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豆 originally pictured a tall, footed ritual vessel — a stemmed offering bowl raised on a pedestal (the 'bean' meaning is a much later borrowing). Combined with the two feet above, the whole forms a compound ideograph: a person stepping up to place an offering on a raised vessel — to mount, ascend, climb. From there
登 generalises to 'climb, board, publish, register' (
登山,
登记,
登录).