'non-symmetric' originates from Modern English, formed from the prefix 'non-' (from Latin 'non') meaning 'not' and 'symmetric' (ultimately from Greek 'summetrikos'/'symmetrikos'), where 'sym-/sum-' meant 'together' and 'metron' meant 'measure'.

'symmetric' changed from Greek 'symmetrikos' into Latin/French forms (via Late Latin/Old French) and then into Middle English as 'symmetric', and the modern negated form 'non-symmetric' was formed in English by adding the prefix 'non-'.

Initially 'symmetrikos' meant 'measured together' or 'commensurate', and over time 'symmetric' came to mean 'having symmetry'; 'non-symmetric' therefore came to mean 'not having symmetry' in modern usage.