Hinton - The Fourth Dimension.pdf

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THE FOURTH DIMENSION
Each of the six surrounding cubes carried on in the same
motion will make a tesseract also, and these will be
grouped around the tesseract formed by A. But will they
enclose it completely?
All the cubes An, Af, etc., lie in our space. But there is
nothing between the cube A and that solid sheet in contact
with which every particle of matter is. When the cube A
moves in the fourth direction it starts from its position,
say Aκ, and ends in a final position Aα (using the words
“ana” and “kata” for up and down in the fourth dimension).
Now the movement in this fourth dimension is
not bounded by any of the cubes An, Af, nor by what
they form when thus moved. The tesseract which A
becomes is bounded in the positive and negative ways in
this new direction by the first position of A and the last
position of A. Or, if we ask how many tesseracts lie
around the tesseract which A forms, there are eight, of
which one meets it by the cube A, and another meets it
by a cube like A at the end of its motion.
We come here to a very curious thing. The whole
solid cube A is to be looked on merely as a boundary of
the tesseract.
Yet this is exactly analogous to what the plane being
would come to in his study of the solid world. The
square A (fig. 96), which the plane being looks on as a
solid existence in his plane world, is merely the boundary
of the cube which he supposes generated by its motion.
The fact is that we have to recognise that, if there is
another dimension of space, our present idea of a solid
body, as one which has three dimensions only, does not
correspond to anything real, but is the abstract idea of a
three-dimensional boundary limiting a four-dimensional
solid, which a four-dimensional being would form. The
plane being’s thought of a square is not the thought
of what we should call a possibly existing real square,