THE MEANS BY WHICH A PLANE-BEING WOULD ACQUIRE A CONCEPTION OF OUR
FIGURES.
Take the Block of twenty-seven Mala cubes, and build up the following
shape (Fig. 18):--
Urna Mala, Moles Mala, Plebs Mala, Pallor Mala, Mora Mala.
If this shape, passed through the vertical plane, the plane-being would
perceive:--
(1) The squares Urna Moena and Moles Moena.
(2) The three squares Plebs Moena, Pallor Moena, Mora Moena,
and then the whole figure would have passed through his plane.
If the whole Block were turned round the Z axis till the Alvus sides
entered, and the figure built up as it would be in that disposition of
the cubes, the plane-being would perceive during its passage through the
plane:--
(1) Urna Alvus;
(2) Moles Alvus, Plebs Alvus, Pallor Alvus, Mora Alvus, which would all
enter on the left side of the Z axis.
Again, if the Block were turned round the X axis, the Syce side would
enter, and the cubes appear in the following order:--
(1) Urna Syce, Moles Syce, Plebs Syce;
(2) Pallor Syce;
(3) Mora Syce.
[Illustration: Fig. 17.]
[Illustration: Fig. 18.]
A comparison of these three sets of appearances would give the
plane-being a full account of the figure. It is that which can produce
these various appearances.
Let us now suppose a glass plate placed in front of the Block in its
first position. On this plate let the axes X and Z be drawn. They divide
the surface into four parts, to which we give the following names (Fig.
17):--
Z X = that quarter defined by the positive Z and positive X axis.
Z [=X] = that quarter defined by the positive Z and negative X axis
(which is called “Z negative X”).
[=Z] [=X] = that quarter defined by the negative Z and negative X axis.
[=Z] X = that quarter defined by the negative Z and positive X axis.
The Block appears in these different quarters or quadrants, as it is
turned round the different axes. In Z X the Moenas appear, in Z [=X] the
Alvus faces, in [=Z] X the Syces. In each quadrant are drawn nine
squares, to receive the faces of the cubes when they enter. For
instance, in Z X we have the Moenas of:--
Z
| Comes Tibicen Vestis
| Ostrum Bidens Scena
| Urna Moles Saltus
+--------------------------------------X
And in Z [=X] we have the Alvus of:--
Z
Mars Spicula Comes |
Ala Uncus Ostrum |
Sector Frenum Urna |
-X-------------------------------------+
And in the [=Z] X we have the Syces of:--
+-----------------------------------X
| Urna Moles Saltus
| Frenum Plebs Sypho
| Sector Hama Remus
-Z
Now, if the shape taken at the beginning of this chapter be looked at
through the glass, and the distance of the second and third walls of the
shape behind the glass be considered of no account--that is, if they be
treated as close up to the glass--we get a plane outline, which occupies
the squares Urna Moena, Moles Moena, Bidens Moena, Tibicen Moena. This
outline is called a projection of the figure. To see it like a
plane-being, we should have to look down on it along the Z axis.
It is obvious that one projection does not give the shape. For instance,
the square Bidens Moena might be filled by either Pallor or Cortis. All
that a square in the room of Bidens Moena denotes, is that there is a
cube somewhere in the Y, or unknown, direction from Bidens Moena. This
view, just taken, we should call the front view in our space; we are
then looking at it along the negative Y axis.
When the same shape is turned round on the Z axis, so as to be projected
on the Z [=X] quadrant, we have the squares--Urna Alvus, Frenum Alvus,
Uncus Alvus, Spicula Alvus. When it is turned round the X axis, and
projected on [=Z] X, we have the squares, Urna Syce, Moles Syce, Plebs
Syce, and no more. This is what is ordinarily called the ground plan;
but we have set it in a different position from that in which it is
usually drawn.
[Illustration: Fig. 19.]
Now, the best method for a plane-being of familiarizing himself with
shapes in our space, would be to practise the realization of them from
their different projections in his plane. Thus, given the three
projections just mentioned, he should be able to construct the figure
from which they are derived. The projections (Fig. 19) are drawn below
the perspective pictures of the shape (Fig. 18). From the front, or
Moena view, he would conclude that the shape was Urna Mala, Moles Mala,
Bidens Mala, Tibicen Mala; or instead of these, or also in addition to
them, any of the cubes running in the Y direction from the plane. That
is, from the Moena projection he might infer the presence of all the
following cubes (the word Mala is omitted for brevity): Urna, Frenum,
Sector, Moles, Plebs, Hama, Bidens, Pallor, Cortis, Tibicen, Mora,
Merces.
Next, the Alvus view or projection might be given by the cubes (the word
Mala being again omitted): Urna, Moles, Saltus, Frenum, Plebs, Sypho,
Uncus, Pallor, Tergum, Spicula, Mora, Oliva. Lastly, looking at the
ground plan or Syce view, he would infer the possible presence of Urna,
Ostrum, Comes, Moles, Bidens, Tibicen, Plebs, Pallor, Mora.
Now, the shape in higher space, which is usually there, is that which is
common to all these three appearances. It can be determined, therefore,
by rejecting those cubes which are not present in all three lists of
cubes possible from the projections. And by this process the plane-being
could arrive at the enumeration of the cubes which belong to the shape
of which he had the projections. After a time, when he had experience of
the cubes (which, though invisible to him as wholes, he could see part
by part in turn entering his space), the projections would have more
meaning to him, and he might comprehend the shape they expressed
fragmentarily in his space. To practise the realization from
projections, we should proceed in this way. First, we should think of
the possibilities involved in the Moena view, and build them up in cubes
before us. Secondly, we should build up the cubes possible from the
Alvus view. Again, taking the shape at the beginning of the chapter, we
should find that the shape of the Alvus possibilities intersected that
of the Moena possibilities in Urna, Moles, Frenum, Plebs, Pallor, Mora;
or, in other words, these cubes are common to both. Thirdly, we should
build up the Syce possibilities, and, comparing their shape with those
of the Moena and Alvus projections, we should find Urna, Moles, Plebs,
Pallor, Mora, of the Syce view coinciding with the same cubes of the
other views, the only cube present in the intersection of the Moena and
Alvus possibilities, and not present in the Syce view, being Frenum.
The determination of the figure denoted by the three projections, may be
more easily effected by treating each projection as an indication of
what cubes are to be cut away. Taking the same shape as before, we have
in the Moena projection Urna, Moles, Bidens, Tibicen; and the
possibilities from them are Urna, Frenum, Sector, Moles, Plebs, Hama,
Bidens, Pallor, Cortis, Tibicen, Mora, Merces. This may aptly be called
the Moena solution. Now, from the Syce projection, we learn at once that
those cubes, which in it would produce Frenum, Sector, Hama, Remus,
Sypho, Saltus, are not in the shape. This absence of Frenum and Sector
in the Syce view proves that their presence in the Moena solution is
superfluous. The absence of Hama removes the possibility of Hama,
Cortis, Merces. The absence of Remus, Sypho, Saltus, makes no
difference, as neither they nor any of their Syce possibilities are
present in the Moena solution. Hence, the result of comparison of the
Moena and Syce projections and possibilities is the shape: Urna, Moles,
Plebs, Bidens, Pallor, Tibicen, Mora. This may be aptly called the
Moena-Syce solution. Now, in the Alvus projection we see that Ostrum,
Comes, Sector, Ala, and Mars are absent. The absence of Sector, Ala, and
Mars has no effect on our Moena-Syce solution; as it does not contain
any of their Alvus possibilities. But the absence of Ostrum and Comes
proves that in the Moena-Syce solution Bidens and Tibicen are
superfluous, since their presence in the original shape would give
Ostrum and Comes in the Alvus projection. Thus we arrive at the
Moena-Alvus-Syce solution, which gives us the shape: Urna, Moles, Plebs,
Pallor, Mora.
It will be obvious on trial that a shape can be instantly recognised
from its three projections, if the Block be thoroughly well known in all
three positions. Any difficulty in the realization of the shapes comes
from the arbitrary habit of associating the cubes with some one
direction in which they happen to go with regard to us. If we remember
Ostrum as above Urna, we are not remembering the Block, but only one
particular relation of the Block to us. That position of Ostrum is a
fact as much related to ourselves as to the Block. There is, of course,
some information about the Block implied in that position; but it is so
mixed with information about ourselves as to be ineffectual knowledge of
the Block. It is of the highest importance to enter minutely into all
the details of solution written above. For, corresponding to every
operation necessary to a plane-being for the comprehension of our world,
there is an operation, with which we have to become familiar, if in our
turn we would enter into some comprehension of a world higher than our
own. Every cube of the Block ought to be thoroughly known in all its
relations. And the Block must be regarded, not as a formless mass out of
which shapes can be made, but as the sum of all possible shapes, from
which any one we may choose is a selection. In fact, to be familiar with
the Block, we ought to know every shape that could be made by any
selection of its cubes; or, in other words, we ought to make an
exhaustive study of it. In the Appendix is given a set of exercises in
the use of these names (which form a language of shape), and in various
kinds of projections. The projections studied in this chapter are not
the only, nor the most natural, projections by which a plane-being would
study higher space. But they suffice as an illustration of our present
purpose. If the reader will go through the exercises in the Appendix,
and form others for himself, he will find every bit of manipulation done
will be of service to him in the comprehension of higher space.
There is one point of view in the study of the Block, by means of slabs,
which is of some interest. The cubes of the Block, and therefore also
the representative slabs of their faces, can be regarded as forming rows
and columns. There are three sets of them. If we take the Moena view,
they represent the views of the three walls of the Block, as they pass
through the plane. To form the Alvus view, we only have to rearrange the
slabs, and form new sets. The first new set is formed by taking the
first, or left-hand, column of each of the Moena sets. The second Alvus
set is formed by taking the second or middle columns of the three Moena
sets. The third will consist of the remaining or right-hand columns of
the Moenas.
Similarly, the three Syce sets may be formed from the three horizontal
rows or floors of the Moena sets.
Hence, it appears that the plane-being would study our space by taking
all the possible combinations of the corresponding rows and columns. He
would break up the first three sets into other sets, and the study of
the Block would practically become to him the study of these various
arrangements.