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There are many different theories about what parallel worlds could be like: The physicist Max Tegmark has developed the idea that there are four different kinds of them, sorted into level I to IV. Level I has the same physical laws we have, while Level II has the same equations of physics, but maybe different dimensions (e.g. the weight of an electron). Level IV is very different, in dimensions and physical laws. Tegmark also suggests that there could be a parallel world just like ours, with exactly the same people in it, that only differs from ours in the decisions that are made. For example, in a Level I multiverse, every way of development exist in parallel in another world, while in a Level III multiverse, there is a branch for every decision we make. When we decide something, a new branch in each parallel world develops, and so there are infinitely many parallel universes, because in every possible moment the possibilities of development of the universe split up into infinitely many variants. And these variants are all part of a big universe, even if we experience only one of these possibilities by our decisions.
David Deutsch, another well-known physicist, supports this theory, but adds another aspect. In his opinion there exists an universe where the Earth developed otherwise, for example there might be a world in which dinosaurs survived. He tries to explain where other universes could be, by proposing that asking 'where' they are leads to thinking about a special location (like when you ask where a city is) and that this way of thinking is wrong, because these universes are not in ours. He says that we have to think in reverse: You have to imagine something even bigger than stars or galaxies. The reality is, according to Deutsch, this big thing, and this is called multiverse. His idea, that there is a big amount of parallel universes that are almost the same as ours, leads to the question of in how many universes we could meet ourselves. Deutsch answers this question by explaining that this would be an extremely large number, and if we counted the universes containing copies of ourselves that are identic to ours, the number would even be infinite. According to Deutsch universes begin developing infinitely just when you throw a coin. Imagine you throw the coin in infinitely many universes that are identical, and additionally in a few others. The identical universes split up into two groups, according to the possible results of throwing a coin: head or tails.
Images found at http://space.mit.edu/home/tegmark.