A FOUR-DIMENSIONAL FIGURE
The method used in the preceding chapter to illustrate the problem
of Kant’s critique, gives a singularly easy and direct mode of
constructing a series of important figures in any number of dimensions.
We have seen that to represent our space a plane being must give up one
of his axes, and similarly to represent the higher shapes we must give
up one amongst our three axes.
But there is another kind of giving up which reduces the construction
of higher shapes to a matter of the utmost simplicity.
Ordinarily we have on a straight line any number of positions. The
wealth of space in position is illimitable, while there are only three
dimensions.
I propose to give up this wealth of positions, and to consider the
figures obtained by taking just as many positions as dimensions.
In this way I consider dimensions and positions as two “kinds,” and
applying the simple rule of selecting every one of one kind with every
other of every other kind, get a series of figures which are noteworthy
because they exactly fill space of any number of dimensions (as the
hexagon fills a plane) by equal repetitions of themselves.
The rule will be made more evident by a simple application.
Let us consider one dimension and one position. I will call the axis
_i_, and the position _o_.
———————————————-_i_
_o_
Here the figure is the position _o_ on the line _i_. Take now two
dimensions and two positions on each.
[Illustration: Fig. 63.]
We have the two positions _o_; 1 on _i_, and the two positions _o_, 1
on _j_, fig. 63. These give rise to a certain complexity. I will let
the two lines _i_ and _j_ meet in the position I call _o_ on each, and
I will consider _i_ as a direction starting equally from every position
on _j_, and _j_ as starting equally from every position on _i_. We thus
obtain the following figure:—A is both _oi_ and _oj_, B is 1_i_ and
_oj_, and so on as shown in fig. 63_b_. The positions on AC are all
_oi_ positions. They are, if we like to consider it in that way, points
at no distance in the _i_ direction from the line AC. We can call the
line AC the _oi_ line. Similarly the points on AB are those no distance
from AB in the _j_ direction, and we can call them _oj_ points and the
line AB the _oj_ line. Again, the line CD can be called the 1_j_ line
because the points on it are at a distance, 1 in the _j_ direction.
[Illustration: Fig. 63_b_.]
We have then four positions or points named as shown, and, considering
directions and positions as “kinds,” we have the combination of two
kinds with two kinds. Now, selecting every one of one kind with every
other of every other kind will mean that we take 1 of the kind _i_ and
with it _o_ of the kind _j_; and then, that we take _o_ of the kind _i_
and with it 1 of the kind _j_.
Thus we get a pair of positions lying in the straight line BC, fig.
64. We can call this pair 10 and 01 if we adopt the plan of mentally,
adding an _i_ to the first and a _j_ to the second of the symbols
written thus—01 is a short expression for O_i_, 1_j_.
[Illustration: Fig. 64.]
Coming now to our space, we have three dimensions, so we take three
positions on each. These positions I will suppose to be at equal
distances along each axis. The three axes and the three positions on
each are shown in the accompanying diagrams, fig. 65, of which the
first represents a cube with the front faces visible, the second the
rear faces of the same cube; the positions I will call 0, 1, 2; the
axes, _i_, _j_, _k_. I take the base ABC as the starting place, from
which to determine distances in the _k_ direction, and hence every
point in the base ABC will be an _ok_ position, and the base ABC can be
called an _ok_ plane.
[Illustration: Fig. 65.]
In the same way, measuring the distances from the face ADC, we see
that every position in the face ADC is an _oi_ position, and the whole
plane of the face may be called an _oi_ plane. Thus we see that with
the introduction of a new dimension the signification of a compound
symbol, such as “_oi_,” alters. In the plane it meant the line AC. In
space it means the whole plane ACD.
Now, it is evident that we have twenty-seven positions, each of them
named. If the reader will follow this nomenclature in respect of the
positions marked in the figures he will have no difficulty in assigning
names to each one of the twenty-seven positions. A is _oi_, _oj_, _ok_.
It is at the distance 0 along _i_, 0 along _j_, 0 along _k_, and _io_
can be written in short 000, where the _ijk_ symbols are omitted.
The point immediately above is 001, for it is no distance in the _i_
direction, and a distance of 1 in the _k_ direction. Again, looking at
B, it is at a distance of 2 from A, or from the plane ADC, in the _i_
direction, 0 in the _j_ direction from the plane ABD, and 0 in the _k_
direction, measured from the plane ABC. Hence it is 200 written for
2_i_, 0_j_, 0_k_.
Now, out of these twenty-seven “things” or compounds of position and
dimension, select those which are given by the rule, every one of one
kind with every other of every other kind.
Take 2 of the _i_ kind. With this we must have a 1 of the _j_ kind, and
then by the rule we can only have a 0 of the _k_ kind, for if we had
any other of the _k_ kind we should repeat one of the kinds we already
had. In 2_i_, 1_j_, 1_k_, for instance, 1 is repeated. The point we
obtain is that marked 210, fig. 66.
[Illustration: Fig. 66.]
Proceeding in this way, we pick out the following cluster of points,
fig. 67. They are joined by lines, dotted where they are hidden by the
body of the cube, and we see that they form a figure—a hexagon which
could be taken out of the cube and placed on a plane. It is a figure
which will fill a plane by equal repetitions of itself. The plane being
representing this construction in his plane would take three squares to
represent the cube. Let us suppose that he takes the _ij_ axes in his
space and _k_ represents the axis running out of his space, fig. 68.
In each of the three squares shown here as drawn separately he could
select the points given by the rule, and he would then have to try to
discover the figure determined by the three lines drawn. The line from
210 to 120 is given in the figure, but the line from 201 to 102 or GK
is not given. He can determine GK by making another set of drawings and
discovering in them what the relation between these two extremities is.
[Illustration: Fig. 67.]
[Illustration: Fig. 68.]
[Illustration: Fig. 69.]
Let him draw the _i_ and _k_ axes in his plane, fig. 69. The _j_ axis
then runs out and he has the accompanying figure. In the first of these
three squares, fig. 69, he can pick out by the rule the two points
201, 102—G, and K. Here they occur in one plane and he can measure the
distance between them. In his first representation they occur at G and
K in separate figures.
Thus the plane being would find that the ends of each of the lines was
distant by the diagonal of a unit square from the corresponding end
of the last and he could then place the three lines in their right
relative position. Joining them he would have the figure of a hexagon.
[Illustration: Fig. 70.]
We may also notice that the plane being could make a representation of
the whole cube simultaneously. The three squares, shown in perspective
in fig. 70, all lie in one plane, and on these the plane being could
pick out any selection of points just as well as on three separate
squares. He would obtain a hexagon by joining the points marked. This
hexagon, as drawn, is of the right shape, but it would not be so if
actual squares were used instead of perspective, because the relation
between the separate squares as they lie in the plane figure is not
their real relation. The figure, however, as thus constructed, would
give him an idea of the correct figure, and he could determine it
accurately by remembering that distances in each square were correct,
but in passing from one square to another their distance in the third
dimension had to be taken into account.
Coming now to the figure made by selecting according to our rule from
the whole mass of points given by four axes and four positions in each,
we must first draw a catalogue figure in which the whole assemblage is
shown.
We can represent this assemblage of points by four solid figures. The
first giving all those positions which are at a distance O from our
space in the fourth dimension, the second showing all those that are at
a distance 1, and so on.
These figures will each be cubes. The first two are drawn showing the
front faces, the second two the rear faces. We will mark the points 0,
1, 2, 3, putting points at those distances along each of these axes,
and suppose all the points thus determined to be contained in solid
models of which our drawings in fig. 71 are representatives. Here we
notice that as on the plane 0_i_ meant the whole line from which the
distances in the _i_ direction was measured, and as in space 0_i_
means the whole plane from which distances in the _i_ direction are
measured, so now 0_h_ means the whole space in which the first cube
stands—measuring away from that space by a distance of one we come to
the second cube represented.
[Illustration: Fig. 71.]
Now selecting according to the rule every one of one kind with every
other of every other kind, we must take, for instance, 3_i_, 2_j_,
1_k_, 0_h_. This point is marked 3210 at the lower star in the figure.
It is 3 in the _i_ direction, 2 in the _j_ direction, 1 in the _k_
direction, 0 in the _h_ direction.
With 3_i_ we must also take 1_j_, 2_k_, 0_h_. This point is shown by
the second star in the cube 0_h_.
[Illustration: Fig. 72.]
In the first cube, since all the points are 0_h_ points, we can only
have varieties in which _i_, _j_, _k_, are accompanied by 3, 2, 1.
The points determined are marked off in the diagram fig. 72, and lines
are drawn joining the adjacent pairs in each figure, the lines being
dotted when they pass within the substance of the cube in the first two
diagrams.
Opposite each point, on one side or the other of each cube, is written
its name. It will be noticed that the figures are symmetrical right and
left; and right and left the first two numbers are simply interchanged.
Now this being our selection of points, what figure do they make when
all are put together in their proper relative positions?
To determine this we must find the distance between corresponding
corners of the separate hexagons.
[Illustration: Fig. 73.]
To do this let us keep the axes _i_, _j_, in our space, and draw _h_
instead of _k_, letting _k_ run out in the fourth dimension, fig. 73.
Here we have four cubes again, in the first of which all the points are
0_k_ points; that is, points at a distance zero in the _k_ direction
from the space of the three dimensions _ijh_. We have all the points
selected before, and some of the distances, which in the last diagram
led from figure to figure are shown here in the same figure, and so
capable of measurement. Take for instance the points 3120 to 3021,
which in the first diagram (fig. 72) lie in the first and second
figures. Their actual relation is shown in fig. 73 in the cube marked
2K, where the points in question are marked with a *. We see that the
distance in question is the diagonal of a unit square. In like manner
we find that the distance between corresponding points of any two
hexagonal figures is the diagonal of a unit square. The total figure
is now easily constructed. An idea of it may be gained by drawing all
the four cubes in the catalogue figure in one (fig. 74). These cubes
are exact repetitions of one another, so one drawing will serve as a
representation of the whole series, if we take care to remember where
we are, whether in a 0_h_, a 1_h_, a 2_h_, or a 3_h_ figure, when we
pick out the points required. Fig. 74 is a representation of all the
catalogue cubes put in one. For the sake of clearness the front faces
and the back faces of this cube are represented separately.
[Illustration: Fig. 74.]
The figure determined by the selected points is shown below.
In putting the sections together some of the outlines in them
disappear. The line TW for instance is not wanted.
We notice that PQTW and TWRS are each the half of a hexagon. Now QV and
VR lie in one straight line. Hence these two hexagons fit together,
forming one hexagon, and the line TW is only wanted when we consider a
section of the whole figure, we thus obtain the solid represented in
the lower part of fig. 74. Equal repetitions of this figure, called a
tetrakaidecagon, will fill up three-dimensional space.
To make the corresponding four-dimensional figure we have to take five
axes mutually at right angles with five points on each. A catalogue of
the positions determined in five-dimensional space can be found thus.
Take a cube with five points on each of its axes, the fifth point is
at a distance of four units of length from the first on any one of
the axes. And since the fourth dimension also stretches to a distance
of four we shall need to represent the successive sets of points at
distances 0, 1, 2, 3, 4, in the fourth dimensions, five cubes. Now
all of these extend to no distance at all in the fifth dimension. To
represent what lies in the fifth dimension we shall have to draw,
starting from each of our cubes, five similar cubes to represent the
four steps on in the fifth dimension. By this assemblage we get a
catalogue of all the points shown in fig. 75, in which _L_ represents
the fifth dimension.
[Illustration: Fig. 75.]
Now, as we saw before, there is nothing to prevent us from putting all
the cubes representing the different stages in the fourth dimension in
one figure, if we take note when we look at it, whether we consider
it as a 0_h_, a 1_h_, a 2_h_, etc., cube. Putting then the 0_h_, 1_h_,
2_h_, 3_h_, 4_h_ cubes of each row in one, we have five cubes with the
sides of each containing five positions, the first of these five cubes
represents the 0_l_ points, and has in it the _i_ points from 0 to 4,
the _j_ points from 0 to 4, the _k_ points from 0 to 4, while we have
to specify with regard to any selection we make from it, whether we
regard it as a 0_h_, a 1_h_, a 2_h_, a 3_h_, or a 4_h_ figure. In fig.
76 each cube is represented by two drawings, one of the front part, the
other of the rear part.
Let then our five cubes be arranged before us and our selection be made
according to the rule. Take the first figure in which all points are
0_l_ points. We cannot have 0 with any other letter. Then, keeping in
the first figure, which is that of the 0_l_ positions, take first of
all that selection which always contains 1_h_. We suppose, therefore,
that the cube is a 1_h_ cube, and in it we take _i_, _j_, _k_ in
combination with 4, 3, 2 according to the rule.
The figure we obtain is a hexagon, as shown, the one in front. The
points on the right hand have the same figures as those on the left,
with the first two numerals interchanged. Next keeping still to the
0_l_ figure let us suppose that the cube before us represents a section
at a distance of 2 in the _h_ direction. Let all the points in it be
considered as 2_h_ points. We then have a 0_l_, 2_h_ region, and have
the sets _ijk_ and 431 left over. We must then pick out in accordance
with our rule all such points as 4_i_, 3_j_, 1_k_.
These are shown in the figure and we find that we can draw them without
confusion, forming the second hexagon from the front. Going on in this
way it will be seen that in each of the five figures a set of hexagons
is picked out, which put together form a three-space figure something
like the tetrakaidecagon.
[Illustration: Fig. 76.]
These separate figures are the successive stages in which the whole
four-dimensional figure in which they cohere can be apprehended.
The first figure and the last are tetrakaidecagons. These are two
of the solid boundaries of the figure. The other solid boundaries
can be traced easily. Some of them are complete from one face in the
figure to the corresponding face in the next, as for instance the
solid which extends from the hexagonal base of the first figure to the
equal hexagonal base of the second figure. This kind of boundary is a
hexagonal prism. The hexagonal prism also occurs in another sectional
series, as for instance, in the square at the bottom of the first
figure, the oblong at the base of the second and the square at the base
of the third figure.
Other solid boundaries can be traced through four of the five sectional
figures. Thus taking the hexagon at the top of the first figure we
find in the next a hexagon also, of which some alternate sides are
elongated. The top of the third figure is also a hexagon with the other
set of alternate rules elongated, and finally we come in the fourth
figure to a regular hexagon.
These four sections are the sections of a tetrakaidecagon as can
be recognised from the sections of this figure which we have had
previously. Hence the boundaries are of two kinds, hexagonal prisms and
tetrakaidecagons.
These four-dimensional figures exactly fill four-dimensional space by
equal repetitions of themselves.