I think Dr.King meant "Self purification" to be a process to cleanse one's mind and body from any sort of anger. This way in King's nonviolent protest, protestors have a mental resistance against the malicious sentiments they're facing. In the letter, King even states that in his workshops he assesses whether they are commited to nonviolence and protesting for a better future. One thing that was muddy to me was understanding Dr.King's different uses of extremist and non extremist and the message he was
