Origins
To be involved is to be rolled up inside something, tangled beyond easy separation.βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ The word comes from Latin involvere β in- ('in') and volvere ('to roll') β meaning 'to wrap up, to envelop, to entangle'.
The image is textile: thread wound tight around a spindle, impossible to pull free without unravelling everything. When you become involved in a project or a relationship, you are etymologically rolling yourself into it, binding yourself to its turns.
Latin volvere generated one of the largest word families in English. Revolve: to roll again, to turn in circles. Evolve: to roll out, to unfold β Darwin's evolution is the unrolling of life. Volume: originally a roll of papyrus, the standard form of ancient books. Vault: an arched ceiling that rolls overhead. Voluble: rolling easily, fluent in speech. Convoluted: rolled together in a confused tangle.
Proto-Indo-European Roots
The word waltz may also trace to this PIE root *wel- ('to turn'), through Germanic rather than Latin β a dance of turning.
Involved gained its modern participatory sense by the 17th century. But the tangling meaning persists: 'it's complicated' and 'I'm involved' carry the same implication of being wrapped up in something not easily escaped.