To dye or color something deeply and thoroughly, or to firmly establish something so that it becomes a deep part of someone's nature or beliefs.
From 'en-' plus 'grain,' where 'grain' refers to the texture of wood or fabric, or possibly to dye (kermes dye was called 'graine'). The term evolved from textile dying practices in the 16th century to metaphorical usage.
When something is 'engrained,' it's colored as deeply as wood grain goes—in medieval textile workshops, engraining was a multi-step process that made dyes permanent and saturating, which is why we still use it metaphorically for deep beliefs.
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