To surround a place (usually a city or castle) with an army and prevent supplies or people from entering or leaving; or to crowd around someone persistently.
From Old French 'besiege,' meaning to seat oneself before, combining 'be-' (around) and 'siege' (from Latin 'sedere,' to sit). The original meaning literally meant to surround and sit at a location to force surrender.
The word 'siege' comes from the root meaning 'to sit'—because medieval armies would literally sit camped around a city for months or years! So 'besiege' means to sit yourself around something in force.
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