The name "Achilles" comes (most probably) from the Greek word Achos (αχος) which means "pain, sorrow, misery". His pain ought to prompt him to mature; to enter into that hero's journey away from the known, the familiar, the comfortable of the civilized world and into the barren wastelands of the desert, there to experience a great trial and eventually return "home" to maturity. This same pattern is parallel to the transformation of the alchemical exchange which, itself, follows the pattern of the developing natural world.
The Pathos stage; Youth; (the pupum) In the beginning of the epic Achilles is Juvenile; everything is about Achilles - he is selfish & thinks he is a god yet he is driven by petty emotions (rage, or menin). Thus he removes himself from the conflict & metaphorically removes himself from humanity (his clan, the Achaeans)
The Logos stage; Transition; (the chrysalis) Once exiled (self-inflicted, but still exile) from the clan Achilles has time to reflect and think about his condition. He reveals to the embassy that he is wrestling with a futility of all action; "The same fate waits for the coward and the brave"; this is the vision of the meaninglessness of life, the same sense that the preacher in Ecclesiastes expresses when he says
It is the desert of the heroic journey and it is not a pleasant place to be. Whilst here Achilles experiences the terrible loss of Patroclus, which prompts the wholesale slaughter of his fellow men. This "challenge" tests whether he is able to control himself and master the desire to enforce his will on the world like a god; whether he will ever be able to return to the human race and share in the common suffering of all men.
The Ethos stage; Maturity: (the imago) Achilles realizes that he is only a small part of a much larger world; he recognizes and accepts that he is part of the human race & thus, guided by reason and respect, he exhibits self-control even to his enemy (Priam). His return is not only to the Achaean group (which he did by reconciling with Agamemnon) but to the human race itself by seeing in his mortal enemy a commonality of suffering and pain.