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which the hyperbolic geodesics are represented as
Euclidean straight line segments), think of this disk or ball as being the projection of a higher-dimensional Euclidean sphere, and lift the line segments vertically onto the upper hemisphere. The resulting representations of hyperbolic geodesics on the hemisphere are always the intersections of the hemisphere with vertical planes, producing exactly semicircles perpendicular to the boundary. —
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is necessary to return this comment or completely remove inappropriate references to this mathematician. When I studied the topic of non-Euclidean spaces at the university (including the history of this branch of mathematics), I never heard that name. I think that says a lot. Attempts to turn
Knowledge into a stronghold of communist propaganda should not be allowed.
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I assumed that this conversation would boil down to insults. I am 67 years old, and I will not allow myself to be treated like that! I have been using
Knowledge for over a dozen years, but I have never seen anyone allow the communists to promote their ideology here. Are you russian or do you work for
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If there are any mentions, it happens only in the context of describing the propaganda by russians/communists, who in every field of science tried to come up with their own "discoverers". This is wildness and cannot be seriously mentioned on
Knowledge outside of the section on historical absurdities.
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removed the note that the mention of russian mathematician-plagiarist has more to do with Soviet Cold War propaganda than historical facts. This name is not used in any serious
English-language source, in extreme cases it is mentioned as an alternative with a reference to the Iron Curtain. I think it
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and, in fact, it is already mentioned several times in the article. I just disagreed with your statement about it being the only way to embed a hyperbolic 2-space into an
Euclidean 3-space. Although now, I am not so sure that you were wrong. Also a small part of the pseudosphere might well look like
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You are thinking of geodesics for the spherical geometry on the hemisphere, I think? It is a different geometry than the one described here and has different geodesics. Another way to think of the hemisphere model of hyperbolic geometry: take the Klein model (a disk or ball in
Euclidean space in
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To JRSpriggs: The usual case is for infinite space and this is what is normally understood. Is there any mention of special finite
Euclidean constructions in the article?. But anyway why are you so much against mentioning the pseudo-sphere? As I said it has historical importance.
921:, it was purely because it had the right number of syllables, and not because of any actual history, right? There is no serious dispute that Lobachevksy should be listed with Gauss and Bolyai as one of the founders of hyperbolic geometry, and the first to publish of the three. —
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This is just nonsense I'm not going to answer. I'm still curious about a reference on the history of geometry that doe not mention
Lobachevsky in relation with hyperbolic geometry or that substitutes your wild claims, let me know if you find or recall one. Cheers,
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to the boundary must either pass through the point (0, 0, +1) or be the intersection of hemisphere with a vertical plane which does not pass through (0. 0. 0). Any geodesic in the hemisphere must be the intersection of the hemisphere with a plane which
662:. Lethe described the geodesics in the hemisphere model as "semicircles orthogonal to boundary". I reverted him saying "some are, but some are not.". Then David Eppstein reverted me with the challenge "Describe one that is not.".
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negative curvature is the pseudo-sphere which looks quite unlike the diagram shown. As far as I can see, the pseudo-sphere is not mentioned in the article although it has some historical importance.
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To JFB80: If the surface is finite in extent, having a boundary, then it does not have to be the pseudo-sphere. If you disagree, please provide a citation or a proof.
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I'm curious about which book would talk about the "history of this branch of mathematics" and not mention
Lobachevsky, do you have any available references? Cheers,
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While it is true that surfaces of negative curvature have a saddle-like appearance, it is known that the only Euclidean 2-dimensional surface of
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To TheKing44: All these are models. Yes we all know models exist. What I mean is a surface in Euclidean space which is a hyperbolic space.
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pointing out my previous swapped curvature signs. Could you please detail how this diagram can be improved to your satisfaction? Cheers,
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There was a feeling that an article on such a neutral topic was written by absolutely biased editors, promoting their POV.
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To David Eppstein: Thank you for clarifying that and correcting my mistake.
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I know I'm late to this discussion, but thanks a lot for a good laugh. –
682:. It will meet the boundary at a 45 degree angle (i.e. not orthogonal).
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Feel free to edit the article attached to this page, join up at the
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You do know that when Lehrer used Lobachevksy's name in the song
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It is fine now that you corrected the signs of the curvature.
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