无故
wú gù · HSK 7
without cause or reason
无故 = 无 (without) + 故 (cause/reason) — happening with no reason behind it
无 means lacking or none; 故 means cause or old-reason (same 故 as in 'reason why'). Together: an action done 无故 has no cause justifying it — punished 无故 means punished for no reason.
无 — top 一 + 林-like middle + fire radical 灬 below
Originally a pictograph of a person dancing while holding ornaments or tassels, meaning 'to dance' — the ancestor of 舞. It was borrowed phonetically to write the negative 'not have / without,' and the dance meaning was reassigned to 舞 with an added foot element. The simplified form 无 comes from an ancient variant.
故 — 古 (old) + 攵 (action/strike)
故 combines 古 'old' as the phonetic with 攵, a variant of 攴 meaning to strike or act. Originally it referred to a cause or reason behind an action. From this it extended to mean 'former, old, deceased' and 'therefore, hence' as a connector.