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![Fifth Element, The]()
Fifth Element, The (1997)
IMDB rating: 7.20
Plot: Two hundred and fifty years in the future, life as we know it is threatened by the arrival of Evil. Only the fifth element (played by Milla Jovovich) can stop the Evil from extinguishing life, as it tries to do every five thousand years. She is helped by ex-soldier, current-cab-driver, Corben Dallas (played by Bruce Willis), who is, in turn, helped by Prince/Arsenio clone, Ruby Rhod. Unfortunately, Evil is being assisted by Mr. Zorg (Gary Oldman), who seeks to profit from the chaos that Evil will bring, and his alien mercenaries.
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Directors: Besson Luc
Actors: Willis Bruce,Holm Ian,Oldman Gary,Tucker Chris,Perry Luke,James Brion,Lister Jr. Tom ‘Tiny’,Evans Lee,Creed-Miles Charlie,Tricky,Neville John,Bluthal John,Kassovitz Mathieu,Fairbank Christopher,Sci-Fi,Action,Adventure,Thriller,
Studying Euclidean geometry in a hyperbolic world?
You have blissfully lived your life as a two-dimensional creature in the hyperbolic plane, but recently someone has doubted the fifth postulate of Poincare’s Elements. Poincare’s famous fifth postulate states that
"given a line L and a point p not on L, there are at least two lines through p that do not meet L."
You decided to engage in the most daring intellectual experiment you’ve ever performed, and you replaced this fifth postulate with a new one:
"given a line L and a point p not on L, there is exactly one line through p that does not meet L."
You developed the logical consequences of this along with Poincare’s four other axioms and published your result in a book, but the great mathematician Lobachevsky derided it as nonsense, declaring that you’d finally gone mad–clearly there is no such thing as this "Euclidean geometry," he said.
But you realize that one way to prove that this Euclidean geometry exists is to find a model for it, to find some subset of the familiar hyperbolic plane that can be realized as having this mystical Euclidean geometry. All you have to do is find the appropriate subset, construct a dictionary that says what each element of Euclidean geometry corresponds to (e.g., it should say what a Euclidean geodesic would correspond to in the ordinary hyperbolic plane), and identify the group of motions. Then everybody will see that Euclidean geometry can in fact be made consistent and even visualized within familiar territory.
You begin your writing your proposal, starting with…
One way to "discover" the upper half-plane model is to notice that the geometry of the pseudosphere is locally hyperbolic and to then try to find a conformal map from the pseudosphere to the Euclidean plane. After a bit of work, this pretty much gives rise to the upper half-plane model as it is known today.
("Locally hyperbolic" is probably not the best term since clearly if you take a small enough piece of the pseudosphere then it will in fact appear Euclidean. But if you take a medium-sized patch–not too large but not too small–then it will appear roughly hyperbolic.)
Maybe something analogous could be profitable here.
You’re quite right that the pseudosphere isn’t a perfect model of hyperbolic geometry, which is why I used the poor term "locally hyperbolic." In fact, I believe the construction of the upper half-plane model from the pseudosphere may be one of the earliest examples in mathematics of the use of a universal cover (to take care of, e.g., the topological differences between the pseudosphere and the plane).
I was more or less just trying to think about how a creature in the hyperbolic plane might try to find a model of Euclidean geometry in a way analogous to how we found a model of hyperbolic geometry.
I agree that this question has a lot of interesting and even deep connections with physics. It makes me wish I was more up to speed with modern physics so that I could explore these connections more fully.
Fascinating problem, thanks for even posting it. Best one in a long time. Let me think about this one for a while, and I’m anxious to see other people’s answers to this one. Not that this is an answer, but I’ve noticed for a long time now that the difference between Euclidean and Lobachevsky geometries is the same as the difference between Newtonian physics and both quantum and relativisitic physics, which is the imaginary number. Like how Cos(ix) = Cosh(x), or the fact that Schrodinger’s equation is just the complex version of the oridinary diffusion equation. Which is what makes your question all the more interesting, it’s kind of like "Through The Looking Glass". Deep stuff here.
Okay, some preliminary thoughts. One of the commonest ways to "illustrate" that Euclidian geometry is but a linear approximation of non-Euclidian geometry is to limit the space to a small local area. What that translates to in any of the mathematically equivalent representations of hyperbolic geometry—Klien, Poincare disk, Poincare half plane, hemispherical—is that the point p be only a small distance from line L. Then the spread of non-parallel lines is bunched up into a single one. As an example, consider the Poincare half-plane, given the "line" L (which is a semicircle), and a point p very close to it, there isn’t much of a range of semicircles passing through p that doesn’t intersect the line L. However, this is no fun nor deep enough for my taste. I think the more interesting way is to find an inverse of the meaning of distance between the line L and p, i..e., what is "far" is really "very near", and vice versa. Lemmie think on this some more. I like this problem very much, it seems to pull together a LOT of things in physics. Suddenly more things are making more sense to me.
Edit: About your additional comments, first of all, the mathematics of non-euclidean & euclidean geometry is true "for all observers", whether or not they exist in either world. Hence, denizens of an hyperbolic should have the ability to "construct" an Euclidean model "with the strange property that there is only one line through point p that does not intersect line L". But I think the spirit of this quesiton here is how can we, denizens of an Eulcidean world, can imagine how an "Euclidean plane" would be modelled by denizens of the hyperbolic world. What is of particular interest to me about this problem is that it suggests that there is a "relativity of reality" here, i.e., we can see that perhaps Euclidean geometry is just a special (local) case of non-Euclidean geometry, but how about trying to flip that around, where the denizens of an hyperbolic world "realize" that they are merely a special case of "something more general", which is Euclidean? This is why I’m needing to spend a lot of time thinking about this one.
Also I’d like to point out that something like the Poincare disk is actually more general than the pseudosphere in representing hyperbolic geometry, because the former doesn’t have a boundary (because the circle represents infinity), while the latter does. I would try to work from suich representations for this problem. The math is easier that way, anyway.
Edit 2: I have to admit that I don’t think I’ll have the time to fully work this one out. It is a massive undertaking for me, because I understand the implications here. There’s weeks, months, and maybe years of thinking involved here, it’s already having an impact on the way I’ve been looking at the mathematics of physics, particularly the matter of "bridging" between classical and so-called "modern" physics. I’ve never really deeply studied what I would call "classical non-Euclidean geometry" (i.e., as originally envisioned in pre-relativity days), but now I realize it deserves a much closer look. Thanks for bringing this to my attention! It has been exciting so far, and I don’t see an bottom to this.
Yes, there are plenty of papers written on this very subject, much of it pretty dense. I’m still wading through some of them.
By the way, see my answer to your next question on this subject, which I think is a more straightfoward question. A good one, however, one that I like because it once again emphasizes the importance of the role i plays.
Edit 3: I fully understand your query about how we could imagine how a "creature of the hyperbolic plane" would go about "constructing the Eucliean plane". But in fact, just like we imagine a "small patch of local geometry which is Euclidean" in a hyperbolic (or any non-Euclidean) space, such a creature would be imagining how a "small patch of local geometry which is hyperbolic" would fit in an Euclidean world. How so? We are already looking at such an example—the Poincare disc, which is represented in Euclidean space. To us, the Poincare disc looks like awesomely warped space (think Escher), but to the creature, it may appear normal, and it is us that looks awesomely warped, the "weird Eucldiean space beyond our local hyperbolic space".
Scythian1950 | Jan 31, 2010
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sammyyy
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