To do this he would only have
to make a little excursion in the fourth dimension.
[Illustration with caption: FIG. 3]
Another curious application of the principle is more purely
geometrical. We have here two triangles, of which the sides and
angles of the one are all equal to corresponding sides and angles
of the other. Euclid takes it for granted that the one triangle
can be laid upon the other so that the two shall fit together. But
this cannot be done unless we lift one up and turn it over. In the
geometry of "flat-land" such a thing as lifting up is
inconceivable; the two triangles could never be fitted together.
[Illustration with caption: FIG 4]
Now let us suppose two pyramids similarly related. All the faces
and angles of the one correspond to the faces and angles of the
other. Yet, lift them about as we please, we could never fit them
together. If we fit the bases together the two will lie on
opposite sides, one being below the other. But the dweller in four
dimensions of space will fit them together without any trouble. By
the mere turning over of one he will convert it into the other
without any change whatever in the relative position of its parts.
What he could do with the pyramids he could also do with one of us
if we allowed him to take hold of us and turn a somersault with us
in the fourth dimension.
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