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I don't think there would be any thickness at all, because there would be 2 dimensions, no up or down. Basic laws of physics would disappear in a 2D world as well, so its really hard to imagine a 2D world and what it would be like.
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i would refer you to a classic, Edwin A. Abbot's "Flatland: A Romance of Many Dimensions," which may be found here: http://www.alcyone.com/max/lit/flatland/
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