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?
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.