NOMENCLATURE AND ANALOGIES PRELIMINARY TO THE STUDY OF FOUR-DIMENSIONAL
FIGURES
In the following pages a method of designating different regions of
space by a systematic colour scheme has been adopted. The explanations
have been given in such a manner as to involve no reference to models,
the diagrams will be found sufficient. But to facilitate the study a
description of a set of models is given in an appendix which the reader
can either make for himself or obtain. If models are used the diagrams
in Chapters XI. and XII. will form a guide sufficient to indicate their
use. Cubes of the colours designated by the diagrams should be picked
out and used to reinforce the diagrams. The reader, in the following
description, should suppose that a board or wall stretches away from
him, against which the figures are placed.
[Illustration: Fig. 77.]
Take a square, one of those shown in Fig. 77 and give it a neutral
colour, let this colour be called “null,” and be such that it makes no
appreciable difference to any colour with which it mixed. If there is
no such real colour let us imagine such a colour, and assign to it the
properties of the number zero, which makes no difference in any number
to which it is added.
Above this square place a red square. Thus we symbolise the going up by
adding red to null.
Away from this null square place a yellow square, and represent going
away by adding yellow to null.
To complete the figure we need a fourth square. Colour this orange,
which is a mixture of red and yellow, and so appropriately represents a
going in a direction compounded of up and away. We have thus a colour
scheme which will serve to name the set of squares drawn. We have two
axes of colours—red and yellow—and they may occupy as in the figure
the direction up and away, or they may be turned about; in any case
they enable us to name the four squares drawn in their relation to one
another.
Now take, in Fig. 78, nine squares, and suppose that at the end of the
going in any direction the colour started with repeats itself.
[Illustration: Fig. 78.]
We obtain a square named as shown.
Let us now, in fig. 79, suppose the number of squares to be increased,
keeping still to the principle of colouring already used.
Here the nulls remain four in number. There are three reds between the
first null and the null above it, three yellows between the first null
and the null beyond it, while the oranges increase in a double way.
[Illustration: Fig. 79.]
Suppose this process of enlarging the number of the squares to be
indefinitely pursued and the total figure obtained to be reduced in
size, we should obtain a square of which the interior was all orange,
while the lines round it were red and yellow, and merely the points
null colour, as in fig. 80. Thus all the points, lines, and the area
would have a colour.
[Illustration: Fig. 80.]
We can consider this scheme to originate thus:—Let a null point move
in a yellow direction and trace out a yellow line and end in a null
point. Then let the whole line thus traced move in a red direction. The
null points at the ends of the line will produce red lines, and end in
null points. The yellow line will trace out a yellow and red, or orange
square.
Now, turning back to fig. 78, we see that these two ways of naming, the
one we started with and the one we arrived at, can be combined.
By its position in the group of four squares, in fig. 77, the null
square has a relation to the yellow and to the red directions. We can
speak therefore of the red line of the null square without confusion,
meaning thereby the line AB, fig. 81, which runs up from the initial
null point A in the figure as drawn. The yellow line of the null square
is its lower horizontal line AC as it is situated in the figure.
[Illustration: Fig. 81.]
If we wish to denote the upper yellow line BD, fig. 81, we can speak
of it as the yellow γ line, meaning the yellow line which is separated
from the primary yellow line by the red movement.
In a similar way each of the other squares has null points, red and
yellow lines. Although the yellow square is all yellow, its line CD,
for instance, can be referred to as its red line.
This nomenclature can be extended.
If the eight cubes drawn, in fig. 82, are put close together, as on
the right hand of the diagram, they form a cube, and in them, as thus
arranged, a going up is represented by adding red to the zero, or
null colour, a going away by adding yellow, a going to the right by
adding white. White is used as a colour, as a pigment, which produces
a colour change in the pigments with which it is mixed. From whatever
cube of the lower set we start, a motion up brings us to a cube showing
a change to red, thus light yellow becomes light yellow red, or light
orange, which is called ochre. And going to the right from the null on
the left we have a change involving the introduction of white, while
the yellow change runs from front to back. There are three colour
axes—the red, the white, the yellow—and these run in the position the
cubes occupy in the drawing—up, to the right, away—but they could be
turned about to occupy any positions in space.
[Illustration: Fig. 82.]
[Illustration: Fig. 83. The three layers.]
We can conveniently represent a block of cubes by three sets of
squares, representing each the base of a cube.
Thus the block, fig. 83, can be represented by the layers on the
right. Here, as in the case of the plane, the initial colours repeat
themselves at the end of the series.
Proceeding now to increase the number of the cubes we obtain fig. 84,
in which the initial letters of the colours are given instead of their
full names.
Here we see that there are four null cubes as before, but the series
which spring from the initial corner will tend to become lines of
cubes, as also the sets of cubes parallel to them, starting from other
corners. Thus, from the initial null springs a line of red cubes, a
line of white cubes, and a line of yellow cubes.
If the number of the cubes is largely increased, and the size of the
whole cube is diminished, we get a cube with null points, and the edges
coloured with these three colours.
[Illustration: Fig. 84.]
The light yellow cubes increase in two ways, forming ultimately a sheet
of cubes, and the same is true of the orange and pink sets. Hence,
ultimately the cube thus formed would have red, white, and yellow
lines surrounding pink, orange, and light yellow faces. The ochre cubes
increase in three ways, and hence ultimately the whole interior of the
cube would be coloured ochre.
We have thus a nomenclature for the points, lines, faces, and solid
content of a cube, and it can be named as exhibited in fig. 85.
[Illustration: Fig. 85.]
We can consider the cube to be produced in the following way. A null
point moves in a direction to which we attach the colour indication
yellow; it generates a yellow line and ends in a null point. The yellow
line thus generated moves in a direction to which we give the colour
indication red. This lies up in the figure. The yellow line traces out
a yellow, red, or orange square, and each of its null points trace out
a red line, and ends in a null point.
This orange square moves in a direction to which we attribute the
colour indication white, in this case the direction is the right. The
square traces out a cube coloured orange, red, or ochre, the red lines
trace out red to white or pink squares, and the yellow lines trace out
light yellow squares, each line ending in a line of its own colour.
While the points each trace out a null + white, or white line to end in
a null point.
Now returning to the first block of eight cubes we can name each point,
line, and square in them by reference to the colour scheme, which they
determine by their relation to each other.
Thus, in fig. 86, the null cube touches the red cube by a light yellow
square; it touches the yellow cube by a pink square, and touches the
white cube by an orange square.
There are three axes to which the colours red, yellow, and white are
assigned, the faces of each cube are designated by taking these colours
in pairs. Taking all the colours together we get a colour name for the
solidity of a cube.
[Illustration: Fig. 86.]
Let us now ask ourselves how the cube could be presented to the plane
being. Without going into the question of how he could have a real
experience of it, let us see how, if we could turn it about and show it
to him, he, under his limitations, could get information about it. If
the cube were placed with its red and yellow axes against a plane, that
is resting against it by its orange face, the plane being would observe
a square surrounded by red and yellow lines, and having null points.
See the dotted square, fig. 87.
[Illustration: Fig. 87.]
We could turn the cube about the red line so that a different face
comes into juxtaposition with the plane.
Suppose the cube turned about the red line. As it is turning from its
first position all of it except the red line leaves the plane—goes
absolutely out of the range of the plane being’s apprehension. But when
the yellow line points straight out from the plane then the pink face
comes into contact with it. Thus the same red line remaining as he saw
it at first, now towards him comes a face surrounded by white and red
lines.
If we call the direction to the right the unknown direction, then
the line he saw before, the yellow line, goes out into this unknown
direction, and the line which before went into the unknown direction,
comes in. It comes in in the opposite direction to that in which the
yellow line ran before; the interior of the face now against the plane
is pink. It is a property of two lines at right angles that, if one
turns out of a given direction and stands at right angles to it, then
the other of the two lines comes in, but runs the opposite way in that
given direction, as in fig. 88.
[Illustration: Fig. 88.]
Now these two presentations of the cube would seem, to the plane
creature like perfectly different material bodies, with only that line
in common in which they both meet.
Again our cube can be turned about the yellow line. In this case the
yellow square would disappear as before, but a new square would come
into the plane after the cube had rotated by an angle of 90° about this
line. The bottom square of the cube would come in thus in figure 89.
The cube supposed in contact with the plane is rotated about the lower
yellow line and then the bottom face is in contact with the plane.
Here, as before, the red line going out into the unknown dimension,
the white line which before ran in the unknown dimension would come
in downwards in the opposite sense to that in which the red line ran
before.
[Illustration: Fig. 89.]
Now if we use _i_, _j_, _k_, for the three space directions, _i_ left
to right, _j_ from near away, _k_ from below up; then, using the colour
names for the axes, we have that first of all white runs _i_, yellow
runs _j_, red runs _k_; then after the first turning round the _k_
axis, white runs negative _j_, yellow runs _i_, red runs _k_; thus we
have the table:—
_i_ _j_ _k_
1st position white yellow red
2nd position yellow white— red
3rd position red yellow white—
Here white with a negative sign after it in the column under _j_ means
that white runs in the negative sense of the _j_ direction.
We may express the fact in the following way:— In the plane there is
room for two axes while the body has three. Therefore in the plane we
can represent any two. If we want to keep the axis that goes in the
unknown dimension always running in the positive sense, then the axis
which originally ran in the unknown dimension (the white axis) must
come in in the negative sense of that axis which goes out of the plane
into the unknown dimension.
It is obvious that the unknown direction, the direction in which the
white line runs at first, is quite distinct from any direction which
the plane creature knows. The white line may come in towards him, or
running down. If he is looking at a square, which is the face of a cube
(looking at it by a line), then any one of the bounding lines remaining
unmoved, another face of the cube may come in, any one of the faces,
namely, which have the white line in them. And the white line comes
sometimes in one of the space directions he knows, sometimes in another.
Now this turning which leaves a line unchanged is something quite
unlike any turning he knows in the plane. In the plane a figure turns
round a point. The square can turn round the null point in his plane,
and the red and yellow lines change places, only of course, as with
every rotation of lines at right angles, if red goes where yellow went,
yellow comes in negative of red’s old direction.
This turning, as the plane creature conceives it, we should call
turning about an axis perpendicular to the plane. What he calls turning
about the null point we call turning about the white line as it stands
out from his plane. There is no such thing as turning about a point,
there is always an axis, and really much more turns than the plane
being is aware of.
Taking now a different point of view, let us suppose the cubes to be
presented to the plane being by being passed transverse to his plane.
Let us suppose the sheet of matter over which the plane being and all
objects in his world slide, to be of such a nature that objects can
pass through it without breaking it. Let us suppose it to be of the
same nature as the film of a soap bubble, so that it closes around
objects pushed through it, and, however the object alters its shape as
it passes through it, let us suppose this film to run up to the contour
of the object in every part, maintaining its plane surface unbroken.
Then we can push a cube or any object through the film and the plane
being who slips about in the film will know the contour of the cube
just and exactly where the film meets it.
[Illustration: Fig. 90.]
Fig. 90 represents a cube passing through a plane film. The plane being
now comes into contact with a very thin slice of the cube somewhere
between the left and right hand faces. This very thin slice he thinks
of as having no thickness, and consequently his idea of it is what we
call a section. It is bounded by him by pink lines front and back,
coming from the part of the pink face he is in contact with, and above
and below, by light yellow lines. Its corners are not null-coloured
points, but white points, and its interior is ochre, the colour of the
interior of the cube.
If now we suppose the cube to be an inch in each dimension, and to pass
across, from right to left, through the plane, then we should explain
the appearances presented to the plane being by saying: First of all
you have the face of a cube, this lasts only a moment; then you have a
figure of the same shape but differently coloured. This, which appears
not to move to you in any direction which you know of, is really moving
transverse to your plane world. Its appearance is unaltered, but each
moment it is something different—a section further on, in the white,
the unknown dimension. Finally, at the end of the minute, a face comes
in exactly like the face you first saw. This finishes up the cube—it is
the further face in the unknown dimension.
The white line, which extends in length just like the red or the
yellow, you do not see as extensive; you apprehend it simply as an
enduring white point. The null point, under the condition of movement
of the cube, vanishes in a moment, the lasting white point is really
your apprehension of a white line, running in the unknown dimension.
In the same way the red line of the face by which the cube is first in
contact with the plane lasts only a moment, it is succeeded by the pink
line, and this pink line lasts for the inside of a minute. This lasting
pink line in your apprehension of a surface, which extends in two
dimensions just like the orange surface extends, as you know it, when
the cube is at rest.
But the plane creature might answer, “This orange object is substance,
solid substance, bounded completely and on every side.”
Here, of course, the difficulty comes in. His solid is our surface—his
notion of a solid is our notion of an abstract surface with no
thickness at all.
We should have to explain to him that, from every point of what he
called a solid, a new dimension runs away. From every point a line
can be drawn in a direction unknown to him, and there is a solidity
of a kind greater than that which he knows. This solidity can only
be realised by him by his supposing an unknown direction, by motion
in which what he conceives to be solid matter instantly disappears.
The higher solid, however, which extends in this dimension as well
as in those which he knows, lasts when a motion of that kind takes
place, different sections of it come consecutively in the plane
of his apprehension, and take the place of the solid which he at
first conceives to be all. Thus, the higher solid—our solid in
contradistinction to his area solid, his two-dimensional solid, must
be conceived by him as something which has duration in it, under
circumstances in which his matter disappears out of his world.
We may put the matter thus, using the conception of motion.
A null point moving in a direction away generates a yellow line, and
the yellow line ends in a null point. We suppose, that is, a point
to move and mark out the products of this motion in such a manner.
Now suppose this whole line as thus produced to move in an upward
direction; it traces out the two-dimensional solid, and the plane being
gets an orange square. The null point moves in a red line and ends in
a null point, the yellow line moves and generates an orange square and
ends in a yellow line, the farther null point generates a red line and
ends in a null point. Thus, by movement in two successive directions
known to him, he can imagine his two-dimensional solid produced with
all its boundaries.
Now we tell him: “This whole two-dimensional solid can move in a third
or unknown dimension to you. The null point moving in this dimension
out of your world generates a white line and ends in a null point. The
yellow line moving generates a light yellow two-dimensional solid and
ends in a yellow line, and this two-dimensional solid, lying end on to
your plane world, is bounded on the far side by the other yellow line.
In the same way each of the lines surrounding your square traces out an
area, just like the orange area you know. But there is something new
produced, something which you had no idea of before; it is that which
is produced by the movement of the orange square. That, than which you
can imagine nothing more solid, itself moves in a direction open to it
and produces a three-dimensional solid. Using the addition of white
to symbolise the products of this motion this new kind of solid will
be light orange or ochre, and it will be bounded on the far side by
the final position of the orange square which traced it out, and this
final position we suppose to be coloured like the square in its first
position, orange with yellow and red boundaries and null corners.”
This product of movement, which it is so easy for us to describe, would
be difficult for him to conceive. But this difficulty is connected
rather with its totality than with any particular part of it.
Any line, or plane of this, to him higher, solid we could show to him,
and put in his sensible world.
We have already seen how the pink square could be put in his world by
a turning of the cube about the red line. And any section which we can
conceive made of the cube could be exhibited to him. You have simply to
turn the cube and push it through, so that the plane of his existence
is the plane which cuts out the given section of the cube, then the
section would appear to him as a solid. In his world he would see the
contour, get to any part of it by digging down into it.
THE PROCESS BY WHICH A PLANE BEING WOULD GAIN A NOTION OF A SOLID.
If we suppose the plane being to have a general idea of the existence
of a higher solid—our solid—we must next trace out in detail the
method, the discipline, by which he would acquire a working familiarity
with our space existence. The process begins with an adequate
realisation of a simple solid figure. For this purpose we will suppose
eight cubes forming a larger cube, and first we will suppose each cube
to be coloured throughout uniformly. Let the cubes in fig. 91 be the
eight making a larger cube.
[Illustration: Fig. 91.]
Now, although each cube is supposed to be coloured entirely through
with the colour, the name of which is written on it, still we can
speak of the faces, edges, and corners of each cube as if the colour
scheme we have investigated held for it. Thus, on the null cube we can
speak of a null point, a red line, a white line, a pink face, and so
on. These colour designations are shown on No. 1 of the views of the
tesseract in the plate. Here these colour names are used simply in
their geometrical significance. They denote what the particular line,
etc., referred to would have as its colour, if in reference to the
particular cube the colour scheme described previously were carried out.
If such a block of cubes were put against the plane and then passed
through it from right to left, at the rate of an inch a minute, each
cube being an inch each way, the plane being would have the following
appearances:—
First of all, four squares null, yellow, red, orange, lasting each a
minute; and secondly, taking the exact places of these four squares,
four others, coloured white, light yellow, pink, ochre. Thus, to make
a catalogue of the solid body, he would have to put side by side in
his world two sets of four squares each, as in fig. 92. The first are
supposed to last a minute, and then the others to come in in place of
them, and also last a minute.
[Illustration: Fig. 92.]
In speaking of them he would have to denote what part of the respective
cube each square represents. Thus, at the beginning he would have null
cube orange face, and after the motion had begun he would have null
cube ochre section. As he could get the same coloured section whichever
way the cube passed through, it would be best for him to call this
section white section, meaning that it is transverse to the white axis.
These colour-names, of course, are merely used as names, and do not
imply in this case that the object is really coloured. Finally, after
a minute, as the first cube was passing beyond his plane he would have
null cube orange face again.
The same names will hold for each of the other cubes, describing what
face or section of them the plane being has before him; and the second
wall of cubes will come on, continue, and go out in the same manner. In
the area he thus has he can represent any movement which we carry out
in the cubes, as long as it does not involve a motion in the direction
of the white axis. The relation of parts that succeed one another in
the direction of the white axis is realised by him as a consecution of
states.
Now, his means of developing his space apprehension lies in this, that
that which is represented as a time sequence in one position of the
cubes, can become a real co-existence, _if something that has a real
co-existence becomes a time sequence_.
We must suppose the cubes turned round each of the axes, the red line,
and the yellow line, then something, which was given as time before,
will now be given as the plane creature’s space; something, which was
given as space before, will now be given as a time series as the cube
is passed through the plane.
The three positions in which the cubes must be studied are the one
given above and the two following ones. In each case the original null
point which was nearest to us at first is marked by an asterisk. In
figs. 93 and 94 the point marked with a star is the same in the cubes
and in the plane view.
[Illustration: Fig. 93. The cube swung round the red line, so that the
white line points towards us.]
In fig. 93 the cube is swung round the red line so as to point towards
us, and consequently the pink face comes next to the plane. As it
passes through there are two varieties of appearance designated by
the figures 1 and 2 in the plane. These appearances are named in the
figure, and are determined by the order in which the cubes come in the
motion of the whole block through the plane.
With regard to these squares severally, however, different names must
be used, determined by their relations in the block.
Thus, in fig. 93, when the cube first rests against the plane the null
cube is in contact by its pink face; as the block passes through we get
an ochre section of the null cube, but this is better called a yellow
section, as it is made by a plane perpendicular to the yellow line.
When the null cube has passed through the plane, as it is leaving it,
we get again a pink face.
[Illustration: Fig. 94. The cube swung round yellow line, with red line
running from left to right, and white line running down.]
The same series of changes take place with the cube appearances which
follow on those of the null cube. In this motion the yellow cube
follows on the null cube, and the square marked yellow in 2 in the
plane will be first “yellow pink face,” then “yellow yellow section,”
then “yellow pink face.”
In fig. 94, in which the cube is turned about the yellow line, we have
a certain difficulty, for the plane being will find that the position
his squares are to be placed in will lie below that which they first
occupied. They will come where the support was on which he stood his
first set of squares. He will get over this difficulty by moving his
support.
Then, since the cubes come upon his plane by the light yellow face, he
will have, taking the null cube as before for an example, null, light
yellow face; null, red section, because the section is perpendicular
to the red line; and finally, as the null cube leaves the plane, null,
light yellow face. Then, in this case red following on null, he will
have the same series of views of the red as he had of the null cube.
[Illustration: Fig. 95.]
There is another set of considerations which we will briefly allude to.
Suppose there is a hollow cube, and a string is stretched across it
from null to null, _r_, _y_, _wh_, as we may call the far diagonal
point, how will this string appear to the plane being as the cube moves
transverse to his plane?
Let us represent the cube as a number of sections, say 5, corresponding
to 4 equal divisions made along the white line perpendicular to it.
We number these sections 0, 1, 2, 3, 4, corresponding to the distances
along the white line at which they are taken, and imagine each section
to come in successively, taking the place of the preceding one.
These sections appear to the plane being, counting from the first, to
exactly coincide each with the preceding one. But the section of the
string occupies a different place in each to that which it does in the
preceding section. The section of the string appears in the position
marked by the dots. Hence the slant of the string appears as a motion
in the frame work marked out by the cube sides. If we suppose the
motion of the cube not to be recognised, then the string appears to the
plane being as a moving point. Hence extension on the unknown dimension
appears as duration. Extension sloping in the unknown direction appears
as continuous movement.