REMARKS ON THE FIGURES
An inspection of above figures will give an answer to many questions
about the tesseract. If we have a tesseract one inch each way, then it
can be represented as a cube—a cube having white, yellow, red axes,
and from this cube as a beginning, a volume extending into the fourth
dimension. Now suppose the tesseract to pass transverse to our space,
the cube of the red, yellow, white axes disappears at once, it is
indefinitely thin in the fourth dimension. Its place is occupied by
those parts of the tesseract which lie further away from our space in
the fourth dimension. Each one of these sections will last only for
one moment, but the whole of them will take up some appreciable time
in passing. If we take the rate of one inch a minute the sections will
take the whole of the minute in their passage across our space, they
will take the whole of the minute except the moment which the beginning
cube and the end cube occupy in their crossing our space. In each one
of the cubes, the section cubes, we can draw lines in all directions
except in the direction occupied by the blue line, the fourth
dimension; lines in that direction are represented by the transition
from one section cube to another. Thus to give ourselves an adequate
representation of the tesseract we ought to have a limitless number of
section cubes intermediate between the first bounding cube, the ochre
cube, and the last bounding cube, the other ochre cube. Practically
three intermediate sectional cubes will be found sufficient for most
purposes. We will take then a series of five figures—two terminal
cubes, and three intermediate sections—and show how the different
regions appear in our space when we take each set of three out of the
four axes of the tesseract as lying in our space.
In fig. 107 initial letters are used for the colours. A reference to
fig. 103 will show the complete nomenclature, which is merely indicated
here.
[Illustration: Fig. 107.]
In this figure the tesseract is shown in five stages distant from our
space: first, zero; second, 1/4 in.; third, 2/4 in.; fourth, 3/4 in.;
fifth, 1 in.; which are called _b_0, _b_1, _b_2, _b_3, _b_4, because
they are sections taken at distances 0, 1, 2, 3, 4 quarter inches along
the blue line. All the regions can be named from the first cube, the
_b_0 cube, as before, simply by remembering that transference along
the b axis gives the addition of blue to the colour of the region in
the ochre, the _b_0 cube. In the final cube _b_4, the colouring of the
original _b_0 cube is repeated. Thus the red line moved along the blue
axis gives a red and blue or purple square. This purple square appears
as the three purple lines in the sections _b_1, _b_2, _b_3, taken at
1/4, 2/4, 3/4 of an inch in the fourth dimension. If the tesseract
moves transverse to our space we have then in this particular region,
first of all a red line which lasts for a moment, secondly a purple
line which takes its place. This purple line lasts for a minute—that
is, all of a minute, except the moment taken by the crossing our space
of the initial and final red line. The purple line having lasted for
this period is succeeded by a red line, which lasts for a moment; then
this goes and the tesseract has passed across our space. The final red
line we call red bl., because it is separated from the initial red
line by a distance along the axis for which we use the colour blue.
Thus a line that lasts represents an area duration; is in this mode
of presentation equivalent to a dimension of space. In the same way
the white line, during the crossing our space by the tesseract, is
succeeded by a light blue line which lasts for the inside of a minute,
and as the tesseract leaves our space, having crossed it, the white bl.
line appears as the final termination.
Take now the pink face. Moved in the blue direction it traces out a
light purple cube. This light purple cube is shown in sections in
_b__{1}, _b__{2}, _b__{3}, and the farther face of this cube in the
blue direction is shown in _b__{4}—a pink face, called pink _b_ because
it is distant from the pink face we began with in the blue direction.
Thus the cube which we colour light purple appears as a lasting square.
The square face itself, the pink face, vanishes instantly the tesseract
begins to move, but the light purple cube appears as a lasting square.
Here also duration is the equivalent of a dimension of space—a lasting
square is a cube. It is useful to connect these diagrams with the views
given in the coloured plate.
Take again the orange face, that determined by the red and yellow axes;
from it goes a brown cube in the blue direction, for red and yellow
and blue are supposed to make brown. This brown cube is shown in three
sections in the faces _b__{1}, _b__{2}, _b__{3}. In _b__{4} is the
opposite orange face of the brown cube, the face called orange _b_,
for it is distant in the blue direction from the orange face. As the
tesseract passes transverse to our space, we have then in this region
an instantly vanishing orange square, followed by a lasting brown
square, and finally an orange face which vanishes instantly.
Now, as any three axes will be in our space, let us send the white
axis out into the unknown, the fourth dimension, and take the blue
axis into our known space dimension. Since the white and blue axes are
perpendicular to each other, if the white axis goes out into the fourth
dimension in the positive sense, the blue axis will come into the
direction the white axis occupied, in the negative sense.
[Illustration: Fig. 108.]
Hence, not to complicate matters by having to think of two senses in
the unknown direction, let us send the white line into the positive
sense of the fourth dimension, and take the blue one as running in the
negative sense of that direction which the white line has left; let the
blue line, that is, run to the left. We have now the row of figures
in fig. 108. The dotted cube shows where we had a cube when the white
line ran in our space—now it has turned out of our space, and another
solid boundary, another cubic face of the tesseract comes into our
space. This cube has red and yellow axes as before; but now, instead
of a white axis running to the right, there is a blue axis running to
the left. Here we can distinguish the regions by colours in a perfectly
systematic way. The red line traces out a purple square in the
transference along the blue axis by which this cube is generated from
the orange face. This purple square made by the motion of the red line
is the same purple face that we saw before as a series of lines in the
sections _b__{1}, _b__{2}, _b__{3}. Here, since both red and blue axes
are in our space, we have no need of duration to represent the area
they determine. In the motion of the tesseract across space this purple
face would instantly disappear.
From the orange face, which is common to the initial cubes in fig. 107
and fig. 108, there goes in the blue direction a cube coloured brown.
This brown cube is now all in our space, because each of its three axes
run in space directions, up, away, to the left. It is the same brown
cube which appeared as the successive faces on the sections _b__{1},
_b__{2}, _b__{3}. Having all its three axes in our space, it is given
in extension; no part of it needs to be represented as a succession.
The tesseract is now in a new position with regard to our space, and
when it moves across our space the brown cube instantly disappears.
In order to exhibit the other regions of the tesseract we must remember
that now the white line runs in the unknown dimension. Where shall we
put the sections at distances along the line? Any arbitrary position in
our space will do: there is no way by which we can represent their real
position.
However, as the brown cube comes off from the orange face to the left,
let us put these successive sections to the left. We can call them
_wh__{0}, _wh__{1}, _wh__{2}, _wh__{3}, _wh__{4}, because they are
sections along the white axis, which now runs in the unknown dimension.
Running from the purple square in the white direction we find the light
purple cube. This is represented in the sections _wh__{1}, _wh__{2},
_wh__{3}, _wh__{4}, fig. 108. It is the same cube that is represented
in the sections _b__{1}, _b__{2}, _b__{3}: in fig. 107 the red and
white axes are in our space, the blue out of it; in the other case, the
red and blue are in our space, the white out of it. It is evident that
the face pink _y_, opposite the pink face in fig. 107, makes a cube
shown in squares in _b__{1}, _b__{2}, _b__{3}, _b__{4}, on the opposite
side to the _l_ purple squares. Also the light yellow face at the base
of the cube _b__{0}, makes a light green cube, shown as a series of
base squares.
The same light green cube can be found in fig. 107. The base square in
_wh__{0} is a green square, for it is enclosed by blue and yellow axes.
From it goes a cube in the white direction, this is then a light green
cube and the same as the one just mentioned as existing in the sections
_b__{0}, _b__{1}, _b__{2}, _b__{3}, _b__{4}.
The case is, however, a little different with the brown cube. This cube
we have altogether in space in the section _wh__{0}, fig. 108, while
it exists as a series of squares, the left-hand ones, in the sections
_b__{0}, _b__{1}, _b__{2}, _b__{3}, _b__{4}. The brown cube exists as a
solid in our space, as shown in fig. 108. In the mode of representation
of the tesseract exhibited in fig. 107, the same brown cube appears as
a succession of squares. That is, as the tesseract moves across space,
the brown cube would actually be to us a square—it would be merely
the lasting boundary of another solid. It would have no thickness at
all, only extension in two dimensions, and its duration would show its
solidity in three dimensions.
It is obvious that, if there is a four-dimensional space, matter in
three dimensions only is a mere abstraction; all material objects
must then have a slight four-dimensional thickness. In this case the
above statement will undergo modification. The material cube which is
used as the model of the boundary of a tesseract will have a slight
thickness in the fourth dimension, and when the cube is presented to
us in another aspect, it would not be a mere surface. But it is most
convenient to regard the cubes we use as having no extension at all in
the fourth dimension. This consideration serves to bring out a point
alluded to before, that, if there is a fourth dimension, our conception
of a solid is the conception of a mere abstraction, and our talking
about real three-dimensional objects would seem to a four-dimensional
being as incorrect as a two-dimensional being’s telling about real
squares, real triangles, etc., would seem to us.
The consideration of the two views of the brown cube shows that any
section of a cube can be looked at by a presentation of the cube in
a different position in four-dimensional space. The brown faces in
_b__{1}, _b__{2}, _b__{3}, are the very same brown sections that would
be obtained by cutting the brown cube, _wh__{0}, across at the right
distances along the blue line, as shown in fig. 108. But as these
sections are placed in the brown cube, _wh__{0}, they come behind one
another in the blue direction. Now, in the sections _wh__{1}, _wh__{2},
_wh__{3}, we are looking at these sections from the white direction—the
blue direction does not exist in these figures. So we see them in
a direction at right angles to that in which they occur behind one
another in _wh__{0}. There are intermediate views, which would come in
the rotation of a tesseract. These brown squares can be looked at from
directions intermediate between the white and blue axes. It must be
remembered that the fourth dimension is perpendicular equally to all
three space axes. Hence we must take the combinations of the blue axis,
with each two of our three axes, white, red, yellow, in turn.
In fig. 109 we take red, white, and blue axes in space, sending yellow
into the fourth dimension. If it goes into the positive sense of the
fourth dimension the blue line will come in the opposite direction to
that in which the yellow line ran before. Hence, the cube determined
by the white, red, blue axes, will start from the pink plane and run
towards us. The dotted cube shows where the ochre cube was. When it is
turned out of space, the cube coming towards from its front face is
the one which comes into our space in this turning. Since the yellow
line now runs in the unknown dimension we call the sections _y__{0},
_y__{1}, _y__{2}, _y__{3}, _y__{4}, as they are made at distances 0, 1,
2, 3, 4, quarter inches along the yellow line. We suppose these cubes
arranged in a line coming towards us—not that that is any more natural
than any other arbitrary series of positions, but it agrees with the
plan previously adopted.
[Illustration: Fig. 109.]
The interior of the first cube, _y__{0}, is that derived from pink by
adding blue, or, as we call it, light purple. The faces of the cube are
light blue, purple, pink. As drawn, we can only see the face nearest to
us, which is not the one from which the cube starts—but the face on the
opposite side has the same colour name as the face towards us.
The successive sections of the series, _y__{0}, _y__{1}, _y__{2}, etc.,
can be considered as derived from sections of the _b__{0} cube made at
distances along the yellow axis. What is distant a quarter inch from
the pink face in the yellow direction? This question is answered by
taking a section from a point a quarter inch along the yellow axis in
the cube _b__{0}, fig. 107. It is an ochre section with lines orange
and light yellow. This section will therefore take the place of the
pink face in _y__{1} when we go on in the yellow direction. Thus, the
first section, _y__{1}, will begin from an ochre face with light yellow
and orange lines. The colour of the axis which lies in space towards
us is blue, hence the regions of this section-cube are determined in
nomenclature, they will be found in full in fig. 105.
There remains only one figure to be drawn, and that is the one in which
the red axis is replaced by the blue. Here, as before, if the red axis
goes out into the positive sense of the fourth dimension, the blue line
must come into our space in the negative sense of the direction which
the red line has left. Accordingly, the first cube will come in beneath
the position of our ochre cube, the one we have been in the habit of
starting with.
[Illustration: Fig. 110.]
To show these figures we must suppose the ochre cube to be on a movable
stand. When the red line swings out into the unknown dimension, and the
blue line comes in downwards, a cube appears below the place occupied
by the ochre cube. The dotted cube shows where the ochre cube was.
That cube has gone and a different cube runs downwards from its base.
This cube has white, yellow, and blue axes. Its top is a light yellow
square, and hence its interior is light yellow + blue or light green.
Its front face is formed by the white line moving along the blue axis,
and is therefore light blue, the left-hand side is formed by the yellow
line moving along the blue axis, and therefore green.
As the red line now runs in the fourth dimension, the successive
sections can he called _r__{0}, _r__{1}, _r__{2}, _r__{3}, _r__{4},
these letters indicating that at distances 0, 1/4, 2/4, 3/4, 1 inch
along the red axis we take all of the tesseract that can be found in a
three-dimensional space, this three-dimensional space extending not at
all in the fourth dimension, but up and down, right and left, far and
near.
We can see what should replace the light yellow face of _r__{0}, when
the section _r__{1} comes in, by looking at the cube _b__{0}, fig. 107.
What is distant in it one-quarter of an inch from the light yellow face
in the red direction? It is an ochre section with orange and pink lines
and red points; see also fig. 103.
This square then forms the top square of _r__{1}. Now we can determine
the nomenclature of all the regions of _r__{1} by considering what
would be formed by the motion of this square along a blue axis.
But we can adopt another plan. Let us take a horizontal section of
_r__{0}, and finding that section in the figures, of fig. 107 or fig.
103, from them determine what will replace it, going on in the red
direction.
A section of the _r__{0} cube has green, light blue, green, light blue
sides and blue points.
Now this square occurs on the base of each of the section figures,
_b__{1}, _b__{2}, etc. In them we see that 1/4 inch in the red
direction from it lies a section with brown and light purple lines and
purple corners, the interior being of light brown. Hence this is the
nomenclature of the section which in _r__{1} replaces the section of
_r__{0} made from a point along the blue axis.
Hence the colouring as given can be derived.
We have thus obtained a perfectly named group of tesseracts. We can
take a group of eighty-one of them 3 × 3 × 3 × 3, in four dimensions,
and each tesseract will have its name null, red, white, yellow, blue,
etc., and whatever cubic view we take of them we can say exactly
what sides of the tesseracts we are handling, and how they touch each
other.[5]
[5] At this point the reader will find it advantageous, if he has the
models, to go through the manipulations described in the appendix.
Thus, for instance, if we have the sixteen tesseracts shown below, we
can ask how does null touch blue.
[Illustration: Fig. 111.]
In the arrangement given in fig. 111 we have the axes white, red,
yellow, in space, blue running in the fourth dimension. Hence we have
the ochre cubes as bases. Imagine now the tesseractic group to pass
transverse to our space—we have first of all null ochre cube, white
ochre cube, etc.; these instantly vanish, and we get the section shown
in the middle cube in fig. 103, and finally, just when the tesseract
block has moved one inch transverse to our space, we have null ochre
cube, and then immediately afterwards the ochre cube of blue comes in.
Hence the tesseract null touches the tesseract blue by its ochre cube,
which is in contact, each and every point of it, with the ochre cube of
blue.
How does null touch white, we may ask? Looking at the beginning A, fig.
111, where we have the ochre cubes, we see that null ochre touches
white ochre by an orange face. Now let us generate the null and white
tesseracts by a motion in the blue direction of each of these cubes.
Each of them generates the corresponding tesseract, and the plane of
contact of the cubes generates the cube by which the tesseracts are
in contact. Now an orange plane carried along a blue axis generates a
brown cube. Hence null touches white by a brown cube.
[Illustration: Fig. 112.]
If we ask again how red touches light blue tesseract, let us rearrange
our group, fig. 112, or rather turn it about so that we have a
different space view of it; let the red axis and the white axis run
up and right, and let the blue axis come in space towards us, then
the yellow axis runs in the fourth dimension. We have then two blocks
in which the bounding cubes of the tesseracts are given, differently
arranged with regard to us—the arrangement is really the same, but it
appears different to us. Starting from the plane of the red and white
axes we have the four squares of the null, white, red, pink tesseracts
as shown in A, on the red, white plane, unaltered, only from them now
comes out towards us the blue axis. Hence we have null, white, red,
pink tesseracts in contact with our space by their cubes which have
the red, white, blue axis in them, that is by the light purple cubes.
Following on these four tesseracts we have that which comes next to
them in the blue direction, that is the four blue, light blue, purple,
light purple. These are likewise in contact with our space by their
light purple cubes, so we see a block as named in the figure, of which
each cube is the one determined by the red, white, blue, axes.
The yellow line now runs out of space; accordingly one inch on in the
fourth dimension we come to the tesseracts which follow on the eight
named in C, fig. 112, in the yellow direction.
These are shown in C.y_{1}, fig. 112. Between figure C and C.y_{1} is
that four-dimensional mass which is formed by moving each of the cubes
in C one inch in the fourth dimension—that is, along a yellow axis; for
the yellow axis now runs in the fourth dimension.
In the block C we observe that red (light purple cube) touches light
blue (light purple cube) by a point. Now these two cubes moving
together remain in contact during the period in which they trace out
the tesseracts red and light blue. This motion is along the yellow
axis, consequently red and light blue touch by a yellow line.
We have seen that the pink face moved in a yellow direction traces out
a cube; moved in the blue direction it also traces out a cube. Let us
ask what the pink face will trace out if it is moved in a direction
within the tesseract lying equally between the yellow and blue
directions. What section of the tesseract will it make?
We will first consider the red line alone. Let us take a cube with the
red line in it and the yellow and blue axes.
The cube with the yellow, red, blue axes is shown in fig. 113. If the
red line is moved equally in the yellow and in the blue direction by
four equal motions of ¼ inch each, it takes the positions 11, 22, 33,
and ends as a red line.
[Illustration: Fig. 113.]
Now, the whole of this red, yellow, blue, or brown cube appears as a
series of faces on the successive sections of the tesseract starting
from the ochre cube and letting the blue axis run in the fourth
dimension. Hence the plane traced out by the red line appears as a
series of lines in the successive sections, in our ordinary way of
representing the tesseract; these lines are in different places in each
successive section.
[Illustration: Fig. 114.]
Thus drawing our initial cube and the successive sections, calling them
_b__{0}, _b__{1}, _b__{2}, _b__{3}, _b__{4}, fig. 115, we have the red
line subject to this movement appearing in the positions indicated.
We will now investigate what positions in the tesseract another line in
the pink face assumes when it is moved in a similar manner.
Take a section of the original cube containing a vertical line, 4,
in the pink plane, fig. 115. We have, in the section, the yellow
direction, but not the blue.
From this section a cube goes off in the fourth dimension, which is
formed by moving each point of the section in the blue direction.
[Illustration: Fig. 115.]
[Illustration: Fig. 116.]
Drawing this cube we have fig. 116.
Now this cube occurs as a series of sections in our original
representation of the tesseract. Taking four steps as before this cube
appears as the sections drawn in _b__{0}, _b__{1}, _b__{2}, _b__{3},
_b__{4}, fig. 117, and if the line 4 is subjected to a movement equal
in the blue and yellow directions, it will occupy the positions
designated by 4, 4_{1}, 4_{2}, 4_{3}, 4_{4}.
[Illustration: Fig. 117.]
Hence, reasoning in a similar manner about every line, it is evident
that, moved equally in the blue and yellow directions, the pink plane
will trace out a space which is shown by the series of section planes
represented in the diagram.
Thus the space traced out by the pink face, if it is moved equally in
the yellow and blue directions, is represented by the set of planes
delineated in Fig. 118, pink face or 0, then 1, 2, 3, and finally pink
face or 4. This solid is a diagonal solid of the tesseract, running
from a pink face to a pink face. Its length is the length of the
diagonal of a square, its side is a square.
Let us now consider the unlimited space which springs from the pink
face extended.
This space, if it goes off in the yellow direction, gives us in it the
ochre cube of the tesseract. Thus, if we have the pink face given and a
point in the ochre cube, we have determined this particular space.
Similarly going off from the pink face in the blue direction is another
space, which gives us the light purple cube of the tesseract in it. And
any point being taken in the light purple cube, this space going off
from the pink face is fixed.
[Illustration: Fig. 118.]
The space we are speaking of can be conceived as swinging round the
pink face, and in each of its positions it cuts out a solid figure from
the tesseract, one of which we have seen represented in fig. 118.
Each of these solid figures is given by one position of the swinging
space, and by one only. Hence in each of them, if one point is taken,
the particular one of the slanting spaces is fixed. Thus we see that
given a plane and a point out of it a space is determined.
Now, two points determine a line.
Again, think of a line and a point outside it. Imagine a plane rotating
round the line. At some time in its rotation it passes through the
point. Thus a line and a point, or three points, determine a plane.
And finally four points determine a space. We have seen that a plane
and a point determine a space, and that three points determine a plane;
so four points will determine a space.
These four points may be any points, and we can take, for instance, the
four points at the extremities of the red, white, yellow, blue axes, in
the tesseract. These will determine a space slanting with regard to the
section spaces we have been previously considering. This space will cut
the tesseract in a certain figure.
One of the simplest sections of a cube by a plane is that in which the
plane passes through the extremities of the three edges which meet in a
point. We see at once that this plane would cut the cube in a triangle,
but we will go through the process by which a plane being would most
conveniently treat the problem of the determination of this shape, in
order that we may apply the method to the determination of the figure
in which a space cuts a tesseract when it passes through the 4 points
at unit distance from a corner.
We know that two points determine a line, three points determine a
plane, and given any two points in a plane the line between them lies
wholly in the plane.
[Illustration: Fig. 119.]
Let now the plane being study the section made by a plane passing
through the null _r_, null _wh_, and null _y_ points, fig. 119. Looking
at the orange square, which, as usual, we suppose to be initially in
his plane, he sees that the line from null _r_ to null _y_, which is
a line in the section plane, the plane, namely, through the three
extremities of the edges meeting in null, cuts the orange face in an
orange line with null points. This then is one of the boundaries of the
section figure.
Let now the cube be so turned that the pink face comes in his plane.
The points null _r_ and null _wh_ are now visible. The line between
them is pink with null points, and since this line is common to the
surface of the cube and the cutting plane, it is a boundary of the
figure in which the plane cuts the cube.
Again, suppose the cube turned so that the light yellow face is in
contact with the plane being’s plane. He sees two points, the null _wh_
and the null _y_. The line between these lies in the cutting plane.
Hence, since the three cutting lines meet and enclose a portion of
the cube between them, he has determined the figure he sought. It is
a triangle with orange, pink, and light yellow sides, all equal, and
enclosing an ochre area.
Let us now determine in what figure the space, determined by the four
points, null _r_, null _y_, null _wh_, null _b_, cuts the tesseract. We
can see three of these points in the primary position of the tesseract
resting against our solid sheet by the ochre cube. These three points
determine a plane which lies in the space we are considering, and this
plane cuts the ochre cube in a triangle, the interior of which is
ochre (fig. 119 will serve for this view), with pink, light yellow and
orange sides, and null points. Going in the fourth direction, in one
sense, from this plane we pass into the tesseract, in the other sense
we pass away from it. The whole area inside the triangle is common to
the cutting plane we see, and a boundary of the tesseract. Hence we
conclude that the triangle drawn is common to the tesseract and the
cutting space.
Now let the ochre cube turn out and the brown cube come in. The dotted
lines show the position the ochre cube has left (fig. 120).
[Illustration: Fig. 120.]
Here we see three out of the four points through which the cutting
plane passes, null _r_, null _y_, and null _b_. The plane they
determine lies in the cutting space, and this plane cuts out of the
brown cube a triangle with orange, purple and green sides, and null
points. The orange line of this figure is the same as the orange line
in the last figure.
Now let the light purple cube swing into our space, towards us, fig.
121.
[Illustration: Fig. 121.]
The cutting space which passes through the four points, null _r_, _y_,
_wh_, _b_, passes through the null _r_, _wh_, _b_, and therefore the
plane these determine lies in the cutting space.
This triangle lies before us. It has a light purple interior and pink,
light blue, and purple edges with null points.
This, since it is all of the plane that is common to it, and this
bounding of the tesseract, gives us one of the bounding faces of our
sectional figure. The pink line in it is the same as the pink line we
found in the first figure—that of the ochre cube.
Finally, let the tesseract swing about the light yellow plane, so that
the light green cube comes into our space. It will point downwards.
The three points, _n.y_, _n.wh_, _n.b_, are in the cutting space, and
the triangle they determine is common to the tesseract and the cutting
space. Hence this boundary is a triangle having a light yellow line,
which is the same as the light yellow line of the first figure, a light
blue line and a green line.
[Illustration: Fig. 122.]
We have now traced the cutting space between every set of three that
can be made out of the four points in which it cuts the tesseract, and
have got four faces which all join on to each other by lines.
[Illustration: Fig. 123.]
The triangles are shown in fig. 123 as they join on to the triangle
in the ochre cube. But they join on each to the other in an exactly
similar manner; their edges are all identical two and two. They form a
closed figure, a tetrahedron, enclosing a light brown portion which is
the portion of the cutting space which lies inside the tesseract.
We cannot expect to see this light brown portion, any more than a plane
being could expect to see the inside of a cube if an angle of it were
pushed through his plane. All he can do is to come upon the boundaries
of it in a different way to that in which he would if it passed
straight through his plane.
Thus in this solid section; the whole interior lies perfectly open in
the fourth dimension. Go round it as we may we are simply looking at
the boundaries of the tesseract which penetrates through our solid
sheet. If the tesseract were not to pass across so far, the triangle
would be smaller; if it were to pass farther, we should have a
different figure, the outlines of which can be determined in a similar
manner.
The preceding method is open to the objection that it depends rather on
our inferring what must be, than our seeing what is. Let us therefore
consider our sectional space as consisting of a number of planes, each
very close to the last, and observe what is to be found in each plane.
The corresponding method in the case of two dimensions is as
follows:—The plane being can see that line of the sectional plane
through null _y_, null _wh_, null _r_, which lies in the orange plane.
Let him now suppose the cube and the section plane to pass half way
through his plane. Replacing the red and yellow axes are lines parallel
to them, sections of the pink and light yellow faces.
[Illustration: Fig. 124.]
Where will the section plane cut these parallels to the red and yellow
axes?
Let him suppose the cube, in the position of the drawing, fig. 124,
turned so that the pink face lies against his plane. He can see the
line from the null _r_ point to the null _wh_ point, and can see
(compare fig. 119) that it cuts AB a parallel to his red axis, drawn
at a point half way along the white line, in a point B, half way up. I
shall speak of the axis as having the length of an edge of the cube.
Similarly, by letting the cube turn so that the light yellow square
swings against his plane, he can see (compare fig. 119) that a parallel
to his yellow axis drawn from a point half-way along the white axis, is
cut at half its length by the trace of the section plane in the light
yellow face.
Hence when the cube had passed half-way through he would have—instead
of the orange line with null points, which he had at first—an ochre
line of half its length, with pink and light yellow points. Thus, as
the cube passed slowly through his plane, he would have a succession
of lines gradually diminishing in length and forming an equilateral
triangle. The whole interior would be ochre, the line from which it
started would be orange. The succession of points at the ends of
the succeeding lines would form pink and light yellow lines and the
final point would be null. Thus looking at the successive lines in
the section plane as it and the cube passed across his plane he would
determine the figure cut out bit by bit.
Coming now to the section of the tesseract, let us imagine that the
tesseract and its cutting _space_ pass slowly across our space; we can
examine portions of it, and their relation to portions of the cutting
space. Take the section space which passes through the four points,
null _r_, _wh_, _y_, _b_; we can see in the ochre cube (fig. 119) the
plane belonging to this section space, which passes through the three
extremities of the red, white, yellow axes.
Now let the tesseract pass half way through our space. Instead of our
original axes we have parallels to them, purple, light blue, and green,
each of the same length as the first axes, for the section of the
tesseract is of exactly the same shape as its ochre cube.
But the sectional space seen at this stage of the transference would
not cut the section of the tesseract in a plane disposed as at first.
To see where the sectional space would cut these parallels to the
original axes let the tesseract swing so that, the orange face
remaining stationary, the blue line comes in to the left.
Here (fig. 125) we have the null _r_, _y_, _b_ points, and of the
sectional space all we see is the plane through these three points in
it.
[Illustration: Fig. 125.]
In this figure we can draw the parallels to the red and yellow axes and
see that, if they started at a point half way along the blue axis, they
would each be cut at a point so as to be half of their previous length.
Swinging the tesseract into our space about the pink face of the ochre
cube we likewise find that the parallel to the white axis is cut at
half its length by the sectional space.
Hence in a section made when the tesseract had passed half across our
space the parallels to the red, white, yellow axes, which are now in
our space, are cut by the section space, each of them half way along,
and for this stage of the traversing motion we should have fig. 126.
The section made of this cube by the plane in which the sectional space
cuts it, is an equilateral triangle with purple, l. blue, green points,
and l. purple, brown, l. green lines.
[Illustration: Fig. 126.]
Thus the original ochre triangle, with null points and pink, orange,
light yellow lines, would be succeeded by a triangle coloured in manner
just described.
This triangle would initially be only a very little smaller than the
original triangle, it would gradually diminish, until it ended in a
point, a null point. Each of its edges would be of the same length.
Thus the successive sections of the successive planes into which we
analyse the cutting space would be a tetrahedron of the description
shown (fig. 123), and the whole interior of the tetrahedron would be
light brown.
[Illustration: Fig. 127. Front view. The rear faces.]
In fig. 127 the tetrahedron is represented by means of its faces as
two triangles which meet in the p. line, and two rear triangles which
join on to them, the diagonal of the pink face being supposed to run
vertically upward.
We have now reached a natural termination. The reader may pursue
the subject in further detail, but will find no essential novelty.
I conclude with an indication as to the manner in which figures
previously given may be used in determining sections by the method
developed above.
Applying this method to the tesseract, as represented in Chapter IX.,
sections made by a space cutting the axes equidistantly at any distance
can be drawn, and also the sections of tesseracts arranged in a block.
If we draw a plane, cutting all four axes at a point six units distance
from null, we have a slanting space. This space cuts the red, white,
yellow axes in the points LMN (fig. 128), and so in the region of our
space before we go off into the fourth dimension, we have the plane
represented by LMN extended. This is what is common to the slanting
space and our space.
[Illustration: Fig. 128.]
This plane cuts the ochre cube in the triangle EFG.
Comparing this with (fig. 72) _oh_, we see that the hexagon there drawn
is part of the triangle EFG.
Let us now imagine the tesseract and the slanting space both together
to pass transverse to our space, a distance of one unit, we have in
1_h_ a section of the tesseract, whose axes are parallels to the
previous axes. The slanting space cuts them at a distance of five units
along each. Drawing the plane through these points in 1_h_ it will be
found to cut the cubical section of the tesseract in the hexagonal
figure drawn. In 2_h_ (fig. 72) the slanting space cuts the parallels
to the axes at a distance of four along each, and the hexagonal figure
is the section of this section of the tesseract by it. Finally when
3_h_ comes in the slanting space cuts the axes at a distance of three
along each, and the section is a triangle, of which the hexagon drawn
is a truncated portion. After this the tesseract, which extends only
three units in each of the four dimensions, has completely passed
transverse of our space, and there is no more of it to be cut. Hence,
putting the plane sections together in the right relations, we have
the section determined by the particular slanting space: namely an
octahedron.