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After the Heptapods give a huge number of circle-letters, Ian has a breakthrough, noting the spacing is even. He gives a number, which if I remember correctly was 0.083, or 0.83. It wasn't clear to me how this point advanced the story. What "breakthrough" has he achieved by discovering this number?

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0.083 refers to the ratio of how much space the circle-letters take on the whole white background (the "negative space"). Incidentally, 0.083 equals 1/12th. There were 12 Heptapod ships around the world. I believe this means the message from Abbott and Costello is a part of a whole that could be completed if we communicated with all the heptapods.

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    Exactly. It made Ian realize they were only seeing 1/12th of the message, and would need to cooperate with the other groups to be able to see the whole picture. – aryxus Jun 2 '17 at 18:43

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