THE SIMPLEST FOUR-DIMENSIONAL SOLID
A plane being, in learning to apprehend solid existence, must first
of all realise that there is a sense of direction altogether wanting
to him. That which we call right and left does not exist in his
perception. He must assume a movement in a direction, and a distinction
of positive and negative in that direction, which has no reality
corresponding to it in the movements he can make. This direction, this
new dimension, he can only make sensible to himself by bringing in
time, and supposing that changes, which take place in time, are due
to objects of a definite configuration in three dimensions passing
transverse to his plane, and the different sections of it being
apprehended as changes of one and the same plane figure.
He must also acquire a distinct notion about his plane world, he must
no longer believe that it is the all of space, but that space extends
on both sides of it. In order, then, to prevent his moving off in this
unknown direction, he must assume a sheet, an extended solid sheet, in
two dimensions, against which, in contact with which, all his movements
take place.
When we come to think of a four-dimensional solid, what are the
corresponding assumptions which we must make?
We must suppose a sense which we have not, a sense of direction
wanting in us, something which a being in a four-dimensional world
has, and which we have not. It is a sense corresponding to a new space
direction, a direction which extends positively and negatively from
every point of our space, and which goes right away from any space
direction we know of. The perpendicular to a plane is perpendicular,
not only to two lines in it, but to every line, and so we must conceive
this fourth dimension as running perpendicularly to each and every line
we can draw in our space.
And as the plane being had to suppose something which prevented his
moving off in the third, the unknown dimension to him, so we have to
suppose something which prevents us moving off in the direction unknown
to us. This something, since we must be in contact with it in every one
of our movements, must not be a plane surface, but a solid; it must be
a solid, which in every one of our movements we are against, not in.
It must be supposed as stretching out in every space dimension that we
know; but we are not in it, we are against it, we are next to it, in
the fourth dimension.
That is, as the plane being conceives himself as having a very small
thickness in the third dimension, of which he is not aware in his
sense experience, so we must suppose ourselves as having a very small
thickness in the fourth dimension, and, being thus four-dimensional
beings, to be prevented from realising that we are such beings by a
constraint which keeps us always in contact with a vast solid sheet,
which stretches on in every direction. We are against that sheet, so
that, if we had the power of four-dimensional movement, we should
either go away from it or through it; all our space movements as we
know them being such that, performing them, we keep in contact with
this solid sheet.
Now consider the exposition a plane being would make for himself as to
the question of the enclosure of a square, and of a cube.
He would say the square A, in Fig. 96, is completely enclosed by the
four squares, A far, A near, A above, A below, or as they are written
A_n_, A_f_, A_a_, A_b_.
[Illustration: Fig. 96.]
If now he conceives the square A to move in the, to him, unknown
dimension it will trace out a cube, and the bounding squares will
form cubes. Will these completely surround the cube generated by A?
No; there will be two faces of the cube made by A left uncovered;
the first, that face which coincides with the square A in its first
position; the next, that which coincides with the square A in its
final position. Against these two faces cubes must be placed in order
to completely enclose the cube A. These may be called the cubes left
and right or A_l_ and A_r_. Thus each of the enclosing squares of the
square A becomes a cube and two more cubes are wanted to enclose the
cube formed by the movement of A in the third dimension.
[Illustration: Fig. 97.]
The plane being could not see the square A with the squares A_n_, A_f_,
etc., placed about it, because they completely hide it from view; and
so we, in the analogous case in our three-dimensional world, cannot
see a cube A surrounded by six other cubes. These cubes we will call A
near A_n_, A far A_f_, A above A_a_, A below A_b_, A left A_l_, A right
A_r_, shown in fig. 97. If now the cube A moves in the fourth dimension
right out of space, it traces out a higher cube—a tesseract, as it may
be called. Each of the six surrounding cubes carried on in the same
motion will make a tesseract also, and these will be grouped around the
tesseract formed by A. But will they enclose it completely?
All the cubes A_n_, A_f_, etc., lie in our space. But there is nothing
between the cube A and that solid sheet in contact with which every
particle of matter is. When the cube A moves in the fourth direction
it starts from its position, say A_k_, and ends in a final position
A_n_ (using the words “ana” and “kata” for up and down in the fourth
dimension). Now the movement in this fourth dimension is not bounded by
any of the cubes A_n_, A_f_, nor by what they form when thus moved. The
tesseract which A becomes is bounded in the positive and negative ways
in this new direction by the first position of A and the last position
of A. Or, if we ask how many tesseracts lie around the tesseract which
A forms, there are eight, of which one meets it by the cube A, and
another meets it by a cube like A at the end of its motion.
We come here to a very curious thing. The whole solid cube A is to be
looked on merely as a boundary of the tesseract.
Yet this is exactly analogous to what the plane being would come to in
his study of the solid world. The square A (fig. 96), which the plane
being looks on as a solid existence in his plane world, is merely the
boundary of the cube which he supposes generated by its motion.
The fact is that we have to recognise that, if there is another
dimension of space, our present idea of a solid body, as one which
has three dimensions only, does not correspond to anything real,
but is the abstract idea of a three-dimensional boundary limiting a
four-dimensional solid, which a four-dimensional being would form. The
plane being’s thought of a square is not the thought of what we should
call a possibly existing real square, but the thought of an abstract
boundary, the face of a cube.
Let us now take our eight coloured cubes, which form a cube in
space, and ask what additions we must make to them to represent
the simplest collection of four-dimensional bodies—namely, a group
of them of the same extent in every direction. In plane space we
have four squares. In solid space we have eight cubes. So we should
expect in four-dimensional space to have sixteen four-dimensional
bodies-bodies which in four-dimensional space correspond to cubes in
three-dimensional space, and these bodies we call tesseracts.
Given then the null, white, red, yellow cubes, and those which make up
the block, we notice that we represent perfectly well the extension
in three directions (fig. 98). From the null point of the null cube,
travelling one inch, we come to the white cube; travelling one inch
away we come to the yellow cube; travelling one inch up we come to the
red cube. Now, if there is a fourth dimension, then travelling from the
same null point for one inch in that direction, we must come to the
body lying beyond the null region.
[Illustration: Fig. 98.]
I say null region, not cube; for with the introduction of the fourth
dimension each of our cubes must become something different from cubes.
If they are to have existence in the fourth dimension, they must be
“filled up from” in this fourth dimension.
Now we will assume that as we get a transference from null to white
going in one way, from null to yellow going in another, so going
from null in the fourth direction we have a transference from null
to blue, using thus the colours white, yellow, red, blue, to denote
transferences in each of the four directions—right, away, up, unknown
or fourth dimension.
[Illustration: Fig. 99.
A plane being’s representation of a block of eight cubes by two sets of
four squares.]
Hence, as the plane being must represent the solid regions, he would
come to by going right, as four squares lying in some position in his
plane, arbitrarily chosen, side by side with his original four squares,
so we must represent those eight four-dimensional regions, which we
should come to by going in the fourth dimension from each of our eight
cubes, by eight cubes placed in some arbitrary position relative to our
first eight cubes.
[Illustration: Fig. 100.]
Our representation of a block of sixteen tesseracts by two blocks of
eight cubes.[3]
[3] The eight cubes used here in 2 can be found in the second of the
model blocks. They can be taken out and used.
Hence, of the two sets of eight cubes, each one will serve us as a
representation of one of the sixteen tesseracts which form one single
block in four-dimensional space. Each cube, as we have it, is a tray,
as it were, against which the real four-dimensional figure rests—just
as each of the squares which the plane being has is a tray, so to
speak, against which the cube it represents could rest.
If we suppose the cubes to be one inch each way, then the original
eight cubes will give eight tesseracts of the same colours, or the
cubes, extending each one inch in the fourth dimension.
But after these there come, going on in the fourth dimension, eight
other bodies, eight other tesseracts. These must be there, if we
suppose the four-dimensional body we make up to have two divisions, one
inch each in each of four directions.
The colour we choose to designate the transference to this second
region in the fourth dimension is blue. Thus, starting from the null
cube and going in the fourth dimension, we first go through one inch of
the null tesseract, then we come to a blue cube, which is the beginning
of a blue tesseract. This blue tesseract stretches one inch farther on
in the fourth dimension.
Thus, beyond each of the eight tesseracts, which are of the same colour
as the cubes which are their bases, lie eight tesseracts whose colours
are derived from the colours of the first eight by adding blue. Thus—
Null gives blue
Yellow ” green
Red ” purple
Orange ” brown
White ” light blue
Pink ” light purple
Light yellow ” light green
Ochre ” light brown
The addition of blue to yellow gives green—this is a natural
supposition to make. It is also natural to suppose that blue added to
red makes purple. Orange and blue can be made to give a brown, by using
certain shades and proportions. And ochre and blue can be made to give
a light brown.
But the scheme of colours is merely used for getting a definite and
realisable set of names and distinctions visible to the eye. Their
naturalness is apparent to any one in the habit of using colours, and
may be assumed to be justifiable, as the sole purpose is to devise a
set of names which are easy to remember, and which will give us a set
of colours by which diagrams may be made easy of comprehension. No
scientific classification of colours has been attempted.
Starting, then, with these sixteen colour names, we have a catalogue of
the sixteen tesseracts, which form a four-dimensional block analogous
to the cubic block. But the cube which we can put in space and look at
is not one of the constituent tesseracts; it is merely the beginning,
the solid face, the side, the aspect, of a tesseract.
We will now proceed to derive a name for each region, point, edge,
plane face, solid and a face of the tesseract.
The system will be clear, if we look at a representation in the plane
of a tesseract with three, and one with four divisions in its side.
The tesseract made up of three tesseracts each way corresponds to the
cube made up of three cubes each way, and will give us a complete
nomenclature.
In this diagram, fig. 101, 1 represents a cube of 27 cubes, each of
which is the beginning of a tesseract. These cubes are represented
simply by their lowest squares, the solid content must be understood. 2
represents the 27 cubes which are the beginnings of the 27 tesseracts
one inch on in the fourth dimension. These tesseracts are represented
as a block of cubes put side by side with the first block, but in
their proper positions they could not be in space with the first set. 3
represents 27 cubes (forming a larger cube) which are the beginnings of
the tesseracts, which begin two inches in the fourth direction from our
space and continue another inch.
[Illustration: Fig. 101.]
[Illustration: Fig. 102[4]]
[4] The coloured plate, figs. 1, 2, 3, shows these relations more
conspicuously.
In fig. 102, we have the representation of a block of 4 × 4 × 4 × 4
or 256 tesseracts. They are given in four consecutive sections, each
supposed to be taken one inch apart in the fourth dimension, and so
giving four blocks of cubes, 64 in each block. Here we see, comparing
it with the figure of 81 tesseracts, that the number of the different
regions show a different tendency of increase. By taking five blocks of
five divisions each way this would become even more clear.
We see, fig. 102, that starting from the point at any corner, the white
coloured regions only extend out in a line. The same is true for the
yellow, red, and blue. With regard to the latter it should be noticed
that the line of blues does not consist in regions next to each other
in the drawing, but in portions which come in in different cubes.
The portions which lie next to one another in the fourth dimension
must always be represented so, when we have a three-dimensional
representation. Again, those regions such as the pink one, go on
increasing in two dimensions. About the pink region this is seen
without going out of the cube itself, the pink regions increase in
length and height, but in no other dimension. In examining these
regions it is sufficient to take one as a sample.
The purple increases in the same manner, for it comes in in a
succession from below to above in block 2, and in a succession from
block to block in 2 and 3. Now, a succession from below to above
represents a continuous extension upwards, and a succession from block
to block represents a continuous extension in the fourth dimension.
Thus the purple regions increase in two dimensions, the upward and
the fourth, so when we take a very great many divisions, and let each
become very small, the purple region forms a two-dimensional extension.
In the same way, looking at the regions marked l. b. or light blue,
which starts nearest a corner, we see that the tesseracts occupying
it increase in length from left to right, forming a line, and that
there are as many lines of light blue tesseracts as there are sections
between the first and last section. Hence the light blue tesseracts
increase in number in two ways—in the right and left, and in the fourth
dimension. They ultimately form what we may call a plane surface.
Now all those regions which contain a mixture of two simple colours,
white, yellow, red, blue, increase in two ways. On the other hand,
those which contain a mixture of three colours increase in three ways.
Take, for instance, the ochre region; this has three colours, white,
yellow, red; and in the cube itself it increases in three ways.
Now regard the orange region; if we add blue to this we get a brown.
The region of the brown tesseracts extends in two ways on the left of
the second block, No. 2 in the figure. It extends also from left to
right in succession from one section to another, from section 2 to
section 3 in our figure.
Hence the brown tesseracts increase in number in three dimensions
upwards, to and fro, fourth dimension. Hence they form a cubic, a
three-dimensional region; this region extends up and down, near
and far, and in the fourth direction, but is thin in the direction
from left to right. It is a cube which, when the complete tesseract
is represented in our space, appears as a series of faces on the
successive cubic sections of the tesseract. Compare fig. 103 in which
the middle block, 2, stands as representing a great number of sections
intermediate between 1 and 3.
In a similar way from the pink region by addition of blue we have
the light purple region, which can be seen to increase in three ways
as the number of divisions becomes greater. The three ways in which
this region of tesseracts extends is up and down, right and left,
fourth dimension. Finally, therefore, it forms a cubic mass of very
small tesseracts, and when the tesseract is given in space sections
it appears on the faces containing the upward and the right and left
dimensions.
We get then altogether, as three-dimensional regions, ochre, brown,
light purple, light green.
Finally, there is the region which corresponds to a mixture of all the
colours; there is only one region such as this. It is the one that
springs from ochre by the addition of blue—this colour we call light
brown.
Looking at the light brown region we see that it increases in four
ways. Hence, the tesseracts of which it is composed increase in
number in each of four dimensions, and the shape they form does not
remain thin in any of the four dimensions. Consequently this region
becomes the solid content of the block of tesseracts, itself; it
is the real four-dimensional solid. All the other regions are then
boundaries of this light brown region. If we suppose the process
of increasing the number of tesseracts and diminishing their size
carried on indefinitely, then the light brown coloured tesseracts
become the whole interior mass, the three-coloured tesseracts become
three-dimensional boundaries, thin in one dimension, and form the
ochre, the brown, the light purple, the light green. The two-coloured
tesseracts become two-dimensional boundaries, thin in two dimensions,
_e.g._, the pink, the green, the purple, the orange, the light blue,
the light yellow. The one-coloured tesseracts become bounding lines,
thin in three dimensions, and the null points become bounding corners,
thin in four dimensions. From these thin real boundaries we can pass in
thought to the abstractions—points, lines, faces, solids—bounding the
four-dimensional solid, which in this case is light brown coloured, and
under this supposition the light brown coloured region is the only real
one, is the only one which is not an abstraction.
It should be observed that, in taking a square as the representation
of a cube on a plane, we only represent one face, or the section
between two faces. The squares, as drawn by a plane being, are not the
cubes themselves, but represent the faces or the sections of a cube.
Thus in the plane being’s diagram a cube of twenty-seven cubes “null”
represents a cube, but is really, in the normal position, the orange
square of a null cube, and may be called null, orange square.
A plane being would save himself confusion if he named his
representative squares, not by using the names of the cubes simply, but
by adding to the names of the cubes a word to show what part of a cube
his representative square was.
Thus a cube null standing against his plane touches it by null orange
face, passing through his plane it has in the plane a square as trace,
which is null white section, if we use the phrase white section to
mean a section drawn perpendicular to the white line. In the same way
the cubes which we take as representative of the tesseract are not
the tesseract itself, but definite faces or sections of it. In the
preceding figures we should say then, not null, but “null tesseract
ochre cube,” because the cube we actually have is the one determined by
the three axes, white, red, yellow.
There is another way in which we can regard the colour nomenclature of
the boundaries of a tesseract.
Consider a null point to move tracing out a white line one inch in
length, and terminating in a null point, see fig. 103 or in the
coloured plate.
Then consider this white line with its terminal points itself to move
in a second dimension, each of the points traces out a line, the line
itself traces out an area, and gives two lines as well, its initial and
its final position.
Thus, if we call “a region” any element of the figure, such as a point,
or a line, etc., every “region” in moving traces out a new kind of
region, “a higher region,” and gives two regions of its own kind, an
initial and a final position. The “higher region” means a region with
another dimension in it.
Now the square can move and generate a cube. The square light yellow
moves and traces out the mass of the cube. Letting the addition of
red denote the region made by the motion in the upward direction we
get an ochre solid. The light yellow face in its initial and terminal
positions give the two square boundaries of the cube above and below.
Then each of the four lines of the light yellow square—white, yellow,
and the white, yellow opposite them—trace out a bounding square. So
there are in all six bounding squares, four of these squares being
designated in colour by adding red to the colour of the generating
lines. Finally, each point moving in the up direction gives rise to
a line coloured null + red, or red, and then there are the initial
and terminal positions of the points giving eight points. The number
of the lines is evidently twelve, for the four lines of this light
yellow square give four lines in their initial, four lines in their
final position, while the four points trace out four lines, that is
altogether twelve lines.
Now the squares are each of them separate boundaries of the cube, while
the lines belong, each of them, to two squares, thus the red line is
that which is common to the orange and pink squares.
Now suppose that there is a direction, the fourth dimension, which is
perpendicular alike to every one of the space dimensions already used—a
dimension perpendicular, for instance, to up and to right hand, so that
the pink square moving in this direction traces out a cube.
A dimension, moreover, perpendicular to the up and away directions,
so that the orange square moving in this direction also traces out
a cube, and the light yellow square, too, moving in this direction
traces out a cube. Under this supposition, the whole cube moving in
the unknown dimension, traces out something new—a new kind of volume,
a higher volume. This higher volume is a four-dimensional volume, and
we designate it in colour by adding blue to the colour of that which by
moving generates it.
It is generated by the motion of the ochre solid, and hence it is
of the colour we call light brown (white, yellow, red, blue, mixed
together). It is represented by a number of sections like 2 in fig. 103.
Now this light brown higher solid has for boundaries: first, the ochre
cube in its initial position, second, the same cube in its final
position, 1 and 3, fig. 103. Each of the squares which bound the cube,
moreover, by movement in this new direction traces out a cube, so we
have from the front pink faces of the cube, third, a pink blue or
light purple cube, shown as a light purple face on cube 2 in fig. 103,
this cube standing for any number of intermediate sections; fourth,
a similar cube from the opposite pink face; fifth, a cube traced out
by the orange face—this is coloured brown and is represented by the
brown face of the section cube in fig. 103; sixth, a corresponding
brown cube on the right hand; seventh, a cube starting from the light
yellow square below; the unknown dimension is at right angles to this
also. This cube is coloured light yellow and blue or light green; and,
finally, eighth, a corresponding cube from the upper light yellow face,
shown as the light green square at the top of the section cube.
The tesseract has thus eight cubic boundaries. These completely enclose
it, so that it would be invisible to a four-dimensional being. Now, as
to the other boundaries, just as the cube has squares, lines, points,
as boundaries, so the tesseract has cubes, squares, lines, points, as
boundaries.
The number of squares is found thus—round the cube are six squares,
these will give six squares in their initial and six in their final
positions. Then each of the twelve lines of the cube trace out a square
in the motion in the fourth dimension. Hence there will be altogether
12 + 12 = 24 squares.
If we look at any one of these squares we see that it is the meeting
surface of two of the cubic sides. Thus, the red line by its movement
in the fourth dimension, traces out a purple square—this is common
to two cubes, one of which is traced out by the pink square moving
in the fourth dimension, and the other is traced out by the orange
square moving in the same way. To take another square, the light yellow
one, this is common to the ochre cube and the light green cube. The
ochre cube comes from the light yellow square by moving it in the up
direction, the light green cube is made from the light yellow square by
moving it in the fourth dimension. The number of lines is thirty-two,
for the twelve lines of the cube give twelve lines of the tesseract
in their initial position, and twelve in their final position, making
twenty-four, while each of the eight points traces out a line, thus
forming thirty-two lines altogether.
The lines are each of them common to three cubes, or to three square
faces; take, for instance, the red line. This is common to the orange
face, the pink face, and that face which is formed by moving the red
line in the sixth dimension, namely, the purple face. It is also common
to the ochre cube, the pale purple cube, and the brown cube.
The points are common to six square faces and to four cubes; thus,
the null point from which we start is common to the three square
faces—pink, light yellow, orange, and to the three square faces made by
moving the three lines white, yellow, red, in the fourth dimension,
namely, the light blue, the light green, the purple faces—that is, to
six faces in all. The four cubes which meet in it are the ochre cube,
the light purple cube, the brown cube, and the light green cube.
[Illustration: Fig. 103.
The tesseract, red, white, yellow axes in space. In the lower line the
three rear faces are shown, the interior being removed.]
[Illustration: Fig. 104.
The tesseract, red, yellow, blue axes in space, the blue axis running
to the left, opposite faces are coloured identically.]
A complete view of the tesseract in its various space presentations
is given in the following figures or catalogue cubes, figs. 103-106.
The first cube in each figure represents the view of a tesseract
coloured as described as it begins to pass transverse to our space.
The intermediate figure represents a sectional view when it is partly
through, and the final figure represents the far end as it is just
passing out. These figures will be explained in detail in the next
chapter.
[Illustration: Fig. 105.
The tesseract, with red, white, blue axes in space. Opposite faces are
coloured identically.]
[Illustration: Fig. 106.
The tesseract, with blue, white, yellow axes in space. The blue axis
runs downward from the base of the ochre cube as it stands originally.
Opposite faces are coloured identically.]
We have thus obtained a nomenclature for each of the regions of a
tesseract; we can speak of any one of the eight bounding cubes, the
twenty square faces, the thirty-two lines, the sixteen points.