Another Crazy Idea: Pattern-Matching The Cosmos

Has anyone ever done a computer analysis of all known stars seen from Earth, to detect if there are any groups of stars that are identical (or linearly transformed) in different parts of the sky? For example, have they looked for identical sets, mirror image sets of stars, flipped image sets, slightly drifted but nearly equivalent sets of stars, etc.? The idea is that if we located two sets of identical, or linearly transformed, star groups then we might conclude that they are in fact the same star system viewed from different directions on curved space-time from our perspective. In other words, it might be possible to see the same set of stars from two directions at once. This could indicate something about the topology of space-time. It would be an interesting journey in “wild speculation” — but who knows, it could yield unexpected results. It shouldn’t be so difficult to do an exhaustive search using a supercomputer of all sets of up to n stars in the known universe. Where it might get subtle and complex however is if we want to account for all angles, gravitational lensing effects in various directions from a given star, and for variations in stellar spectral lines and position over long periods of time. We would probably need to start with a simple search — looking for nearly identical patterns in different places — such as “same stars, same relative positions to each other,” or “same stars, slightly different relative positions,” or “slightly different stars, same relative positions,” or “slightly different stars, slightly different relative positions.” Getting more exotic would require more computational power — for example, if we broaden the range of variations we would consider in the positions of the stars relative to one another, or the range of spectral changes that we would consider for each star (for example in billions of years, for a given star X, given what we know about it today, how would it appear to us?).