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.
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