Subject: Re: [HM] Parallel Postulate
From: Imre Toth (HAYMERIC@aol.com)
Date: Tue Jul 04 2000 - 13:11:31 EDT
Dans un courrier daté du 04/07/2000 09:53:24é), weiss@math.tu-dresden.de a
écrit :
<< Subject: Parallel Postulate/hyperbolic geometry >>
To the attention of Gunter Weiss, Dresden
Dear Gunter Weiss, you mentioned the Klein-Model of the Hyp-Geo: its Absolute
is a circle in the Euclid-Plane, and therefore is a Euclidean map of the
Hyp-Geo. It seems to me easy to transform this Euclidean in a model of the
Hyp-Geo in the plane of the Absolute Geometry of Bolyai. It is sufficient to
substitute the Cycles of the AGB to the circles of the EG. I remember :
Cycles are lines of the AGB defined by three of their points and they can be
either normal circles or Paracycles (one point at the infinity, homeomorph
with a Parable - therefore the term Para-cycle) or Hypercycles (twoo point at
the infinity, homeomorf with a Hyperbola, therefore the term Hyper-cycle).
The most convenient seems to me to choose a Paracycle as representing the
Absolute and orthogonal segments of Cycles representing the straight-lines of
the Hyperbolic plane in the plane map of the AGB. The same Model is, of
course, a Model, a map of the Hyperbolic plane in the same Hyperbolic plane.
By the same substitution of Euclidean circle with absolute Cycles it is also
possible to construct a map of the EG in the plane of the AGB (and the plane
of the NEG too). This is in fact the AGB Model of Moebius (the set of all the
absolute Cycles having a unique point in common, representing the Absolute
of the EG-plane in the plane of the AGB (or the NEG). This Moebius-Model is
also a map of the EG-plane in the plane of the Hyperbolic Geometry. The moral
of the fable seems to me to be the following: if either the EG or the
Hyperbolic Geometry is inconsistent, the AGB too, is necessarily
inconsistent; and: if the AGB is in itself consistent, the logical
conjunctions: (1) AGB&E and (2) AGB&non-E are also consistent. In other
words: the logical solidarity concerns not only the relation between EG and
NEG, but already both of the separately and the AB. A quite strange
historical remark: the first complete and correct picture of the
Klein-Poincaré Model is to be find in Orontius Finæus, De Mundi Sphæra, sive
Cosmographia, Lutetiæ Parisiorum, 1555, p. 24. An extremely rare book, I
succeed to find a copy only in the City Library of Bordeaux. Oronce Finée
proposed it to the navigators as a Map of the World. He was of course
ridiculised, and not without good reason. Oronce Finée was the first
professor of Cosmography at the Collège Royal, Paris (Collège de France
today), which somewhat later became the chair of Poincaré. Poincaré had of
course no idea of Finés mapamonde. A good subject for a science-fiction
story: in a negative time Oronce Finée plagiarized the Klein-Poincaré map of
a NEG World. Another remark. You mentioned the visual space - corresponding
to the German Anschauungsraum. But there is another Anschauungsraum too, the
plane as seen in perspective by an observer. He is placed in a fix point of
the plane, O, being the center of the Horizont line, the Absolute of his
plane. The Absolute is visibly an infinite Circle. The point O belongs to a
straight line connecting the infinite point North, with the infinite point
South. What is the geometrical structure of his plane, EG or NEG? It is easy
to decide. The Observer sees a ship navigating in the direction North. When
the ship arrives to the point North (the Observer has of course an infinite
life and, which is more important, an infinite patience) it suddenly
disappears in the Nothing of the dark North Sea of the infinity. If, exactly
at the same instance, the ship reappears, emerging from the Void of the
infinite point South and moves, from backwards, in the direction of the
Observer, than, the alternative is also decided: the plane is EG. If the ship
disappeared in North and does not reappear but only after an other infinity
of time (the Observer, of course, must have a double infinite patience) and
emerges not necessarily in the infinite of South, but on any infinite point
of the Horizont, than the alternative is decided too: the plane is NEG. But
who can decide it, where on the Horizont the ship will emerge again? The
Observer? Is there an Observe who could have a double infinite life? It is
clear that the alternative is undecidable - excepting, perhaps, by God. Thus
the perspective plane of the Observer seems rather to be that of the AGB. If
the point O is the common intersection point of a family of straight lines,
we can consider them all as Asymptotes of an infinity of Hyperbolas. On a
screen, the Observer sees the Absolute as a straight line dividing the plane
of the screen in two half-planes: superior and inferior.He sees the
Asymptotes as a family of straight lines orthogonal to the Horizont line. The
Hyperbolas are projected in the form of half-Ellipses. One and only one of
them is a half-Circle intersecting the Horizont on twoo infinite points, let
them be again North and South and a ship is navigating along the Circle from
South to North. The Captain mails to the Observer that he is navigating on a
straight trajectory from South to North. So the sea of the captain is
certainly NEG. How can the Observer decide - looking only to the perseptive
map of his World - whether his plane is EG or NEG? If the plane is EG than
the orthogonal half-circle of his perspective map is the image of an
euclidean Hyperbola (the equation of this peculiar family of conics can
easily be determined; I did it many years ago, but - not being a
mathematician - never published it). If the plane is NEG, than the half
circle is the perspective image of a straight line, being - in the Hyperbolic
plane - the third asymptote of the infinite set of Hyperbolas having in
common the two straight asymptotes, represented by the two straight lines both
coorthogonal to the line of the Horizont of the screen. Yours sincerely, Imre
Toth (Paris)
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