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Wikipedia contradiction in re: hyperbolic geometry; and in general,
the geometries
Posted:
Nov 3, 2009 10:05 PM
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First, Wikipedia's article on non-euclidean geometry contradicts the
article on hyperbolic geometry in that the former says that, in a
hyperbolic geometry, given a
line l and a point P, there are *infinitely* many distinct lines
passing
thru P and parallel to l. On the other hand, the article on hyperbolic
geometry says that for such a geometry *at least two* distinct lines
pass thru P and are parallel to l.
(BTW the Mathworld article on hyperbolic geometry makes the
oddly imprecise definition that in a hyperbolic geometry *many*
distinct lines pass thru P and are parallel to l. This offends my
nascent mathematical susceptibilities).
So which is hyperbolic - infinitely many, or > 2, or 'many', and
pls supply a def. of 'many' - yeah I know, 'many' means '> 2'.
Isn't 'many' generally to be avoided in mathematics?
Secondly, the three types of geometry define the case for all lines l,
and
for each line l, all points P not on it, belonging to a geometry of
that
type.
There is now I discover 'absolute' geometry which does not
invoke ANY version of the parallel postulate.
Am I correct in assuming that an absolute geometry allows for
parallel lines, but makes no universal claims relative to
lines l, points P not on them, and lines passing thru P
and possibly parallel to l? So there could be
lines l and points P with 0, 1, 2, or oo many lines passing
thru P parallel to l, all in the same geometry?
BTW I can find no way to make sense of the notion of
euclidean geometry being a 'union' of hyperbolic and
elliptic geometry. It seems to be like claiming a set
of apples is a union of a set of peaches and a set of
watermelons.
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