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The Accidental Astrophysicists
An anonymous reader recommends a ScienceNews story that begins: "Dmitry Khavinson and Genevra Neumann didn't know anything about astrophysics. They were just doing mathematics, like they always do, following their curiosity. But five days after they posted one of their results on a preprint server, they got an email that said 'Congratulations! You've proven Sun Hong Rhie's astrophysics conjecture on gravitational lensing!'... Turns out that when gravity causes light rays to bend, it can make one star look like many. But until Khavinson and Nuemann's work, astrophysicists weren't sure just how many. Their proof in mathematics settled the question."
The wikipedia article on gravitational lensing has a neat animation produced with a numerical model. I wouldn't make it your desktop background though because it might warp your file icons.
n refers to the number of massive objects causing the lensing as I understand it, but I could be wrong. I'm slightly drunk while posting and my previous existence as an astronomy grad student eludes me.
I think the article says that this equation is used to find the maximum amount of star images that could be created by a massive object, so in the case of one massive object, according to the equation, you could get a maximum of 5 images of the star.
n = # of massive objects in the way, not light sources.
I'm not actually an astrophysicist, but I may be able to sort of explain. Take a look at the diagram in TFA: it's just in two dimensions, specifically the plane defined by the distant star, the massive object, and the observer. We see two images that are in that plane, because only light rays from the star that are traveling in that plane can be bent by the massive object so that they can reach us; rays traveling in any other plane would be bent to arrive at some other location. And the star is effectively a point source, so we see exactly two point images. With multiple massive objects, there are more planes, but the planes are still discrete, so there are still discrete images. The only exception is when the star, the massive object, and the observer are exactly in line, in which case we see a circle.
Galaxies, on the other hand, are not point sources, which is why when we see gravitationally lensed galaxies they often look stretched out along arcs -- different points in the galaxy line up differently, and thus can look farther apart from each other than they would if we were seeing them without lensing.
So the number you see doesn't have to be a multiple of 5 always, even for n>1.
The development of non-Euclidean Geometry argues against your point, rather than supporting it.
Non-Euclidean geometry arose out of pure mathematical attempts to correct a "flaw" in Euclidean geometry. Namely, that the parallel postulate was so big and complex that it didn't seem like a proper axiom, not like Euclid's other four axioms.
Lots of mathematicians had tried various ways to prove that the parallel postulate wasn't necessary, that it could be derived from the other four, and many flawed "proofs" were constructed. A few mathematicians, notably Saccheri, decided to take a less constructive route and try to disprove the necessity of the parallel postulate by contradiction.
The idea was: Replace the parallel postulate with something else that means the opposite, and then show that geometry breaks down, that logical contradictions can be shown. Saccheri thought he succeeded because he was able to prove some things that made no sense within the Euclidean framework.
Later mathematicians realized that, in fact, Saccheri had "failed" to find a contradictions, that his results resulted in a geometry that was weird and non-Euclidean, but perfectly consistent, and in fact made perfect sense if you applied it in the context of a hyperbolic surface. Under a different modification of the parallel postulate, you get a geometry that makes perfect sense on the surface of a sphere.
Later still, physicists picked up on these alternative geometries and began applying them to great benefit. Notably, Einstein's notion of spacetime as non-Euclidean.
It goes both ways, of course. Physics often motivates math, and pure math is often adopted and applied by physics. Neither would be as rich without the independent work of the other.
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There's no guarantee that you can see the 'straight through' image, because the object doing the lensing might be in the way.
And for n objects lensing, the effect is multiplicative.
What's so difficult about that?
One swallow does not a fellatrix make