Geometry
Non-Euclidean geometry
Non-Euclidean geometry is the subject of study
that results by making certain assumptions about points, lines,
planes, and space and then drawing conclusions generally consistent
with one's spatial intuition concerning objects of moderate
size and yet rich in certain relationships that affront the
intuition, particularly relationships concerning the concept
of parallelism extended to large distances. For instance, similar
figures (the same shape) are necessarily congruent (the same
shape and size): no plan or model or map can be truly accurate.
The two principal types of such a geometry are vividly distinguished
by referring to the following imaginary construction. Two rays
in a plane are drawn from points A and B,
perpendicular to
and on the same side of the line that connects A and
B (Figure 48). Instead
of remaining equidistant they become farther apart or closer
together. In the former case, when the rays diverge, the non-Euclidean
geometry is said to be
hyperbolic (from the Greek
hyperballein, "to throw beyond"). In the latter case,
when the rays converge and ultimately intersect, the geometry
is said to be
elliptic (from elleipein,
"to fall short"). Because it is impossible in practice to
measure how far apart the rays will be when extended millions
of miles, it is quite conceivable that man is living in a non-Euclidean
universe. (Because intuition is developed from relatively limited
observations, it is not to be trusted in this regard.) In such
a world, railroad tracks can still be equidistant, but then
they will not be perfectly straight.
In the language of
axiomatic mathematics,
non-Euclidean geometry satisfies all of Euclid's axioms except
either the fifth or the second. The axioms of Euclid are stated
fully in the section above Euclid's
work. The second axiom states that an interval
can be prolonged indefinitely. The fifth states that, if a line
meets two other lines so as to make the angles a and
b on one side of it together less than two right angles,
the other lines, if prolonged indefinitely, will meet on this
side (in Figure 49, the right
side).
In hyperbolic geometry, axiom 5 is denied, because, if the
ray from A in Figure 48
is replaced by one making a very slightly smaller angle with
AB, the new ray from A and the old one from B
may converge at first, attain a minimal distance, and then
diverge (Figure 50). In elliptic
geometry, axiom 5 is satisfied trivially, but axiom 2 (interpreted
as giving the line an infinite length) is denied, because now
the line is closed, like a circle.
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