Where the shape comes from
Two strokes turning their backs on one another. The word first meant "to split apart", and that sense is still doing the work at the top of 分 — a thing divided with a knife — and in 公 and 半. The graph was then borrowed to write the number eight, which sounded the same, and the number kept it. Sitting on top of a character it is written 丷, with the two strokes leaning in rather than out, and characters capped that way are filed here: 关, 半, 兰, 弟.