Hinton - The Fourth Dimension.pdf

THE SIMPLEST FOUR-DIMENSIONAL SOLID 159
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, An, Af, Aa, Ab,
If now he conceives the square A
to move in the, to him, unknown
dimension it will trace out a cube,
and the bounding spaces 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 coincident 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 Al and Ar. 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.
The plane being could not see the square A with the
Al
A n
A a
A
A b
Fig. 96.
An
Aa
Ab
Fig, 97.
Ar
A f
A f
squares An, Af, 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 surrounded by
six other cubes. These cubes we
will call A near An, A far Af, A above
Aa, A below Ab, A left Al, A right Ar,
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.