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## The Question

(Submitted November 13, 1997)

I saw a reference to this subject on The Imagine the Universe! site, but I'm afraid I just don't get it. The example given stated that if a massive object (such as a galaxy) lay directly between us and another object (such as a quasar) then the light from that quasar would be bent around the massive object to form a ring. That part makes sense. However, if the massive object is not in a direct line between us and the object being lensed, then two or more identical images of the object would appear around the massive object (like Einstein's cross). I just don't understand why this is the case. Why would it not form an ellipse? What focuses the light to two, or four points?

Vague answer: It has to do with optics, the cardinality of points versus lines, and symmetry arguments.

Assume, for the sake of argument, that off-center objects behind the lens were imaged as ellipses. Suppose that two objects were (in actual, rather than apparent position) in directions e.g.,
a) 1 degree north of the lens and
b) 1 degree east of the lens
You would expect that the two ellipses that formed would be the same size and shape, but rotated with respect to each other. They would therefore cross in at least two places.

Since light is reversible in simple optical systems like this one, imagine a ray of light traveling from your eye towards one of the points where the ellipses appear to cross. By reversibility, that ray has to hit object a), since it is going towards a's image. It also has to hit b), since it is going towards b's image. Since simple optics (without half-silvered mirrors and the like) does not allow rays to split, this is impossible.

Since the assumption that the image would be elliptical leads to an impossible result, that assumption must be wrong. (This is the 'reduction ad absurdum' technique).

I'm not a mathematician, but it is probably provable that any mapping of all points in real_space to lines in image_space leads to crossings of the image lines, and therefore is optically impossible. However, that does not mean that you can't map points to multiple point images.

David Palmer
for Ask an Astrophysicist

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