People and
things can pass around each other, but cannot step over anything.
These dwellers in "flatland" could construct a plane geometry
which would be exactly like ours in being based on the axioms of
Euclid. Two parallel straight lines would never meet, though
continued indefinitely.
But suppose that the surface on which these beings live, instead
of being an infinitely extended plane, is really the surface of an
immense globe, like the earth on which we live. It needs no
knowledge of geometry, but only an examination of any globular
object--an apple, for example--to show that if we draw a line as
straight as possible on a sphere, and parallel to it draw a small
piece of a second line, and continue this in as straight a line as
we can, the two lines will meet when we proceed in either
direction one-quarter of the way around the sphere. For our "flat-
land" people these lines would both be perfectly straight, because
the only curvature would be in the direction downward, which they
could never either perceive or discover. The lines would also
correspond to the definition of straight lines, because any
portion of either contained between two of its points would be the
shortest distance between those points.
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