. If they were free to move in every space
direction, they would have a three-dimensional motion; hence they must
be physically limited, and the only way in which we can conceive such
a limitation to exist is by means of a material surface against which
they slide. The existence of this surface could only be known to them
indirectly. It does not lie in any direction from them in which the
kinds of motion they know of leads them. If it were perfectly smooth
and always in contact with every material object, there would be no
difference in their relations to it which would direct their attention
to it.
But if this surface were curved—if it were, say, in the form of a vast
sphere—the triangles they drew would really be triangles of a sphere,
and when these triangles are large enough the angles diverge from
the magnitudes they would have for the same lengths of sides if the
surface were plane. Hence by the measurement of triangles of very great
magnitude a plane being might detect a difference from the laws of a
plane world in his physical world, and so be led to the conclusion that
there was in reality another dimension to space—a third dimension—as
well as the two which his ordinary experience made him familiar with.
Now, astronomers have thought it worth while to examine the
measurements of vast triangles drawn from one celestial body to another
with a view to determine if there is anything like a curvature in our
space—that is to say, they have tried astronomical measurements to
find out if the vast solid sheet against which, on the supposition of
a fourth dimension, everything slides is curved or not. These results
have been negative. The solid sheet, if it exists, is not curved or,
being curved, has not a sufficient curvature to cause any observable
deviation from the theoretical value of the angles calculated.
Hence the examination of the infinitely great leads to no decisive
criterion. If it did we should have to decide between the present
theory and that of metageometry.
Coming now to the prosecution of the inquiry in the direction of
the infinitely small, we have to state the question thus: Our laws
of movement are derived from the examination of bodies which move
in three-dimensional space. All our conceptions are founded on the
supposition of a space which is represented analytically by three
independent axes and variations along them—that is, it is a space in
which there are three independent movements. Any motion possible in it
can be compounded out of these three movements, which we may call: up,
right, away.
To examine the actions of the very small portions of matter with the
view of ascertaining if there is any evidence in the phenomena for
the supposition of a fourth dimension of space, we must commence by
clearly defining what the laws of mechanics would be on the supposition
of a fourth dimension. It is of no use asking if the phenomena of the
smallest particles of matter are like—we do not know what. We must
have a definite conception of what the laws of motion would be on the
supposition of the fourth dimension, and then inquire if the phenomena
of the activity of the smaller particles of matter resemble the
conceptions which we have elaborated.
Now, the task of forming these conceptions is by no means one to be
lightly dismissed. Movement in space has many features which differ
entirely from movement on a plane; and when we set about to form the
conception of motion in four dimensions, we find that there is at least
as great a step as from the plane to three-dimensional space.
I do not say that the step is difficult, but I want to point out
that it must be taken. When we have formed the conception of
four-dimensional motion, we can ask a rational question of Nature.
Before we have elaborated our conceptions we are asking if an unknown
is like an unknown—a futile inquiry.
As a matter of fact, four-dimensional movements are in every way simple
and more easy to calculate than three-dimensional movements, for
four-dimensional movements are simply two sets of plane movements put
together.
Without the formation of an experience of four-dimensional bodies,
their shapes and motions, the subject can be but formal—logically
conclusive, not intuitively evident. It is to this logical apprehension
that I must appeal.
It is perfectly simple to form an experiential familiarity with the
facts of four-dimensional movement. The method is analogous to that
which a plane being would have to adopt to form an experiential
familiarity with three-dimensional movements, and may be briefly summed
up as the formation of a compound sense by means of which duration is
regarded as equivalent to extension.
Consider a being confined to a plane. A square enclosed by four lines
will be to him a solid, the interior of which can only be examined by
breaking through the lines. If such a square were to pass transverse to
his plane, it would immediately disappear. It would vanish, going in no
direction to which he could point.
If, now, a cube be placed in contact with his plane, its surface of
contact would appear like the square which we have just mentioned.
But if it were to pass transverse to his plane, breaking through it,
it would appear as a lasting square. The three-dimensional matter will
give a lasting appearance in circumstances under which two-dimensional
matter will at once disappear.
Similarly, a four-dimensional cube, or, as we may call it, a tesseract,
which is generated from a cube by a movement of every part of the cube
in a fourth direction at right angles to each of the three visible
directions in the cube, if it moved transverse to our space, would
appear as a lasting cube.
A cube of three-dimensional matter, since it extends to no distance at
all in the fourth dimension, would instantly disappear, if subjected
to a motion transverse to our space. It would disappear and be gone,
without it being possible to point to any direction in which it had
moved.
All attempts to visualise a fourth dimension are futile. It must be
connected with a time experience in three space.
The most difficult notion for a plane being to acquire would be that of
rotation about a line. Consider a plane being facing a square. If he
were told that rotation about a line were possible, he would move his
square this way and that. A square in a plane can rotate about a point,
but to rotate about a line would seem to the plane being perfectly
impossible. How could those parts of his square which were on one side
of an edge come to the other side without the edge moving? He could
understand their reflection in the edge. He could form an idea of the
looking-glass image of his square lying on the opposite side of the
line of an edge, but by no motion that he knows of can he make the
actual square assume that position. The result of the rotation would be
like reflection in the edge, but it would be a physical impossibility
to produce it in the plane.
The demonstration of rotation about a line must be to him purely
formal. If he conceived the notion of a cube stretching out in an
unknown direction away from his plane, then he can see the base of
it, his square in the plane, rotating round a point. He can likewise
apprehend that every parallel section taken at successive intervals in
the unknown direction rotates in like manner round a point. Thus he
would come to conclude that the whole body rotates round a line—the
line consisting of the succession of points round which the plane
sections rotate. Thus, given three axes, _x_, _y_, _z_, if _x_ rotates
to take the place of _y_, and _y_ turns so as to point to negative
_x_, then the third axis remaining unaffected by this turning is the
axis about which the rotation takes place. This, then, would have to be
his criterion of the axis of a rotation—that which remains unchanged
when a rotation of every plane section of a body takes place.
There is another way in which a plane being can think about
three-dimensional movements; and, as it affords the type by which we
can most conveniently think about four-dimensional movements, it will
be no loss of time to consider it in detail.
[Illustration: Fig. 1 (129).]
We can represent the plane being and his object by figures cut out of
paper, which slip on a smooth surface. The thickness of these bodies
must be taken as so minute that their extension in the third dimension
escapes the observation of the plane being, and he thinks about them
as if they were mathematical plane figures in a plane instead of being
material bodies capable of moving on a plane surface. Let A_x_, A_y_
be two axes and ABCD a square. As far as movements in the plane are
concerned, the square can rotate about a point A, for example. It
cannot rotate about a side, such as AC.
But if the plane being is aware of the existence of a third dimension
he can study the movements possible in the ample space, taking his
figure portion by portion.
His plane can only hold two axes. But, since it can hold two, he is
able to represent a turning into the third dimension if he neglects one
of his axes and represents the third axis as lying in his plane. He can
make a drawing in his plane of what stands up perpendicularly from his
plane. Let A_z_ be the axis, which stands perpendicular to his plane at
A. He can draw in his plane two lines to represent the two axes, A_x_
and A_z_. Let Fig. 2 be this drawing. Here the _z_ axis has taken the
place of the _y_ axis, and the plane of A_x_ A_z_ is represented in his
plane. In this figure all that exists of the square ABCD will be the
line AB.
[Illustration: Fig. 2 (130).]
The square extends from this line in the _y_ direction, but more of
that direction is represented in Fig. 2. The plane being can study the
turning of the line AB in this diagram. It is simply a case of plane
turning around the point A. The line AB occupies intermediate portions
like AB_{1} and after half a revolution will lie on A_x_ produced
through A.
Now, in the same way, the plane being can take another point, A´, and
another line, A´B´, in his square. He can make the drawing of the two
directions at A´, one along A´B´, the other perpendicular to his plane.
He will obtain a figure precisely similar to Fig. 2, and will see that,
as AB can turn around A, so A´C´ around A.
In this turning AB and A´B´ would not interfere with each other, as
they would if they moved in the plane around the separate points A and
A´.
Hence the plane being would conclude that a rotation round a line was
possible. He could see his square as it began to make this turning. He
could see it half way round when it came to lie on the opposite side of
the line AC. But in intermediate portions he could not see it, for it
runs out of the plane.
Coming now to the question of a four-dimensional body, let us conceive
of it as a series of cubic sections, the first in our space, the rest
at intervals, stretching away from our space in the unknown direction.
We must not think of a four-dimensional body as formed by moving a
three-dimensional body in any direction which we can see.
Refer for a moment to Fig. 3. The point A, moving to the right, traces
out the line AC. The line AC, moving away in a new direction, traces
out the square ACEG at the base of the cube. The square AEGC, moving
in a new direction, will trace out the cube ACEGBDHF. The vertical
direction of this last motion is not identical with any motion possible
in the plane of the base of the cube. It is an entirely new direction,
at right angles to every line that can be drawn in the base. To trace
out a tesseract the cube must move in a new direction—a direction at
right angles to any and every line that can be drawn in the space of
the cube.
The cubic sections of the tesseract are related to the cube we see, as
the square sections of the cube are related to the square of its base
which a plane being sees.
Let us imagine the cube in our space, which is the base of a tesseract,
to turn about one of its edges. The rotation will carry the whole body
with it, and each of the cubic sections will rotate. The axis we see
in our space will remain unchanged, and likewise the series of axes
parallel to it about which each of the parallel cubic sections rotates.
The assemblage of all of these is a plane.
Hence in four dimensions a body rotates about a plane. There is no such
thing as rotation round an axis.
We may regard the rotation from a different point of view. Consider
four independent axes each at right angles to all the others, drawn in
a four-dimensional body. Of these four axes we can see any three. The
fourth extends normal to our space.
Rotation is the turning of one axis into a second, and the second
turning to take the place of the negative of the first. It involves
two axes. Thus, in this rotation of a four-dimensional body, two axes
change and two remain at rest. Four-dimensional rotation is therefore a
turning about a plane.
As in the case of a plane being, the result of rotation about a
line would appear as the production of a looking-glass image of the
original object on the other side of the line, so to us the result
of a four-dimensional rotation would appear like the production of a
looking-glass image of a body on the other side of a plane. The plane
would be the axis of the rotation, and the path of the body between its
two appearances would be unimaginable in three-dimensional space.
[Illustration: Fig. 3 (131).]
Let us now apply the method by which a plane being could examine
the nature of rotation about a line in our examination of rotation
about a plane. Fig. 3 represents a cube in our space, the three axes
_x_, _y_, _z_ denoting its three dimensions. Let _w_ represent the
fourth dimension. Now, since in our space we can represent any three
dimensions, we can, if we choose, make a representation of what is
in the space determined by the three axes _x_, _z_, _w_. This is a
three-dimensional space determined by two of the axes we have drawn,
_x_ and _z_, and in place of _y_ the fourth axis, _w_. We cannot,
keeping _x_ and _z_, have both _y_ and _w_ in our space; so we will
let _y_ go and draw _w_ in its place. What will be our view of the cube?
Evidently we shall have simply the square that is in the plane of _xz_,
the square ACDB. The rest of the cube stretches in the _y_ direction,
and, as we have none of the space so determined, we have only the face
of the cube. This is represented in fig. 4.
[Illustration: Fig. 4 (132).]
Now, suppose the whole cube to be turned from the _x_ to the _w_
direction. Conformably with our method, we will not take the whole of
the cube into consideration at once, but will begin with the face ABCD.
Let this face begin to turn. Fig. 5 represents one of the positions it
will occupy; the line AB remains on the _z_ axis. The rest of the face
extends between the _x_ and the _w_ direction.
[Illustration: Fig. 5 (133).]
Now, since we can take any three axes, let us look at what lies in the
space of _zyw_, and examine the turning there. We must now let the _z_
axis disappear and let the _w_ axis run in the direction in which the
_z_ ran.
Making this representation, what do we see of the cube? Obviously we
see only the lower face. The rest of the cube lies in the space of
_xyz_. In the space of _xyz_ we have merely the base of the cube lying
in the plane of _xy_, as shown in fig. 6.
[Illustration: Fig. 6 (134).]
Now let the _x_ to _w_ turning take place. The square ACEG will turn
about the line AE. This edge will remain along the _y_ axis and will be
stationary, however far the square turns.
Thus, if the cube be turned by an _x_ to _w_ turning, both the edge AB
and the edge AC remain stationary; hence the whole face ABEF in the
_yz_ plane remains fixed. The turning has taken place about the face
ABEF.
[Illustration: Fig. 7 (135).]
Suppose this turning to continue till AC runs to the left from
A. The cube will occupy the position shown in fig. 8. This is
the looking-glass image of the cube in fig. 3. By no rotation in
three-dimensional space can the cube be brought from the position in
fig. 3 to that shown in fig. 8.
[Illustration: Fig. 8 (136).]
We can think of this turning as a turning of the face ABCD about AB,
and a turning of each section parallel to ABCD round the vertical line
in which it intersects the face ABEF, the space in which the turning
takes place being a different one from that in which the cube lies.
One of the conditions, then, of our inquiry in the direction of the
infinitely small is that we form the conception of a rotation about
a plane. The production of a body in a state in which it presents
the appearance of a looking-glass image of its former state is the
criterion for a four-dimensional rotation.
There is some evidence for the occurrence of such transformations
of bodies in the change of bodies from those which produce a
right-handed polarisation of light to those which produce a left-handed
polarisation; but this is not a point to which any very great
importance can be attached.
Still, in this connection, let me quote a remark from Prof. John G.
McKendrick’s address on Physiology before the British Association
at Glasgow. Discussing the possibility of the hereditary production
of characteristics through the material structure of the ovum, he
estimates that in it there exist 12,000,000,000 biophors, or ultimate
particles of living matter, a sufficient number to account for
hereditary transmission, and observes: “Thus it is conceivable that
vital activities may also be determined by the kind of motion that
takes place in the molecules of that which we speak of as living
matter. It may be different in kind from some of the motions known to
physicists, and it is conceivable that life may be the transmission
to dead matter, the molecules of which have already a special kind of
motion, of a form of motion _sui generis_.”
Now, in the realm of organic beings symmetrical structures—those with a
right and left symmetry—are everywhere in evidence. Granted that four
dimensions exist, the simplest turning produces the image form, and by
a folding-over structures could be produced, duplicated right and left,
just as is the case of symmetry in a plane.
Thus one very general characteristic of the forms of organisms could
be accounted for by the supposition that a four-dimensional motion was
involved in the process of life.
But whether four-dimensional motions correspond in other respects to
the physiologist’s demand for a special kind of motion, or not, I
do not know. Our business is with the evidence for their existence
in physics. For this purpose it is necessary to examine into the
significance of rotation round a plane in the case of extensible and of
fluid matter.
Let us dwell a moment longer on the rotation of a rigid body. Looking
at the cube in fig. 3, which turns about the face of ABFE, we see that
any line in the face can take the place of the vertical and horizontal
lines we have examined. Take the diagonal line AF and the section
through it to GH. The portions of matter which were on one side of AF
in this section in fig. 3 are on the opposite side of it in fig. 8.
They have gone round the line AF. Thus the rotation round a face can be
considered as a number of rotations of sections round parallel lines in
it.
The turning about two different lines is impossible in
three-dimensional space. To take another illustration, suppose A and
B are two parallel lines in the _xy_ plane, and let CD and EF be two
rods crossing them. Now, in the space of _xyz_ if the rods turn round
the lines A and B in the same direction they will make two independent
circles.
When the end F is going down the end C will be coming up. They will
meet and conflict.
[Illustration: Fig. 9 (137).]
But if we rotate the rods about the plane of AB by the _z_ to _w_
rotation these movements will not conflict. Suppose all the figure
removed with the exception of the plane _xz_, and from this plane draw
the axis of _w_, so that we are looking at the space of _xzw_.
Here, fig. 10, we cannot see the lines A and B. We see the points G and
H, in which A and B intercept the _x_ axis, but we cannot see the lines
themselves, for they run in the _y_ direction, and that is not in our
drawing.
Now, if the rods move with the _z_ to _w_ rotation they will turn in
parallel planes, keeping their relative positions. The point D, for
instance, will describe a circle. At one time it will be above the line
A, at another time below it. Hence it rotates round A.
[Illustration: Fig. 10 (138).]
Not only two rods but any number of rods crossing the plane will move
round it harmoniously. We can think of this rotation by supposing the
rods standing up from one line to move round that line and remembering
that it is not inconsistent with this rotation for the rods standing up
along another line also to move round it, the relative positions of all
the rods being preserved. Now, if the rods are thick together, they may
represent a disk of matter, and we see that a disk of matter can rotate
round a central plane.
Rotation round a plane is exactly analogous to rotation round an axis
in three dimensions. If we want a rod to turn round, the ends must be
free; so if we want a disk of matter to turn round its central plane
by a four-dimensional turning, all the contour must be free. The whole
contour corresponds to the ends of the rod. Each point of the contour
can be looked on as the extremity of an axis in the body, round each
point of which there is a rotation of the matter in the disk.
If the one end of a rod be clamped, we can twist the rod, but not turn
it round; so if any part of the contour of a disk is clamped we can
impart a twist to the disk, but not turn it round its central plane. In
the case of extensible materials a long, thin rod will twist round its
axis, even when the axis is curved, as, for instance, in the case of a
ring of India rubber.
In an analogous manner, in four dimensions we can have rotation round
a curved plane, if I may use the expression. A sphere can be turned
inside out in four dimensions.
[Illustration: Fig. 11 (139).]
Let fig. 11 represent a spherical surface, on each side of which a
layer of matter exists. The thickness of the matter is represented by
the rods CD and EF, extending equally without and within.
[Illustration: Fig. 12 (140).]
Now, take the section of the sphere by the _yz_ plane we have a
circle—fig. 12. Now, let the _w_ axis be drawn in place of the _x_ axis
so that we have the space of _yzw_ represented. In this space all that
there will be seen of the sphere is the circle drawn.
Here we see that there is no obstacle to prevent the rods turning
round. If the matter is so elastic that it will give enough for the
particles at E and C to be separated as they are at F and D, they
can rotate round to the position D and F, and a similar motion is
possible for all other particles. There is no matter or obstacle to
prevent them from moving out in the _w_ direction, and then on round
the circumference as an axis. Now, what will hold for one section will
hold for all, as the fourth dimension is at right angles to all the
sections which can be made of the sphere.
We have supposed the matter of which the sphere is composed to be
three-dimensional. If the matter had a small thickness in the fourth
dimension, there would be a slight thickness in fig. 12 above the
plane of the paper—a thickness equal to the thickness of the matter
in the fourth dimension. The rods would have to be replaced by thin
slabs. But this would make no difference as to the possibility of the
rotation. This motion is discussed by Newcomb in the first volume of
the _American Journal of Mathematics_.
Let us now consider, not a merely extensible body, but a liquid one. A
mass of rotating liquid, a whirl, eddy, or vortex, has many remarkable
properties. On first consideration we should expect the rotating mass
of liquid immediately to spread off and lose itself in the surrounding
liquid. The water flies off a wheel whirled round, and we should expect
the rotating liquid to be dispersed. But see the eddies in a river
strangely persistent. The rings that occur in puffs of smoke and last
so long are whirls or vortices curved round so that their opposite ends
join together. A cyclone will travel over great distances.
Helmholtz was the first to investigate the properties of vortices.
He studied them as they would occur in a perfect fluid—that is, one
without friction of one moving portion or another. In such a medium
vortices would be indestructible. They would go on for ever, altering
their shape, but consisting always of the same portion of the fluid.
But a straight vortex could not exist surrounded entirely by the fluid.
The ends of a vortex must reach to some boundary inside or outside the
fluid.
A vortex which is bent round so that its opposite ends join is capable
of existing, but no vortex has a free end in the fluid. The fluid
round the vortex is always in motion, and one produces a definite
movement in another.
Lord Kelvin has proposed the hypothesis that portions of a fluid
segregated in vortices account for the origin of matter. The properties
of the ether in respect of its capacity of propagating disturbances
can be explained by the assumption of vortices in it instead of by a
property of rigidity. It is difficult to conceive, however, of any
arrangement of the vortex rings and endless vortex filaments in the
ether.
Now, the further consideration of four-dimensional rotations shows the
existence of a kind of vortex which would make an ether filled with a
homogeneous vortex motion easily thinkable.
To understand the nature of this vortex, we must go on and take a
step by which we accept the full significance of the four-dimensional
hypothesis. Granted four-dimensional axes, we have seen that a rotation
of one into another leaves two unaltered, and these two form the axial
plane about which the rotation takes place. But what about these two?
Do they necessarily remain motionless? There is nothing to prevent a
rotation of these two, one into the other, taking place concurrently
with the first rotation. This possibility of a double rotation deserves
the most careful attention, for it is the kind of movement which is
distinctly typical of four dimensions.
Rotation round a plane is analogous to rotation round an axis. But in
three-dimensional space there is no motion analogous to the double
rotation, in which, while axis 1 changes into axis 2, axis 3 changes
into axis 4.
Consider a four-dimensional body, with four independent axes, _x_,
_y_, _z_, _w_. A point in it can move in only one direction at a given
moment. If the body has a velocity of rotation by which the _x_ axis
changes into the _y_ axis and all parallel sections move in a similar
manner, then the point will describe a circle. If, now, in addition
to the rotation by which the _x_ axis changes into the _y_ axis the
body has a rotation by which the _z_ axis turns into the _w_ axis, the
point in question will have a double motion in consequence of the two
turnings. The motions will compound, and the point will describe a
circle, but not the same circle which it would describe in virtue of
either rotation separately.
We know that if a body in three-dimensional space is given two
movements of rotation they will combine into a single movement of
rotation round a definite axis. It is in no different condition
from that in which it is subjected to one movement of rotation. The
direction of the axis changes; that is all. The same is not true about
a four-dimensional body. The two rotations, _x_ to _y_ and _z_ to _w_,
are independent. A body subject to the two is in a totally different
condition to that which it is in when subject to one only. When subject
to a rotation such as that of _x_ to _y_, a whole plane in the body,
as we have seen, is stationary. When subject to the double rotation
no part of the body is stationary except the point common to the two
planes of rotation.
If the two rotations are equal in velocity, every point in the body
describes a circle. All points equally distant from the stationary
point describe circles of equal size.
We can represent a four-dimensional sphere by means of two diagrams,
in one of which we take the three axes, _x_, _y_, _z_; in the
other the axes _x_, _w_, and _z_. In fig. 13 we have the view of a
four-dimensional sphere in the space of _xyz_. Fig. 13 shows all that
we can see of the four sphere in the space of _xyz_, for it represents
all the points in that space, which are at an equal distance from the
centre.
Let us now take the _xz_ section, and let the axis of _w_ take the
place of the _y_ axis. Here, in fig. 14, we have the space of _xzw_.
In this space we have to take all the points which are at the same
distance from the centre, consequently we have another sphere. If we
had a three-dimensional sphere, as has been shown before, we should
have merely a circle in the _xzw_ space, the _xz_ circle seen in the
space of _xzw_. But now, taking the view in the space of _xzw_, we have
a sphere in that space also. In a similar manner, whichever set of
three axes we take, we obtain a sphere.
[Illustration: _Showing axes xyz_
Fig. 13 (141).]
[Illustration: _Showing axes xwz_
Fig. 14 (142).]
In fig. 13, let us imagine the rotation in the direction _xy_ to be
taking place. The point _x_ will turn to _y_, and _p_ to _p´_. The axis
_zz´_ remains stationary, and this axis is all of the plane _zw_ which
we can see in the space section exhibited in the figure.
In fig. 14, imagine the rotation from _z_ to _w_ to be taking place.
The _w_ axis now occupies the position previously occupied by the _y_
axis. This does not mean that the _w_ axis can coincide with the _y_
axis. It indicates that we are looking at the four-dimensional sphere
from a different point of view. Any three-space view will show us three
axes, and in fig. 14 we are looking at _xzw_.
The only part that is identical in the two diagrams is the circle of
the _x_ and _z_ axes, which axes are contained in both diagrams. Thus
the plane _zxz´_ is the same in both, and the point _p_ represents the
same point in both diagrams. Now, in fig. 14 let the _zw_ rotation
take place, the _z_ axis will turn toward the point _w_ of the _w_
axis, and the point _p_ will move in a circle about the point _x_.
Thus in fig. 13 the point _p_ moves in a circle parallel to the _xy_
plane; in fig. 14 it moves in a circle parallel to the _zw_ plane,
indicated by the arrow.
Now, suppose both of these independent rotations compounded, the point
_p_ will move in a circle, but this circle will coincide with neither
of the circles in which either one of the rotations will take it. The
circle the point _p_ will move in will depend on its position on the
surface of the four sphere.
In this double rotation, possible in four-dimensional space, there
is a kind of movement totally unlike any with which we are familiar
in three-dimensional space. It is a requisite preliminary to the
discussion of the behaviour of the small particles of matter,
with a view to determining whether they show the characteristics
of four-dimensional movements, to become familiar with the main
characteristics of this double rotation. And here I must rely on a
formal and logical assent rather than on the intuitive apprehension,
which can only be obtained by a more detailed study.
In the first place this double rotation consists in two varieties or
kinds, which we will call the A and B kinds. Consider four axes, _x_,
_y_, _z_, _w_. The rotation of _x_ to _y_ can be accompanied with the
rotation of _z_ to _w_. Call this the A kind.
But also the rotation of _x_ to _y_ can be accompanied by the rotation,
of not _z_ to _w_, but _w_ to _z_. Call this the B kind.
They differ in only one of the component rotations. One is not the
negative of the other. It is the semi-negative. The opposite of an
_x_ to _y_, _z_ to _w_ rotation would be _y_ to _x_, _w_ to _z_. The
semi-negative is _x_ to _y_ and _w_ to _z_.
If four dimensions exist and we cannot perceive them, because the
extension of matter is so small in the fourth dimension that all
movements are withheld from direct observation except those which are
three-dimensional, we should not observe these double rotations, but
only the effects of them in three-dimensional movements of the type
with which we are familiar.
If matter in its small particles is four-dimensional, we should expect
this double rotation to be a universal characteristic of the atoms
and molecules, for no portion of matter is at rest. The consequences
of this corpuscular motion can be perceived, but only under the form
of ordinary rotation or displacement. Thus, if the theory of four
dimensions is true, we have in the corpuscles of matter a whole world
of movement, which we can never study directly, but only by means of
inference.
The rotation A, as I have defined it, consists of two equal
rotations—one about the plane of _zw_, the other about the plane
of _xy_. It is evident that these rotations are not necessarily
equal. A body may be moving with a double rotation, in which these
two independent components are not equal; but in such a case we can
consider the body to be moving with a composite rotation—a rotation of
the A or B kind and, in addition, a rotation about a plane.
If we combine an A and a B movement, we obtain a rotation about a
plane; for, the first being _x_ to _y_ and _z_ to _w_, and the second
being _x_ to _y_ and _w_ to _z_, when they are put together the _z_
to _w_ and _w_ to _z_ rotations neutralise each other, and we obtain
an _x_ to _y_ rotation only, which is a rotation about the plane of
_zw_. Similarly, if we take a B rotation, _y_ to _x_ and _z_ to _w_,
we get, on combining this with the A rotation, a rotation of _z_ to
_w_ about the _xy_ plane. In this case the plane of rotation is in the
three-dimensional space of _xyz_, and we have—what has been described
before—a twisting about a plane in our space.
Consider now a portion of a perfect liquid having an A motion. It
can be proved that it possesses the properties of a vortex. It
forms a permanent individuality—a separated-out portion of the
liquid—accompanied by a motion of the surrounding liquid. It has
properties analogous to those of a vortex filament. But it is not
necessary for its existence that its ends should reach the boundary of
the liquid. It is self-contained and, unless disturbed, is circular in
every section.
[Illustration: Fig. 15 (143).]
If we suppose the ether to have its properties of transmitting
vibration given it by such vortices, we must inquire how they lie
together in four-dimensional space. Placing a circular disk on a plane
and surrounding it by six others, we find that if the central one is
given a motion of rotation, it imparts to the others a rotation which
is antagonistic in every two adjacent ones. If A goes round, as shown
by the arrow, B and C will be moving in opposite ways, and each tends
to destroy the motion of the other.
Now, if we suppose spheres to be arranged in a corresponding manner
in three-dimensional space, they will be grouped in figures which are
for three-dimensional space what hexagons are for plane space. If a
number of spheres of soft clay be pressed together, so as to fill up
the interstices, each will assume the form of a fourteen-sided figure
called a tetrakaidecagon.
Now, assuming space to be filled with such tetrakaidecagons, and
placing a sphere in each, it will be found that one sphere is touched
by eight others. The remaining six spheres of the fourteen which
surround the central one will not touch it, but will touch three of
those in contact with it. Hence, if the central sphere rotates, it
will not necessarily drive those around it so that their motions will
be antagonistic to each other, but the velocities will not arrange
themselves in a systematic manner.
In four-dimensional space the figure which forms the next term of the
series hexagon, tetrakaidecagon, is a thirty-sided figure. It has for
its faces ten solid tetrakaidecagons and twenty hexagonal prisms. Such
figures will exactly fill four-dimensional space, five of them meeting
at every point. If, now, in each of these figures we suppose a solid
four-dimensional sphere to be placed, any one sphere is surrounded by
thirty others. Of these it touches ten, and, if it rotates, it drives
the rest by means of these. Now, if we imagine the central sphere to be
given an A or a B rotation, it will turn the whole mass of sphere round
in a systematic manner. Suppose four-dimensional space to be filled
with such spheres, each rotating with a double rotation, the whole mass
would form one consistent system of motion, in which each one drove
every other one, with no friction or lagging behind.
Every sphere would have the same kind of rotation. In three-dimensional
space, if one body drives another round the second body rotates
with the opposite kind of rotation; but in four-dimensional space
these four-dimensional spheres would each have the double negative
of the rotation of the one next it, and we have seen that the
double negative of an A or B rotation is still an A or B rotation.
Thus four-dimensional space could be filled with a system of
self-preservative living energy. If we imagine the four-dimensional
spheres to be of liquid and not of solid matter, then, even if the
liquid were not quite perfect and there were a slight retarding effect
of one vortex on another, the system would still maintain itself.
In this hypothesis we must look on the ether as possessing energy,
and its transmission of vibrations, not as the conveying of a motion
imparted from without, but as a modification of its own motion.
We are now in possession of some of the conceptions of four-dimensional
mechanics, and will turn aside from the line of their development
to inquire if there is any evidence of their applicability to the
processes of nature.
Is there any mode of motion in the region of the minute which, giving
three-dimensional movements for its effect, still in itself escapes the
grasp of our mechanical theories? I would point to electricity. Through
the labours of Faraday and Maxwell we are convinced that the phenomena
of electricity are of the nature of the stress and strain of a medium;
but there is still a gap to be bridged over in their explanation—the
laws of elasticity, which Maxwell assumes, are not those of ordinary
matter. And, to take another instance: a magnetic pole in the
neighbourhood of a current tends to move. Maxwell has shown that the
pressures on it are analogous to the velocities in a liquid which would
exist if a vortex took the place of the electric current: but we cannot
point out the definite mechanical explanation of these pressures. There
must be some mode of motion of a body or of the medium in virtue of
which a body is said to be electrified.
Take the ions which convey charges of electricity 500 times greater in
proportion to their mass than are carried by the molecules of hydrogen
in electrolysis. In respect of what motion can these ions be said to
be electrified? It can be shown that the energy they possess is not
energy of rotation. Think of a short rod rotating. If it is turned
over it is found to be rotating in the opposite direction. Now, if
rotation in one direction corresponds to positive electricity, rotation
in the opposite direction corresponds to negative electricity, and the
smallest electrified particles would have their charges reversed by
being turned over—an absurd supposition.
If we fix on a mode of motion as a definition of electricity, we must
have two varieties of it, one for positive and one for negative; and a
body possessing the one kind must not become possessed of the other by
any change in its position.
All three-dimensional motions are compounded of rotations and
translations, and none of them satisfy this first condition for serving
as a definition of electricity.
But consider the double rotation of the A and B kinds. A body rotating
with the A motion cannot have its motion transformed into the B kind
by being turned over in any way. Suppose a body has the rotation _x_
to _y_ and _z_ to _w_. Turning it about the _xy_ plane, we reverse the
direction of the motion _x_ to _y_. But we also reverse the _z_ to _w_
motion, for the point at the extremity of the positive _z_ axis is
now at the extremity of the negative _z_ axis, and since we have not
interfered with its motion it goes in the direction of position _w_.
Hence we have _y_ to _x_ and _w_ to _z_, which is the same as _x_ to
_y_ and _z_ to _w_. Thus both components are reversed, and there is the
A motion over again. The B kind is the semi-negative, with only one
component reversed.
Hence a system of molecules with the A motion would not destroy it in
one another, and would impart it to a body in contact with them. Thus A
and B motions possess the first requisite which must be demanded in any
mode of motion representative of electricity.
Let us trace out the consequences of defining positive electricity as
an A motion and negative electricity as a B motion. The combination of
positive and negative electricity produces a current. Imagine a vortex
in the ether of the A kind and unite with this one of the B kind. An
A motion and B motion produce rotation round a plane, which is in the
ether a vortex round an axial surface. It is a vortex of the kind we
represent as a part of a sphere turning inside out. Now such a vortex
must have its rim on a boundary of the ether—on a body in the ether.
Let us suppose that a conductor is a body which has the property of
serving as the terminal abutment of such a vortex. Then the conception
we must form of a closed current is of a vortex sheet having its edge
along the circuit of the conducting wire. The whole wire will then be
like the centres on which a spindle turns in three-dimensional space,
and any interruption of the continuity of the wire will produce a
tension in place of a continuous revolution.
As the direction of the rotation of the vortex is from a three-space
direction into the fourth dimension and back again, there will be no
direction of flow to the current; but it will have two sides, according
to whether _z_ goes to _w_ or _z_ goes to negative _w_.
We can draw any line from one part of the circuit to another; then the
ether along that line is rotating round its points.
This geometric image corresponds to the definition of an electric
circuit. It is known that the action does not lie in the wire, but in
the medium, and it is known that there is no direction of flow in the
wire.
No explanation has been offered in three-dimensional mechanics of how
an action can be impressed throughout a region and yet necessarily
run itself out along a closed boundary, as is the case in an electric
current. But this phenomenon corresponds exactly to the definition of a
four-dimensional vortex.
If we take a very long magnet, so long that one of its poles is
practically isolated, and put this pole in the vicinity of an electric
circuit, we find that it moves.
Now, assuming for the sake of simplicity that the wire which determines
the current is in the form of a circle, if we take a number of small
magnets and place them all pointing in the same direction normal to
the plane of the circle, so that they fill it and the wire binds them
round, we find that this sheet of magnets has the same effect on
the magnetic pole that the current has. The sheet of magnets may be
curved, but the edge of it must coincide with the wire. The collection
of magnets is then equivalent to the vortex sheet, and an elementary
magnet to a part of it. Thus, we must think of a magnet as conditioning
a rotation in the ether round the plane which bisects at right angles
the line joining its poles.
If a current is started in a circuit, we must imagine vortices like
bowls turning themselves inside out, starting from the contour. In
reaching a parallel circuit, if the vortex sheet were interrupted and
joined momentarily to the second circuit by a free rim, the axis plane
would lie between the two circuits, and a point on the second circuit
opposite a point on the first would correspond to a point opposite
to it on the first; hence we should expect a current in the opposite
direction in the second circuit. Thus the phenomena of induction are
not inconsistent with the hypothesis of a vortex about an axial plane.
In four-dimensional space, in which all four dimensions were
commensurable, the intensity of the action transmitted by the medium
would vary inversely as the cube of the distance. Now, the action of
a current on a magnetic pole varies inversely as the square of the
distance; hence, over measurable distances the extension of the ether
in the fourth dimension cannot be assumed as other than small in
comparison with those distances.
If we suppose the ether to be filled with vortices in the shape of
four-dimensional spheres rotating with the A motion, the B motion would
correspond to electricity in the one-fluid theory. There would thus
be a possibility of electricity existing in two forms, statically,
by itself, and, combined with the universal motion, in the form of a
current.
To arrive at a definite conclusion it will be necessary to investigate
the resultant pressures which accompany the collocation of solid
vortices with surface ones.
To recapitulate:
The movements and mechanics of four-dimensional space are definite and
intelligible. A vortex with a surface as its axis affords a geometric
image of a closed circuit, and there are rotations which by their
polarity afford a possible definition of statical electricity.[7]
[7] These double rotations of the A and B kinds I should like to call
Hamiltons and co-Hamiltons, for it is a singular fact that in his
“Quaternions” Sir Wm. Rowan Hamilton has given the theory of either
the A or the B kind. They follow the laws of his symbols, I, J, K.
Hamiltons and co-Hamiltons seem to be natural units of geometrical
expression. In the paper in the “Proceedings of the Royal Irish
Academy,” Nov. 1903, already alluded to, I have shown something of the
remarkable facility which is gained in dealing with the composition of
three- and four-dimensional rotations by an alteration in Hamilton’s
notation, which enables his system to be applied to both the A and B
kinds of rotations.
The objection which has been often made to Hamilton’s system, namely,
that it is only under special conditions of application that his
processes give geometrically interpretable results, can be removed, if
we assume that he was really dealing with a four-dimensional motion,
and alter his notation to bring this circumstance into explicit
recognition.
APPENDIX I
THE MODELS
In Chapter XI. a description has been given which will enable any
one to make a set of models illustrative of the tesseract and its
properties. The set here supposed to be employed consists of:—
1. Three sets of twenty-seven cubes each.
2. Twenty-seven slabs.
3. Twelve cubes with points, lines, faces, distinguished by colours,
which will be called the catalogue cubes.
The preparation of the twelve catalogue cubes involves the expenditure
of a considerable amount of time. It is advantageous to use them, but
they can be replaced by the drawing of the views of the tesseract or by
a reference to figs. 103, 104, 105, 106 of the text.
The slabs are coloured like the twenty-seven cubes of the first cubic
block in fig. 101, the one with red, white, yellow axes.
The colours of the three sets of twenty-seven cubes are those of the
cubes shown in fig. 101.
The slabs are used to form the representation of a cube in a plane, and
can well be dispensed with by any one who is accustomed to deal with
solid figures. But the whole theory depends on a careful observation of
how the cube would be represented by these slabs.
In the first step, that of forming a clear idea how a plane being
would represent three-dimensional space, only one of the catalogue
cubes and one of the three blocks is needed.
APPLICATION TO THE STEP FROM PLANE TO SOLID.
Look at fig. 1 of the views of the tesseract, or, what comes to the
same thing, take catalogue cube No. 1 and place it before you with the
red line running up, the white line running to the right, the yellow
line running away. The three dimensions of space are then marked out
by these lines or axes. Now take a piece of cardboard, or a book, and
place it so that it forms a wall extending up and down not opposite to
you, but running away parallel to the wall of the room on your left
hand.
Placing the catalogue cube against this wall we see that it comes into
contact with it by the red and yellow lines, and by the included orange
face.
In the plane being’s world the aspect he has of the cube would be a
square surrounded by red and yellow lines with grey points.
Now, keeping the red line fixed, turn the cube about it so that the
yellow line goes out to the right, and the white line comes into
contact with the plane.
In this case a different aspect is presented to the plane being, a
square, namely, surrounded by red and white lines and grey points. You
should particularly notice that when the yellow line goes out, at right
angles to the plane, and the white comes in, the latter does not run in
the same sense that the yellow did.
From the fixed grey point at the base of the red line the yellow line
ran away from you. The white line now runs towards you. This turning
at right angles makes the line which was out of the plane before, come
into it in an opposite sense to that in which the line ran which has
just left the plane. If the cube does not break through the plane this
is always the rule.
Again turn the cube back to the normal position with red running up,
white to the right, and yellow away, and try another turning.
You can keep the yellow line fixed, and turn the cube about it. In this
case the red line going out to the right the white line will come in
pointing downwards.
You will be obliged to elevate the cube from the table in order to
carry out this turning. It is always necessary when a vertical axis
goes out of a space to imagine a movable support which will allow the
line which ran out before to come in below.
Having looked at the three ways of turning the cube so as to present
different faces to the plane, examine what would be the appearance if
a square hole were cut in the piece of cardboard, and the cube were to
pass through it. A hole can be actually cut, and it will be seen that
in the normal position, with red axis running up, yellow away, and
white to the right, the square first perceived by the plane being—the
one contained by red and yellow lines—would be replaced by another
square of which the line towards you is pink—the section line of the
pink face. The line above is light yellow, below is light yellow and on
the opposite side away from you is pink.
In the same way the cube can be pushed through a square opening in the
plane from any of the positions which you have already turned it into.
In each case the plane being will perceive a different set of contour
lines.
Having observed these facts about the catalogue cube, turn now to the
first block of twenty-seven cubes.
You notice that the colour scheme on the catalogue cube and that of
this set of blocks is the same.
Place them before you, a grey or null cube on the table, above it a
red cube, and on the top a null cube again. Then away from you place a
yellow cube, and beyond it a null cube. Then to the right place a white
cube and beyond it another null. Then complete the block, according to
the scheme of the catalogue cube, putting in the centre of all an ochre
cube.
You have now a cube like that which is described in the text. For the
sake of simplicity, in some cases, this cubic block can be reduced to
one of eight cubes, by leaving out the terminations in each direction.
Thus, instead of null, red, null, three cubes, you can take null, red,
two cubes, and so on.
It is useful, however, to practise the representation in a plane of a
block of twenty-seven cubes. For this purpose take the slabs, and build
them up against the piece of cardboard, or the book in such a way as to
represent the different aspects of the cube.
Proceed as follows:—
First, cube in normal position.
Place nine slabs against the cardboard to represent the nine cubes
in the wall of the red and yellow axes, facing the cardboard; these
represent the aspect of the cube as it touches the plane.
Now push these along the cardboard and make a different set of nine
slabs to represent the appearance which the cube would present to a
plane being, if it were to pass half way through the plane.
There would be a white slab, above it a pink one, above that another
white one, and six others, representing what would be the nature of a
section across the middle of the block of cubes. The section can be
thought of as a thin slice cut out by two parallel cuts across the
cube. Having arranged these nine slabs, push them along the plane, and
make another set of nine to represent what would be the appearance of
the cube when it had almost completely gone through. This set of nine
will be the same as the first set of nine.
Now we have in the plane three sets of nine slabs each, which represent
three sections of the twenty-seven block.
They are put alongside one another. We see that it does not matter in
what order the sets of nine are put. As the cube passes through the
plane they represent appearances which follow the one after the other.
If they were what they represented, they could not exist in the same
plane together.
This is a rather important point, namely, to notice that they should
not co-exist on the plane, and that the order in which they are placed
is indifferent. When we represent a four-dimensional body our solid
cubes are to us in the same position that the slabs are to the plane
being. You should also notice that each of these slabs represents only
the very thinnest slice of a cube. The set of nine slabs first set up
represents the side surface of the block. It is, as it were, a kind
of tray—a beginning from which the solid cube goes off. The slabs
as we use them have thickness, but this thickness is a necessity of
construction. They are to be thought of as merely of the thickness of a
line.
If now the block of cubes passed through the plane at the rate of an
inch a minute the appearance to a plane being would be represented by:—
1. The first set of nine slabs lasting for one minute.
2. The second set of nine slabs lasting for one minute.
3. The third set of nine slabs lasting for one minute.
Now the appearances which the cube would present to the plane being
in other positions can be shown by means of these slabs. The use of
such slabs would be the means by which a plane being could acquire a
familiarity with our cube. Turn the catalogue cube (or imagine the
coloured figure turned) so that the red line runs up, the yellow line
out to the right, and the white line towards you. Then turn the block
of cubes to occupy a similar position.
The block has now a different wall in contact with the plane. Its
appearance to a plane being will not be the same as before. He has,
however, enough slabs to represent this new set of appearances. But he
must remodel his former arrangement of them.
He must take a null, a red, and a null slab from the first of his sets
of slabs, then a white, a pink, and a white from the second, and then a
null, a red, and a null from the third set of slabs.
He takes the first column from the first set, the first column from the
second set, and the first column from the third set.
To represent the half-way-through appearance, which is as if a very
thin slice were cut out half way through the block, he must take the
second column of each of his sets of slabs, and to represent the final
appearance, the third column of each set.
Now turn the catalogue cube back to the normal position, and also the
block of cubes.
There is another turning—a turning about the yellow line, in which the
white axis comes below the support.
You cannot break through the surface of the table, so you must imagine
the old support to be raised. Then the top of the block of cubes in its
new position is at the level at which the base of it was before.
Now representing the appearance on the plane, we must draw a horizontal
line to represent the old base. The line should be drawn three inches
high on the cardboard.
Below this the representative slabs can be arranged.
It is easy to see what they are. The old arrangements have to be
broken up, and the layers taken in order, the first layer of each for
the representation of the aspect of the block as it touches the plane.
Then the second layers will represent the appearance half way through,
and the third layers will represent the final appearance.
It is evident that the slabs individually do not represent the same
portion of the cube in these different presentations.
In the first case each slab represents a section or a face
perpendicular to the white axis, in the second case a face or a section
which runs perpendicularly to the yellow axis, and in the third case a
section or a face perpendicular to the red axis.
But by means of these nine slabs the plane being can represent the
whole of the cubic block. He can touch and handle each portion of the
cubic block, there is no part of it which he cannot observe. Taking it
bit by bit, two axes at a time, he can examine the whole of it.
OUR REPRESENTATION OF A BLOCK OF TESSERACTS.
Look at the views of the tesseract 1, 2, 3, or take the catalogue cubes
1, 2, 3, and place them in front of you, in any order, say running from
left to right, placing 1 in the normal position, the red axis running
up, the white to the right, and yellow away.
Now notice that in catalogue cube 2 the colours of each region are
derived from those of the corresponding region of cube 1 by the
addition of blue. Thus null + blue = blue, and the corners of number 2
are blue. Again, red + blue = purple, and the vertical lines of 2 are
purple. Blue + yellow = green, and the line which runs away is coloured
green.
By means of these observations you may be sure that catalogue cube 2
is rightly placed. Catalogue cube 3 is just like number 1.
Having these cubes in what we may call their normal position, proceed
to build up the three sets of blocks.
This is easily done in accordance with the colour scheme on the
catalogue cubes.
The first block we already know. Build up the second block, beginning
with a blue corner cube, placing a purple on it, and so on.
Having these three blocks we have the means of representing the
appearances of a group of eighty-one tesseracts.
Let us consider a moment what the analogy in the case of the plane
being is.
He has his three sets of nine slabs each. We have our three sets of
twenty-seven cubes each.
Our cubes are like his slabs. As his slabs are not the things which
they represent to him, so our cubes are not the things they represent
to us.
The plane being’s slabs are to him the faces of cubes.
Our cubes then are the faces of tesseracts, the cubes by which they are
in contact with our space.
As each set of slabs in the case of the plane being might be considered
as a sort of tray from which the solid contents of the cubes came out,
so our three blocks of cubes may be considered as three-space trays,
each of which is the beginning of an inch of the solid contents of the
four-dimensional solids starting from them.
We want now to use the names null, red, white, etc., for tesseracts.
The cubes we use are only tesseract faces. Let us denote that fact
by calling the cube of null colour, null face; or, shortly, null f.,
meaning that it is the face of a tesseract.
To determine which face it is let us look at the catalogue cube 1 or
the first of the views of the tesseract, which can be used instead of
the models. It has three axes, red, white, yellow, in our space. Hence
the cube determined by these axes is the face of the tesseract which we
now have before us. It is the ochre face. It is enough, however, simply
to say null f., red f. for the cubes which we use.
To impress this in your mind, imagine that tesseracts do actually run
from each cube. Then, when you move the cubes about, you move the
tesseracts about with them. You move the face but the tesseract follows
with it, as the cube follows when its face is shifted in a plane.
The cube null in the normal position is the cube which has in it the
red, yellow, white axes. It is the face having these, but wanting the
blue. In this way you can define which face it is you are handling. I
will write an “f.” after the name of each tesseract just as the plane
being might call each of his slabs null slab, yellow slab, etc., to
denote that they were representations.
We have then in the first block of twenty-seven cubes, the
following—null f., red f., null f., going up; white f., null f., lying
to the right, and so on. Starting from the null point and travelling
up one inch we are in the null region, the same for the away and the
right-hand directions. And if we were to travel in the fourth dimension
for an inch we should still be in a null region. The tesseract
stretches equally all four ways. Hence the appearance we have in this
first block would do equally well if the tesseract block were to move
across our space for a certain distance. For anything less than an inch
of their transverse motion we should still have the same appearance.
You must notice, however, that we should not have null face after the
motion had begun.
When the tesseract, null for instance, had moved ever so little we
should not have a face of null but a section of null in our space.
Hence, when we think of the motion across our space we must call our
cubes tesseract sections. Thus on null passing across we should see
first null f., then null s., and then, finally, null f. again.
Imagine now the whole first block of twenty-seven tesseracts to have
moved tranverse to our space a distance of one inch. Then the second
set of tesseracts, which originally were an inch distant from our
space, would be ready to come in.
Their colours are shown in the second block of twenty-seven cubes which
you have before you. These represent the tesseract faces of the set of
tesseracts that lay before an inch away from our space. They are ready
now to come in, and we can observe their colours. In the place which
null f. occupied before we have blue f., in place of red f. we have
purple f., and so on. Each tesseract is coloured like the one whose
place it takes in this motion with the addition of blue.
Now if the tesseract block goes on moving at the rate of an inch a
minute, this next set of tesseracts will occupy a minute in passing
across. We shall see, to take the null one for instance, first of all
null face, then null section, then null face again.
At the end of the second minute the second set of tesseracts has gone
through, and the third set comes in. This, as you see, is coloured just
like the first. Altogether, these three sets extend three inches in the
fourth dimension, making the tesseract block of equal magnitude in all
dimensions.
We have now before us a complete catalogue of all the tesseracts in our
group. We have seen them all, and we shall refer to this arrangement
of the blocks as the “normal position.” We have seen as much of each
tesseract at a time as could be done in a three-dimensional space. Each
part of each tesseract has been in our space, and we could have touched
it.
The fourth dimension appeared to us as the duration of the block.
If a bit of our matter were to be subjected to the same motion it
would be instantly removed out of our space. Being thin in the fourth
dimension it is at once taken out of our space by a motion in the
fourth dimension.
But the tesseract block we represent having length in the fourth
dimension remains steadily before our eyes for three minutes, when it
is subjected to this transverse motion.
We have now to form representations of the other views of the same
tesseract group which are possible in our space.
Let us then turn the block of tesseracts so that another face of it
comes into contact with our space, and then by observing what we have,
and what changes come when the block traverses our space, we shall have
another view of it. The dimension which appeared as duration before
will become extension in one of our known dimensions, and a dimension
which coincided with one of our space dimensions will appear as
duration.
Leaving catalogue cube 1 in the normal position, remove the other two,
or suppose them removed. We have in space the red, the yellow, and the
white axes. Let the white axis go out into the unknown, and occupy the
position the blue axis holds. Then the blue axis, which runs in that
direction now will come into space. But it will not come in pointing
in the same way that the white axis does now. It will point in the
opposite sense. It will come in running to the left instead of running
to the right as the white axis does now.
When this turning takes place every part of the cube 1 will disappear
except the left-hand face—the orange face.
And the new cube that appears in our space will run to the left from
this orange face, having axes, red, yellow, blue.
Take models 4, 5, 6. Place 4, or suppose No. 4 of the tesseract views
placed, with its orange face coincident with the orange face of 1, red
line to red line, and yellow line to yellow line, with the blue line
pointing to the left. Then remove cube 1 and we have the tesseract face
which comes in when the white axis runs in the positive unknown, and
the blue axis comes into our space.
Now place catalogue cube 5 in some position, it does not matter which,
say to the left; and place it so that there is a correspondence of
colour corresponding to the colour of the line that runs out of space.
The line that runs out of space is white, hence, every part of this
cube 5 should differ from the corresponding part of 4 by an alteration
in the direction of white.
Thus we have white points in 5 corresponding to the null points in
4. We have a pink line corresponding to a red line, a light yellow
line corresponding to a yellow line, an ochre face corresponding to
an orange face. This cube section is completely named in Chapter XI.
Finally cube 6 is a replica of 1.
These catalogue cubes will enable us to set up our models of the block
of tesseracts.
First of all for the set of tesseracts, which beginning in our space
reach out one inch in the unknown, we have the pattern of catalogue
cube 4.
We see that we can build up a block of twenty-seven tesseract faces
after the colour scheme of cube 4, by taking the left-hand wall of
block 1, then the left-hand wall of block 2, and finally that of block
3. We take, that is, the three first walls of our previous arrangement
to form the first cubic block of this new one.
This will represent the cubic faces by which the group of tesseracts in
its new position touches our space. We have running up, null f., red
f., null f. In the next vertical line, on the side remote from us, we
have yellow f., orange f., yellow f., and then the first colours over
again. Then the three following columns are, blue f., purple f., blue
f.; green f., brown f., green f.; blue f., purple f., blue f. The last
three columns are like the first.
These tesseracts touch our space, and none of them are by any part of
them distant more than an inch from it. What lies beyond them in the
unknown?
This can be told by looking at catalogue cube 5. According to its
scheme of colour we see that the second wall of each of our old
arrangements must be taken. Putting them together we have, as the
corner, white f. above it, pink f. above it, white f. The column next
to this remote from us is as follows:—light yellow f., ochre f., light
yellow f., and beyond this a column like the first. Then for the middle
of the block, light blue f., above it light purple, then light blue.
The centre column has, at the bottom, light green f., light brown f.
in the centre and at the top light green f. The last wall is like the
first.
The third block is made by taking the third walls of our previous
arrangement, which we called the normal one.
You may ask what faces and what sections our cubes represent. To answer
this question look at what axes you have in our space. You have red,
yellow, blue. Now these determine brown. The colours red, yellow, blue
are supposed by us when mixed to produce a brown colour. And that cube
which is determined by the red, yellow, blue axes we call the brown
cube.
When the tesseract block in its new position begins to move across our
space each tesseract in it gives a section in our space. This section
is transverse to the white axis, which now runs in the unknown.
As the tesseract in its present position passes across our space, we
should see first of all the first of the blocks of cubic faces we have
put up—these would last for a minute, then would come the second block
and then the third. At first we should have a cube of tesseract faces,
each of which would be brown. Directly the movement began, we should
have tesseract sections transverse to the white line.
There are two more analogous positions in which the block of tesseracts
can be placed. To find the third position, restore the blocks to the
normal arrangement.
Let us make the yellow axis go out into the positive unknown, and let
the blue axis, consequently, come in running towards us. The yellow ran
away, so the blue will come in running towards us.
Put catalogue cube 1 in its normal position. Take catalogue cube 7
and place it so that its pink face coincides with the pink face of
cube 1, making also its red axis coincide with the red axis of 1 and
its white with the white. Moreover, make cube 7 come towards us from
cube 1. Looking at it we see in our space, red, white, and blue axes.
The yellow runs out. Place catalogue cube 8 in the neighbourhood
of 7—observe that every region in 8 has a change in the direction
of yellow from the corresponding region in 7. This is because it
represents what you come to now in going in the unknown, when the
yellow axis runs out of our space. Finally catalogue cube 9, which is
like number 7, shows the colours of the third set of tesseracts. Now
evidently, starting from the normal position, to make up our three
blocks of tesseract faces we have to take the near wall from the first
block, the near wall from the second, and then the near wall from the
third block. This gives us the cubic block formed by the faces of the
twenty-seven tesseracts which are now immediately touching our space.
Following the colour scheme of catalogue cube 8, we make the next set
of twenty-seven tesseract faces, representing the tesseracts, each of
which begins one inch off from our space, by putting the second walls
of our previous arrangement together, and the representation of the
third set of tesseracts is the cubic block formed of the remaining
three walls.
Since we have red, white, blue axes in our space to begin with, the
cubes we see at first are light purple tesseract faces, and after the
transverse motion begins we have cubic sections transverse to the
yellow line.
Restore the blocks to the normal position, there remains the case in
which the red axis turns out of space. In this case the blue axis will
come in downwards, opposite to the sense in which the red axis ran.
In this case take catalogue cubes 10, 11, 12. Lift up catalogue cube 1
and put 10 underneath it, imagining that it goes down from the previous
position of 1.
We have to keep in space the white and the yellow axes, and let the red
go out, the blue come in.
Now, you will find on cube 10 a light yellow face; this should coincide
with the base of 1, and the white and yellow lines on the two cubes
should coincide. Then the blue axis running down you have the catalogue
cube correctly placed, and it forms a guide for putting up the first
representative block.
Catalogue cube 11 will represent what lies in the fourth dimension—now
the red line runs in the fourth dimension. Thus the change from 10 to
11 should be towards red, corresponding to a null point is a red point,
to a white line is a pink line, to a yellow line an orange line, and so
on.
Catalogue cube 12 is like 10. Hence we see that to build up our blocks
of tesseract faces we must take the bottom layer of the first block,
hold that up in the air, underneath it place the bottom layer of the
second block, and finally underneath this last the bottom layer of the
last of our normal blocks.
Similarly we make the second representative group by taking the middle
courses of our three blocks. The last is made by taking the three
topmost layers. The three axes in our space before the transverse
motion begins are blue, white, yellow, so we have light green tesseract
faces, and after the motion begins sections transverse to the red light.
These three blocks represent the appearances as the tesseract group in
its new position passes across our space. The cubes of contact in this
case are those determinal by the three axes in our space, namely, the
white, the yellow, the blue. Hence they are light green.
It follows from this that light green is the interior cube of the first
block of representative cubic faces.
Practice in the manipulations described, with a realization in each
case of the face or section which is in our space, is one of the best
means of a thorough comprehension of the subject.
We have to learn how to get any part of these four-dimensional figures
into space, so that we can look at them. We must first learn to swing a
tesseract, and a group of tesseracts about in any way.
When these operations have been repeated and the method of arrangement
of the set of blocks has become familiar, it is a good plan to rotate
the axes of the normal cube 1 about a diagonal, and then repeat the
whole series of turnings.
Thus, in the normal position, red goes up, white to the right, yellow
away. Make white go up, yellow to the right, and red away. Learn the
cube in this position by putting up the set of blocks of the normal
cube, over and over again till it becomes as familiar to you as in the
normal position. Then when this is learned, and the corresponding
changes in the arrangements of the tesseract groups are made, another
change should be made: let, in the normal cube, yellow go up, red to
the right, and white away.
Learn the normal block of cubes in this new position by arranging them
and re-arranging them till you know without thought where each one
goes. Then carry out all the tesseract arrangements and turnings.
If you want to understand the subject, but do not see your way clearly,
if it does not seem natural and easy to you, practise these turnings.
Practise, first of all, the turning of a block of cubes round, so that
you know it in every position as well as in the normal one. Practise by
gradually putting up the set of cubes in their new arrangements. Then
put up the tesseract blocks in their arrangements. This will give you
a working conception of higher space, you will gain the feeling of it,
whether you take up the mathematical treatment of it or not.
APPENDIX II
A LANGUAGE OF SPACE
The mere naming the parts of the figures we consider involves a certain
amount of time and attention. This time and attention leads to no
result, for with each new figure the nomenclature applied is completely
changed, every letter or symbol is used in a different significance.
Surely it must be possible in some way to utilise the labour thus at
present wasted!
Why should we not make a language for space itself, so that every
position we want to refer to would have its own name? Then every time
we named a figure in order to demonstrate its properties we should be
exercising ourselves in the vocabulary of place.
If we use a definite system of names, and always refer to the same
space position by the same name, we create as it were a multitude of
little hands, each prepared to grasp a special point, position, or
element, and hold it for us in its proper relations.
We make, to use another analogy, a kind of mental paper, which has
somewhat of the properties of a sensitive plate, in that it will
register, without effort, complex, visual, or tactual impressions.
But of far more importance than the applications of a space language to
the plane and to solid space is the facilitation it brings with it to
the study of four-dimensional shapes.
I have delayed introducing a space language because all the systems I
made turned out, after giving them a fair trial, to be intolerable. I
have now come upon one which seems to present features of permanence,
and I will here give an outline of it, so that it can be applied to the
subject of the text, and in order that it may be subjected to criticism.
The principle on which the language is constructed is to sacrifice
every other consideration for brevity.
It is indeed curious that we are able to talk and converse on every
subject of thought except the fundamental one of space. The only way of
speaking about the spatial configurations that underlie every subject
of discursive thought is a co-ordinate system of numbers. This is so
awkward and incommodious that it is never used. In thinking also, in
realising shapes, we do not use it; we confine ourselves to a direct
visualisation.
Now, the use of words corresponds to the storing up of our experience
in a definite brain structure. A child, in the endless tactual, visual,
mental manipulations it makes for itself, is best left to itself, but
in the course of instruction the introduction of space names would
make the teachers work more cumulative, and the child’s knowledge more
social.
Their full use can only be appreciated, if they are introduced early
in the course of education; but in a minor degree any one can convince
himself of their utility, especially in our immediate subject of
handling four-dimensional shapes. The sum total of the results obtained
in the preceding pages can be compendiously and accurately expressed in
nine words of the Space Language.
In one of Plato’s dialogues Socrates makes an experiment on a slave boy
standing by. He makes certain perceptions of space awake in the mind
of Meno’s slave by directing his close attention on some simple facts
of geometry.
By means of a few words and some simple forms we can repeat Plato’s
experiment on new ground.
Do we by directing our close attention on the facts of four dimensions
awaken a latent faculty in ourselves? The old experiment of Plato’s, it
seems to me, has come down to us as novel as on the day he incepted it,
and its significance not better understood through all the discussion
of which it has been the subject.
Imagine a voiceless people living in a region where everything had
a velvety surface, and who were thus deprived of all opportunity of
experiencing what sound is. They could observe the slow pulsations
of the air caused by their movements, and arguing from analogy, they
would no doubt infer that more rapid vibrations were possible. From
the theoretical side they could determine all about these more rapid
vibrations. They merely differ, they would say, from slower ones,
by the number that occur in a given time; there is a merely formal
difference.
But suppose they were to take the trouble, go to the pains of producing
these more rapid vibrations, then a totally new sensation would fall
on their rudimentary ears. Probably at first they would only be dimly
conscious of Sound, but even from the first they would become aware
that a merely formal difference, a mere difference in point of number
in this particular respect, made a great difference practically, as
related to them. And to us the difference between three and four
dimensions is merely formal, numerical. We can tell formally all about
four dimensions, calculate the relations that would exist. But that
the difference is merely formal does not prove that it is a futile and
empty task, to present to ourselves as closely as we can the phenomena
of four dimensions. In our formal knowledge of it, the whole question
of its actual relation to us, as we are, is left in abeyance.
Possibly a new apprehension of nature may come to us through the
practical, as distinguished from the mathematical and formal, study
of four dimensions. As a child handles and examines the objects with
which he comes in contact, so we can mentally handle and examine
four-dimensional objects. The point to be determined is this. Do we
find something cognate and natural to our faculties, or are we merely
building up an artificial presentation of a scheme only formally
possible, conceivable, but which has no real connection with any
existing or possible experience?
This, it seems to me, is a question which can only be settled by
actually trying. This practical attempt is the logical and direct
continuation of the experiment Plato devised in the “Meno.”
Why do we think true? Why, by our processes of thought, can we predict
what will happen, and correctly conjecture the constitution of the
things around us? This is a problem which every modern philosopher has
considered, and of which Descartes, Leibnitz, Kant, to name a few,
have given memorable solutions. Plato was the first to suggest it.
And as he had the unique position of being the first devisor of the
problem, so his solution is the most unique. Later philosophers have
talked about consciousness and its laws, sensations, categories. But
Plato never used such words. Consciousness apart from a conscious being
meant nothing to him. His was always an objective search. He made man’s
intuitions the basis of a new kind of natural history.
In a few simple words Plato puts us in an attitude with regard to
psychic phenomena—the mind—the ego—“what we are,” which is analogous
to the attitude scientific men of the present day have with regard
to the phenomena of outward nature. Behind this first apprehension
of ours of nature, there is an infinite depth to be learned and
known. Plato said that behind the phenomena of mind that Meno’s slave
boy exhibited, there was a vast, an infinite perspective. And his
singularity, his originality, comes out most strongly marked in this,
that the perspective, the complex phenomena beyond were, according to
him, phenomena of personal experience. A footprint in the sand means a
man to a being that has the conception of a man. But to a creature that
has no such conception, it means a curious mark, somehow resulting from
the concatenation of ordinary occurrences. Such a being would attempt
merely to explain how causes known to him could so coincide as to
produce such a result; he would not recognise its significance.
Plato introduced the conception which made a new kind of natural
history possible. He said that Meno’s slave boy thought true about
things he had never learned, because his “soul” had experience. I
know this will sound absurd to some people, and it flies straight in
the face of the maxim, that explanation consists in showing how an
effect depends on simple causes. But what a mistaken maxim that is!
Can any single instance be shown of a simple cause? Take the behaviour
of spheres for instance; say those ivory spheres, billiard balls,
for example. We can explain their behaviour by supposing they are
homogeneous elastic solids. We can give formulæ which will account for
their movements in every variety. But are they homogeneous elastic
solids? No, certainly not. They are complex in physical and molecular
structure, and atoms and ions beyond open an endless vista. Our simple
explanation is false, false as it can be. The balls act as if they
were homogeneous elastic spheres. There is a statistical simplicity in
the resultant of very complex conditions, which makes that artificial
conception useful. But its usefulness must not blind us to the fact
that it is artificial. If we really look deep into nature, we find a
much greater complexity than we at first suspect. And so behind this
simple “I,” this myself, is there not a parallel complexity? Plato’s
“soul” would be quite acceptable to a large class of thinkers, if by
“soul” and the complexity he attributes to it, he meant the product of
a long course of evolutionary changes, whereby simple forms of living
matter endowed with rudimentary sensation had gradually developed into
fully conscious beings.
But Plato does not mean by “soul” a being of such a kind. His soul is
a being whose faculties are clogged by its bodily environment, or at
least hampered by the difficulty of directing its bodily frame—a being
which is essentially higher than the account it gives of itself through
its organs. At the same time Plato’s soul is not incorporeal. It is a
real being with a real experience. The question of whether Plato had
the conception of non-spatial existence has been much discussed. The
verdict is, I believe, that even his “ideas” were conceived by him as
beings in space, or, as we should say, real. Plato’s attitude is that
of Science, inasmuch as he thinks of a world in Space. But, granting
this, it cannot be denied that there is a fundamental divergence
between Plato’s conception and the evolutionary theory, and also an
absolute divergence between his conception and the genetic account of
the origin of the human faculties. The functions and capacities of
Plato’s “soul” are not derived by the interaction of the body and its
environment.
Plato was engaged on a variety of problems, and his religious and
ethical thoughts were so keen and fertile that the experimental
investigation of his soul appears involved with many other motives.
In one passage Plato will combine matter of thought of all kinds and
from all sources, overlapping, interrunning. And in no case is he more
involved and rich than in this question of the soul. In fact, I wish
there were two words, one denoting that being, corporeal and real, but
with higher faculties than we manifest in our bodily actions, which is
to be taken as the subject of experimental investigation; and the other
word denoting “soul” in the sense in which it is made the recipient and
the promise of so much that men desire. It is the soul in the former
sense that I wish to investigate, and in a limited sphere only. I wish
to find out, in continuation of the experiment in the Meno, what the
“soul” in us thinks about extension, experimenting on the grounds laid
down by Plato. He made, to state the matter briefly, the hypothesis
with regard to the thinking power of a being in us, a “soul.” This
soul is not accessible to observation by sight or touch, but it can be
observed by its functions; it is the object of a new kind of natural
history, the materials for constructing which lie in what it is natural
to us to think. With Plato “thought” was a very wide-reaching term, but
still I would claim in his general plan of procedure a place for the
particular question of extension.
The problem comes to be, “What is it natural to us to think about
matter _qua_ extended?”
First of all, I find that the ordinary intuition of any simple object
is extremely imperfect. Take a block of differently marked cubes, for
instance, and become acquainted with them in their positions. You may
think you know them quite well, but when you turn them round—rotate
the block round a diagonal, for instance—you will find that you have
lost track of the individuals in their new positions. You can mentally
construct the block in its new position, by a rule, by taking the
remembered sequences, but you don’t know it intuitively. By observation
of a block of cubes in various positions, and very expeditiously
by a use of Space names applied to the cubes in their different
presentations, it is possible to get an intuitive knowledge of the
block of cubes, which is not disturbed by any displacement. Now, with
regard to this intuition, we moderns would say that I had formed it by
my tactual visual experiences (aided by hereditary pre-disposition).
Plato would say that the soul had been stimulated to recognise an
instance of shape which it knew. Plato would consider the operation
of learning merely as a stimulus; we as completely accounting for
the result. The latter is the more common-sense view. But, on the
other hand, it presupposes the generation of experience from physical
changes. The world of sentient experience, according to the modern
view, is closed and limited; only the physical world is ample and large
and of ever-to-be-discovered complexity. Plato’s world of soul, on the
other hand, is at least as large and ample as the world of things.
Let us now try a crucial experiment. Can I form an intuition of a
four-dimensional object? Such an object is not given in the physical
range of my sense contacts. All I can do is to present to myself the
sequences of solids, which would mean the presentation to me under my
conditions of a four-dimensional object. All I can do is to visualise
and tactualise different series of solids which are alternative sets of
sectional views of a four-dimensional shape.
If now, on presenting these sequences, I find a power in me of
intuitively passing from one of these sets of sequences to another, of,
being given one, intuitively constructing another, not using a rule,
but directly apprehending it, then I have found a new fact about my
soul, that it has a four-dimensional experience; I have observed it by
a function it has.
I do not like to speak positively, for I might occasion a loss of time
on the part of others, if, as may very well be, I am mistaken. But for
my own part, I think there are indications of such an intuition; from
the results of my experiments, I adopt the hypothesis that that which
thinks in us has an ample experience, of which the intuitions we use in
dealing with the world of real objects are a part; of which experience,
the intuition of four-dimensional forms and motions is also a part. The
process we are engaged in intellectually is the reading the obscure
signals of our nerves into a world of reality, by means of intuitions
derived from the inner experience.
The image I form is as follows. Imagine the captain of a modern
battle-ship directing its course. He has his charts before him; he
is in communication with his associates and subordinates; can convey
his messages and commands to every part of the ship, and receive
information from the conning-tower and the engine-room. Now suppose the
captain immersed in the problem of the navigation of his ship over the
ocean, to have so absorbed himself in the problem of the direction of
his craft over the plane surface of the sea that he forgets himself.
All that occupies his attention is the kind of movement that his ship
makes. The operations by which that movement is produced have sunk
below the threshold of his consciousness, his own actions, by which
he pushes the buttons, gives the orders, are so familiar as to be
automatic, his mind is on the motion of the ship as a whole. In such a
case we can imagine that he identifies himself with his ship; all that
enters his conscious thought is the direction of its movement over the
plane surface of the ocean.
Such is the relation, as I imagine it, of the soul to the body. A
relation which we can imagine as existing momentarily in the case
of the captain is the normal one in the case of the soul with its
craft. As the captain is capable of a kind of movement, an amplitude
of motion, which does not enter into his thoughts with regard to the
directing the ship over the plane surface of the ocean, so the soul is
capable of a kind of movement, has an amplitude of motion, which is
not used in its task of directing the body in the three-dimensional
region in which the body’s activity lies. If for any reason it became
necessary for the captain to consider three-dimensional motions with
regard to his ship, it would not be difficult for him to gain the
materials for thinking about such motions; all he has to do is to
call his own intimate experience into play. As far as the navigation
of the ship, however, is concerned, he is not obliged to call on
such experience. The ship as a whole simply moves on a surface. The
problem of three-dimensional movement does not ordinarily concern its
steering. And thus with regard to ourselves all those movements and
activities which characterise our bodily organs are three-dimensional;
we never need to consider the ampler movements. But we do more than
use the movements of our body to effect our aims by direct means; we
have now come to the pass when we act indirectly on nature, when we
call processes into play which lie beyond the reach of any explanation
we can give by the kind of thought which has been sufficient for the
steering of our craft as a whole. When we come to the problem of what
goes on in the minute, and apply ourselves to the mechanism of the
minute, we find our habitual conceptions inadequate.
The captain in us must wake up to his own intimate nature, realise
those functions of movement which are his own, and in virtue of his
knowledge of them apprehend how to deal with the problems he has come
to.
Think of the history of man. When has there been a time, in which his
thoughts of form and movement were not exclusively of such varieties as
were adapted for his bodily performance? We have never had a demand to
conceive what our own most intimate powers are. But, just as little as
by immersing himself in the steering of his ship over the plane surface
of the ocean, a captain can lose the faculty of thinking about what he
actually does, so little can the soul lose its own nature. It can be
roused to an intuition that is not derived from the experience which
the senses give. All that is necessary is to present some few of those
appearances which, while inconsistent with three-dimensional matter,
are yet consistent with our formal knowledge of four-dimensional
matter, in order for the soul to wake up and not begin to learn, but of
its own intimate feeling fill up the gaps in the presentiment, grasp
the full orb of possibilities from the isolated points presented to
it. In relation to this question of our perceptions, let me suggest
another illustration, not taking it too seriously, only propounding it
to exhibit the possibilities in a broad and general way.
In the heavens, amongst the multitude of stars, there are some which,
when the telescope is directed on them, seem not to be single stars,
but to be split up into two. Regarding these twin stars through a
spectroscope, an astronomer sees in each a spectrum of bands of colour
and black lines. Comparing these spectrums with one another, he finds
that there is a slight relative shifting of the dark lines, and from
that shifting he knows that the stars are rotating round one another,
and can tell their relative velocity with regard to the earth. By
means of his terrestrial physics he reads this signal of the skies.
This shifting of lines, the mere slight variation of a black line in a
spectrum, is very unlike that which the astronomer knows it means. But
it is probably much more like what it means than the signals which the
nerves deliver are like the phenomena of the outer world.
No picture of an object is conveyed through the nerves. No picture of
motion, in the sense in which we postulate its existence, is conveyed
through the nerves. The actual deliverances of which our consciousness
takes account are probably identical for eye and ear, sight and touch.
If for a moment I take the whole earth together and regard it as a
sentient being, I find that the problem of its apprehension is a very
complex one, and involves a long series of personal and physical
events. Similarly the problem of our apprehension is a very complex
one. I only use this illustration to exhibit my meaning. It has this
especial merit, that, as the process of conscious apprehension takes
place in our case in the minute, so, with regard to this earth being,
the corresponding process takes place in what is relatively to it very
minute.
Now, Plato’s view of a soul leads us to the hypothesis that that
which we designate as an act of apprehension may be a very complex
event, both physically and personally. He does not seek to explain
what an intuition is; he makes it a basis from whence he sets out on
a voyage of discovery. Knowledge means knowledge; he puts conscious
being to account for conscious being. He makes an hypothesis of the
kind that is so fertile in physical science—an hypothesis making no
claim to finality, which marks out a vista of possible determination
behind determination, like the hypothesis of space itself, the type of
serviceable hypotheses.
And, above all, Plato’s hypothesis is conducive to experiment. He
gives the perspective in which real objects can be determined; and,
in our present enquiry, we are making the simplest of all possible
experiments—we are enquiring what it is natural to the soul to think of
matter as extended.
Aristotle says we always use a “phantasm” in thinking, a phantasm of
our corporeal senses a visualisation or a tactualisation. But we can
so modify that visualisation or tactualisation that it represents
something not known by the senses. Do we by that representation wake
up an intuition of the soul? Can we by the presentation of these
hypothetical forms, that are the subject of our present discussion,
wake ourselves up to higher intuitions? And can we explain the world
around by a motion that we only know by our souls?
Apart from all speculation, however, it seems to me that the interest
of these four-dimensional shapes and motions is sufficient reason for
studying them, and that they are the way by which we can grow into a
fuller apprehension of the world as a concrete whole.
SPACE NAMES.
If the words written in the squares drawn in fig. 1 are used as the
names of the squares in the positions in which they are placed, it is
evident that a combination of these names will denote a figure composed
of the designated squares. It is found to be most convenient to take as
the initial square that marked with an asterisk, so that the directions
of progression are towards the observer and to his right. The
directions of progression, however, are arbitrary, and can be chosen at
will.
[Illustration: Fig. 1.]
Thus _et_, _at_, _it_, _an_, _al_ will denote a figure in the form of a
cross composed of five squares.
Here, by means of the double sequence, _e_, _a_, _i_ and _n_, _t_, _l_,
it is possible to name a limited collection of space elements.
The system can obviously be extended by using letter sequences of more
members.
But, without introducing such a complexity, the principles of a space
language can be exhibited, and a nomenclature obtained adequate to all
the considerations of the preceding pages.
1. _Extension._
Call the large squares in fig. 2 by the name written in them. It is
evident that each can be divided as shown in fig. 1. Then the small
square marked 1 will be “en” in “En,” or “Enen.” The square marked 2
will be “et” in “En” or “Enet,” while the square marked 4 will be “en”
in “Et” or “Eten.” Thus the square 5 will be called “Ilil.”
[Illustration: Fig. 2.]
This principle of extension can be applied in any number of dimensions.
2. _Application to Three-Dimensional Space._
To name a three-dimensional collocation of cubes take the upward
direction first, secondly the direction towards the observer, thirdly
the direction to his right hand.
[Illustration]
These form a word in which the first letter gives the place of the cube
upwards, the second letter its place towards the observer, the third
letter its place to the right.
We have thus the following scheme, which represents the set of cubes of
column 1, fig. 101, page 165.
We begin with the remote lowest cube at the left hand, where the
asterisk is placed (this proves to be by far the most convenient origin
to take for the normal system).
Thus “nen” is a “null” cube, “ten” a red cube on it, and “len” a “null”
cube above “ten.”
By using a more extended sequence of consonants and vowels a larger set
of cubes can be named.
To name a four-dimensional block of tesseracts it is simply necessary
to prefix an “e,” an “a,” or an “i” to the cube names.
Thus the tesseract blocks schematically represented on page 165, fig.
101 are named as follows:—
[Illustration: 1 2 3]
2. DERIVATION OF POINT, LINE, FACE, ETC., NAMES.
[Illustration]
The principle of derivation can be shown as follows: Taking the square
of squares the number of squares in it can be enlarged and the whole
kept the same size.
[Illustration]
Compare fig. 79, p. 138, for instance, or the bottom layer of fig. 84.
Now use an initial “s” to denote the result of carrying this process on
to a great extent, and we obtain the limit names, that is the point,
line, area names for a square. “Sat” is the whole interior. The corners
are “sen,” “sel,” “sin,” “sil,” while the lines are “san,” “sal,”
“set,” “sit.”
[Illustration]
I find that by the use of the initial “s” these names come to be
practically entirely disconnected with the systematic names for the
square from which they are derived. They are easy to learn, and when
learned can be used readily with the axes running in any direction.
To derive the limit names for a four-dimensional rectangular figure,
like the tesseract, is a simple extension of this process. These point,
line, etc., names include those which apply to a cube, as will be
evident on inspection of the first cube of the diagrams which follow.
All that is necessary is to place an “s” before each of the names given
for a tesseract block. We then obtain apellatives which, like the
colour names on page 174, fig. 103, apply to all the points, lines,
faces, solids, and to the hyper-solid of the tesseract. These names
have the advantage over the colour marks that each point, line, etc.,
has its own individual name.
In the diagrams I give the names corresponding to the positions shown
in the coloured plate or described on p. 174. By comparing cubes 1, 2,
3 with the first row of cubes in the coloured plate, the systematic
names of each of the points, lines, faces, etc., can be determined. The
asterisk shows the origin from which the names run.
These point, line, face, etc., names should be used in connection with
the corresponding colours. The names should call up coloured images of
the parts named in their right connection.
[Illustration]
It is found that a certain abbreviation adds vividness of distinction
to these names. If the final “en” be dropped wherever it occurs the
system is improved. Thus instead of “senen,” “seten,” “selen,” it is
preferable to abbreviate to “sen,” “set,” “sel,” and also use “san,”
“sin” for “sanen,” “sinen.”
[Illustration]
[Illustration]
We can now name any section. Take _e.g._ the line in the first cube
from senin to senel, we should call the line running from senin to
senel, senin senat senel, a line light yellow in colour with null
points.
[Illustration]
Here senat is the name for all of the line except its ends. Using
“senat” in this way does not mean that the line is the whole of senat,
but what there is of it is senat. It is a part of the senat region.
Thus also the triangle, which has its three vertices in senin, senel,
selen, is named thus:
Area: setat.
Sides: setan, senat, setet.
Vertices: senin, senel, sel.
The tetrahedron section of the tesseract can be thought of as a series
of plane sections in the successive sections of the tesseract shown in
fig. 114, p. 191. In b_{0} the section is the one written above. In
b_{1} the section is made by a plane which cuts the three edges from
sanen intermediate of their lengths and thus will be:
Area: satat.
Sides: satan, sanat, satet.
Vertices: sanan, sanet, sat.
The sections in b_{2}, b_{3} will be like the section in b_{1} but
smaller.
Finally in b_{4} the section plane simply passes through the corner
named sin.
Hence, putting these sections together in their right relation, from
the face setat, surrounded by the lines and points mentioned above,
there run:
3 faces: satan, sanat, satet
3 lines: sanan, sanet, sat
and these faces and lines run to the point sin. Thus the tetrahedron is
completely named.
The octahedron section of the tesseract, which can be traced from fig.
72, p. 129 by extending the lines there drawn, is named:
Front triangle selin, selat, selel, setal, senil, setit, selin with
area setat.
The sections between the front and rear triangle, of which one is shown
in 1b, another in 2b, are thus named, points and lines, salan, salat,
salet, satet, satel, satal, sanal, sanat, sanit, satit, satin, satan,
salan.
The rear triangle found in 3b by producing lines is sil, sitet, sinel,
sinat, sinin, sitan, sil.
The assemblage of sections constitute the solid body of the octahedron
satat with triangular faces. The one from the line selat to the point
sil, for instance, is named selin, selat, selel, salet, salat, salan,
sil. The whole interior is salat.
Shapes can easily be cut out of cardboard which, when folded together,
form not only the tetrahedron and the octahedron, but also samples of
all the sections of the tesseract taken as it passes cornerwise through
our space. To name and visualise with appropriate colours a series of
these sections is an admirable exercise for obtaining familiarity with
the subject.
EXTENSION AND CONNECTION WITH NUMBERS.
By extending the letter sequence it is of course possible to name a
larger field. By using the limit names the corners of each square can
be named.
Thus “en sen,” “an sen,” etc., will be the names of the points nearest
the origin in “en” and in “an.”
A field of points of which each one is indefinitely small is given by
the names written below.
[Illustration]
The squares are shown in dotted lines, the names denote the points.
These points are not mathematical points, but really minute areas.
Instead of starting with a set of squares and naming them, we can start
with a set of points.
By an easily remembered convention we can give names to such a region
of points.
Let the space names with a final “e” added denote the mathematical
points at the corner of each square nearest the origin. We have then
for the set of mathematical points indicated. This system is really
completely independent of the area system and is connected with it
merely for the purpose of facilitating the memory processes. The word
“ene” is pronounced like “eny,” with just sufficient attention to the
final vowel to distinguish it from the word “en.”
[Illustration]
Now, connecting the numbers 0, 1, 2 with the sequence e, a, i, and
also with the sequence n, t, l, we have a set of points named as with
numbers in a co-ordinate system. Thus “ene” is (0, 0) “ate” is (1,
1) “ite” is (2, 1). To pass to the area system the rule is that the
name of the square is formed from the name of its point nearest to the
origin by dropping the final e.
By using a notation analogous to the decimal system a larger field of
points can be named. It remains to assign a letter sequence to the
numbers from positive 0 to positive 9, and from negative 0 to negative
9, to obtain a system which can be used to denote both the usual
co-ordinate system of mapping and a system of named squares. The names
denoting the points all end with e. Those that denote squares end with
a consonant.
There are many considerations which must be attended to in extending
the sequences to be used, such as uniqueness in the meaning of the
words formed, ease of pronunciation, avoidance of awkward combinations.
I drop “s” altogether from the consonant series and short “u” from
the vowel series. It is convenient to have unsignificant letters at
disposal. A double consonant like “st” for instance can be referred to
without giving it a local significance by calling it “ust.” I increase
the number of vowels by considering a sound like “ra” to be a vowel,
using, that is, the letter “r” as forming a compound vowel.
The series is as follows:—
CONSONANTS.
0 1 2 3 4 5 6 7 8 9
positive n t l p f sh k ch nt st
negative z d th b v m g j nd sp
VOWELS.
0 1 2 3 4 5 6 7 8 9
positive e a i ee ae ai ar ra ri ree
negative er o oo io oe iu or ro roo rio
_Pronunciation._—e as in men; a as in man; i as in in; ee as in
between; ae as ay in may; ai as i in mine; ar as in art; er as ear in
earth; o as in on; oo as oo in soon; io as in clarion; oe as oa in oat;
iu pronounced like yew.
To name a point such as (23, 41) it is considered as (3, 1) on from
(20, 40) and is called “ifeete.” It is the initial point of the square
ifeet of the area system.
The preceding amplification of a space language has been introduced
merely for the sake of completeness. As has already been said nine
words and their combinations, applied to a few simple models suffice
for the purposes of our present enquiry.
_Printed by Hazell, Watson & Viney, Ld., London and Aylesbury._