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.
> 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.
If there are two or more then one can prove that there are infinitely many.
-- Which of the seven heavens / Was responsible her smile / Wouldn't be sure but attested / That, whoever it was, a god / Worth kneeling-to for a while / Had tabernacled and rested.
In article <c28bde2b-e7b7-413f-95bb-2ed64559f...@a32g2000yqm.googlegroups.com>, Ken Quirici <kquir...@yahoo.com> writes:
>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 >BTW I can find no way to make sense of the notion of >euclidean geometry being a 'union' of hyperbolic and >elliptic geometry.
That notion does seem nonsensical. Where did it come from?
-- Michael F. Stemper #include <Standard_Disclaimer> No animals were harmed in the composition of this message.
mstem...@walkabout.empros.com (Michael Stemper) writes: > In article <c28bde2b-e7b7-413f-95bb-2ed64559f...@a32g2000yqm.googlegroups.com>, Ken Quirici <kquir...@yahoo.com> writes: >>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
>>BTW I can find no way to make sense of the notion of >>euclidean geometry being a 'union' of hyperbolic and >>elliptic geometry.
> That notion does seem nonsensical. Where did it come from?
Nonsensical? How could it be? This discovery is just another in a long line of breakthroughs by the King of Science, Archimedes Plutonium.
I bet you're embarrassed now.
-- Jesse F. Hughes "There's a thrill that's gone that I'll probably not have in quite the same way again. After all, FLT was a unique animal, and we had a great dance." -J.S. Harris on "proving" Fermat's last theorem
In article <87hbtaf9la....@phiwumbda.org>, "Jesse F. Hughes" <je...@phiwumbda.org> writes:
>mstem...@walkabout.empros.com (Michael Stemper) writes: >> In article <c28bde2b-e7b7-413f-95bb-2ed64559f...@a32g2000yqm.googlegroups.com>, Ken Quirici <kquir...@yahoo.com> writes: >>>BTW I can find no way to make sense of the notion of >>>euclidean geometry being a 'union' of hyperbolic and >>>elliptic geometry.
>> That notion does seem nonsensical. Where did it come from?
>Nonsensical? How could it be? This discovery is just another in a >long line of breakthroughs by the King of Science, Archimedes >Plutonium.
D'oh!
-- Michael F. Stemper #include <Standard_Disclaimer> A bad day sailing is better than a good day at the office.
> > 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.
> If there are two or more then one can prove that there are infinitely > many.
> -- > Which of the seven heavens / Was responsible her smile / > Wouldn't be sure but attested / That, whoever it was, a god / > Worth kneeling-to for a while / Had tabernacled and rested.
OK, little steps here. First some clarifications about nomenclature (upon reading more of the Wikipedia article about hyperbolic geometry):
Can we define the asymptotic lines thru the point P parallel to the line l which does not contain P as follows - i.e. is this what the article in question means:
consider the point B on l such that PB is perpendicular to l.
extend the line PB in both directions as the line m.
are the asymptotic lines thru P the following:
1. the line l1 parallel to l which has the smallest interior angle á counterclockwise from m to l 2. the line l1 parallel to l which has the smallest interior angle â clockwise from m to l
?
So all the lines between l1 and l2 are also parallel and are called ultraparallel - so since there are an infinite number of angles between the two angles á and â, there are an infinite number of parallel lines?
OK.
l1 and l2 must exist since they are at least the two 'given' parallel lines thru P to l, right?
But all right angles are the same according to the first four postulates of any of the (three) geometries. This must mean that the asymptotic and ultraparallel lines thru P are not at right angles to the line PB, right?
In article <c28bde2b-e7b7-413f-95bb-2ed64559f...@a32g2000yqm.googlegroups.com>, Ken Quirici <kquir...@yahoo.com> wrote:
> 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. > ....
Frederick Williams explained that. Theories can have alternative axioms; so an author may ask for infinitely many parallels or may just ask for more than one.
> .... > 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?
Yes. But Bolyai's absolute geometry also includes everything in Euclid which doesn't need the parallel postulate, in particular Euclid I.1-28 and 31. So you have, for example, the usual Euclidean facts about congruent triangles.
Absolute geometry comprises just those theorems which Euclidean and hyperbolic geometry have in common.
> 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?
Euclid I.31 (which is absolute) proves the existence of parallels, so in absolute geometry your number can only be 1 or infinity.
You can get 0 (i.e. no parallels at all) by going to elliptic geometry, but that isn't a special case of absolute geometry. Various propositions from Euclid I.16 onward don't hold in the elliptic plane.
> BTW I can find no way to make sense of the notion of > euclidean geometry being a 'union' of hyperbolic and > elliptic geometry....
Neither can I. It's more like a borderline case between the two, although even that takes a bit of explaining. :-)
In article <c28bde2b-e7b7-413f-95bb-2ed64559f...@a32g2000yqm.googlegroups.com>, Ken Quirici <kquir...@yahoo.com> wrote:
> 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?
There's a useful discussion of absolute geometry (which they prefer to call "neutral geometry") in Chapter 3 of Prenowitz and Jordan, Basic Concepts of Geometry. Actually, the whole book is excellent.
-- Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
> In article > <c28bde2b-e7b7-413f-95bb-2ed64559f...@a32g2000yqm.googlegroups.com>, Ken > Quirici <kquir...@yahoo.com> writes: >>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
>>BTW I can find no way to make sense of the notion of >>euclidean geometry being a 'union' of hyperbolic and >>elliptic geometry.
> That notion does seem nonsensical. Where did it come from?
I could almost accept the 'intersection' of the geometries, if we define hyperbolic geometries as those with curvature <= 0 and elliptical as those with curvature >= 0.
"Peter Webb" <webbfam...@DIESPAMDIEoptusnet.com.au> writes: > "Michael Stemper" <mstem...@walkabout.empros.com> wrote in message > news:hcse0v$191$1@news.eternal-september.org... >> In article >> <c28bde2b-e7b7-413f-95bb-2ed64559f...@a32g2000yqm.googlegroups.com>, Ken >> Quirici <kquir...@yahoo.com> writes: >>>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
>>>BTW I can find no way to make sense of the notion of >>>euclidean geometry being a 'union' of hyperbolic and >>>elliptic geometry.
>> That notion does seem nonsensical. Where did it come from?
> I could almost accept the 'intersection' of the geometries, if we define > hyperbolic geometries as those with curvature <= 0 and elliptical as those > with curvature >= 0.
Look, it's very simple. Euclidean is the union of hyperbolic and elliptic geometry. After all, | = ) + (. That's all there is to it.
Am I the only one reading Archimedes Plutonium these days? Sheesh.
-- "All intelligent men are cowards. The Chinese are the world's worst fighters because they are an intelligent race[...] An average Chinese child knows what the European gray-haired statesmen do not know, that by fighting one gets killed or maimed." -- Lin Yutang
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