FURTHER APPEARANCES OF A CUBE TO A PLANE-BEING.
Before leaving the observation of the cube, it is well to look at it for
a moment as it would appear to a plane-being, in whose world there was
such a fact as attraction. To do this, let the cube rest on the table,
so that its Dark-blue face is perpendicular in front of us. Now, let a
sheet of paper be placed in contact with the Dark-blue square. Let up
and sideways be the two dimensions of the plane-being, and away the
unknown direction. Let the line where the paper meets the table,
represent the surface of his earth. Then, there is to him, as all that
he can apprehend of the cube, a Dark-blue square standing upright; and,
when we look over the edge of the paper, and regard merely the part in
contact with the paper, we see what the plane-being would see.
If the cube be turned round the up line, the Brown line, the Orange line
will pass to the near side of the paper, and the section made by the
cube in the paper will be an oblong. Such an oblong can be cut out; and
when the cube is fitted into it, it can be seen that it is bounded by a
Brown line and a Blue-green line opposite thereto, while the other
boundaries are Black and White lines. Next, if we take a section
half-way between the Black and White squares, we shall have a square
cutting the plane of the aforesaid paper in a single line. With regard
to this section, all we have to inquire is, What will take the place of
this line as the cube turns? Obviously, the line will elongate. From a
Dark-blue line it will change to a Light-buff line, the colour of the
inside of the section, and will terminate in a Blue-green point instead
of a French-grey. Again, it is obvious that, if the cube turns round the
Orange line, it will give rise to a series of oblongs, stretching
upwards. This turning can be continued till the cube is wholly on the
near side of the paper, and only the Orange line remains. And, when the
cube has made half a revolution, the Dark-blue square will return into
the plane; but it will run downwards instead of upwards as at first.
Thereafter, if the cube turn further, a series of oblongs will appear,
all running downwards from the Orange line. Hence, if all the
appearances produced by the revolution of the cube have to be shown, it
must be supposed to be raised some distance above the plane-being’s
earth, so that those appearances may be shown which occur when it is
turned round the Orange line downwards, as well as when it is turned
upwards. The unknown direction comes into the plane either upwards or
downwards, but there is no special connection between it and either of
these directions. If it come in upwards, the Brown line goes nearwards
or -Y; if it come in downwards, or -Z, the Brown line goes away, or Y.
Let us consider more closely the directions which the plane-being would
have. Firstly, he would have up-and-down, that is, away from his earth
and towards it on the plane of the paper, the surface of his earth being
the line where the paper meets the table. Then, if he moved along the
surface of his earth, there would only be a line for him to move in, the
line running right and left. But, being the direction of his movement,
he would say it ran forwards and backwards. Thus he would simply have
the words up and down, forwards and backwards, and the expressions right
and left would have no meaning for him. If he were to frame a notion of
a world in higher dimensions, he must invent new words for distinctions
not within his experience.
To repeat the observations already made, let the cube be held in front
of the observer, and suppose the Dark-blue square extended on every side
so as to form a plane. Then let this plane be considered as independent
of the Dark-blue square. Now, holding the Brown line between finger and
thumb, and touching its extremities, the Gold and Light-blue points,
turn the cube round the Brown line. The Dark-blue square will leave the
plane, the Orange line will tend towards the -Y direction, and the Blue
line will finally come into the plane pointing in the +X direction. If
we move the cube so that the line which leaves the plane runs +Y, then
the line which before ran +Y will come into the plane in the direction
opposite to that of the line which has left the plane. The Blue line,
which runs in the unknown direction can come into either of the two
known directions of the plane. It can take the place of the Orange line
by turning the cube round the Brown line, or the place of the Brown line
by turning it round the Orange line. If the plane-being made models to
represent these two appearances of the cube, he would have identically
the same line, the Blue line, running in one of his known directions in
the first model, and in the other of his known directions in the second.
In studying the cube he would find it best to turn it so that the line
of unknown direction ran in that direction in the positive sense. In
that case, it would come into the plane in the negative sense of the
known directions.
Starting with the cube in front of the observer, there are two ways in
which the Vermilion square can be brought into the imaginary plane, that
is the extension of the Dark-blue square. If the cube turn round the
Brown line so that the Orange line goes away, (_i.e._ +Y), the Vermilion
square comes in on the left of the Brown line. If it turn in the
opposite direction, the Vermilion square comes in on the right of the
Brown line. Thus, if we identify the plane-being with the Brown line,
the Vermilion square would appear either behind or before him. These two
appearances of the Vermilion square would seem identical, but they could
not be made to coincide by any movement in the plane. The diagram (Fig.
5.) shows the difference in them. It is obvious that no turn in the
plane could put one in the place of the other, part for part. Thus the
plane-being apprehends the reversal of the unknown direction by the
disposition of his figures. If a figure, which lay on one side of a
line, changed into an identical figure on the other side of it, he could
be sure that a line of the figure, which at first ran in the positive
unknown direction, now ran in the negative unknown direction.
We have dwelt at great length on the appearances, which a cube would
present to a plane-being, and it will be found that all the points which
would be likely to cause difficulty hereafter, have been explained in
this obvious case.
There is, however, one other way, open to a plane-being of studying a
cube, to which we must attend. This is, by steady motion. Let the cube
come into the imaginary plane, which is the extension of the Dark-blue
square, _i.e._ let it touch the piece of paper which is standing
vertical on the table. Then let it travel through this plane at right
angles to it at the rate of an inch a minute. The plane-being would
first perceive a Dark-blue square, that is, he would see the coloured
lines bounding that square, and enclosed therein would be what he would
call a Dark-blue solid. In the movement of the cube, however, this
Dark-blue square would not last for more than a flash of time. (The
edges and points on the models are made very large; in reality they must
be supposed very minute.) This Dark-blue square would be succeeded by
one of the colour of the cube’s interior, _i.e._ by a Light-buff square.
But this colour of the interior would not be visible to the plane-being.
He would go round the square on his plane, and would see the bounding
lines, _viz._ Vermilion, White, Blue-green, Black. And at the corners he
would see Deep-yellow, Bright-blue, Crimson, and Blue points. These
lines and points would really be those parts of the faces and lines of
the cube, which were on the point of passing through his plane. Now,
there would be one difference between the Dark-blue square and the
Light-buff with their respective boundaries. The first only lasted for a
flash; the second would last for a minute or all but a minute. Consider
the Vermilion square. It appears to the plane-being as a line. The Brown
line also appears to him as a line. But there is a difference between
them. The Brown line only lasts for a flash, whereas the Vermilion line
lasts for a minute. Hence, in this mode of presentation, we may say that
for a plane-being a lasting line is the mode of apprehending a plane,
and a lasting plane (which is a plane-being’s solid) is the mode of
apprehending our solids. In the same way, the Blue line, as it passes
through his plane, gives rise to a point. This point lasts for a minute,
whereas the Gold point only lasted for a flash.