Who He Is
Polyphemus is the Cyclops who herds alone, in a cave closed by a stone that twenty-two wagons could not move, far from the others, and who knows only lawless deeds. He is a son of Poseidon. The Odyssey introduces him before recounting the encounter: in the first book, he is the reason for the god's anger against Odysseus, who blinded him. He appears on stage in the account that Odysseus himself gives to the Phaeacians, and he is the adversary who best shows what the poem means by hospitality, since he is the one who denies it.
Trajectory
In Book I, the poem mentions him through Zeus, who explains to Athena that Poseidon pursues Odysseus because he blinded the god's son. The punishment is wandering.
ἐκ τοῦ δὴ Ὀδυσῆα Ποσειδάων ἐνοσίχθων / οὔ τι κατακτείνει, πλάζει δʼ ἀπὸ πατρίδος αἴης.(1.74-75), since then Poseidon, shaker of the earth, does not kill Odysseus, but makes him wander far from his native land.
In Book II, the old Egyptian mourns his son Antiphus, who went to Troy with Odysseus and was devoured by the Cyclops.
In Book IX, Polyphemus is the center of the account. Odysseus and twelve companions enter the cave while he is out. Odysseus wants to see the owner and receive a guest's gifts, despite his companions' pleas to leave. When the Cyclops returns, Odysseus asks for hospitality and invokes Zeus Xenios, avenger of guests. Polyphemus replies that the Cyclopes do not care about Zeus, because they are stronger, and devours two companions, dashing them against the ground like puppies.
οὐ γὰρ Κύκλωπες Διὸς αἰγιόχου ἀλέγουσιν / οὐδὲ θεῶν μακάρων, ἐπεὶ ἦ πολὺ φέρτεροί εἰμεν·(9.275-276), for the Cyclopes do not care for aegis-bearing Zeus, nor for the blessed gods, since we are far stronger.
σὺν δὲ δύω μάρψας ὥς τε σκύλακας ποτὶ γαίῃ / κόπτʼ· ἐκ δʼ ἐγκέφαλος χαμάδις ῥέε, δεῦε δὲ γαῖαν.(9.289-290), seizing two of them together, as if they were puppies, he dashed them against the ground; the brains ran out onto the earth and wet the ground.
On the next night, he drinks the wine Odysseus brings, asks his name, and Odysseus says he is Nobody. The Cyclops replies that he will eat Nobody last, after his companions, and that this will be the guest-gift, and then he sleeps. The men thrust the olive stake into his eye. When the other Cyclopes come, Polyphemus says from inside the cave that Nobody is killing him by trickery, not by violence, and they conclude that the affliction comes from Zeus.
Οὖτιν ἐγὼ πύματον ἔδομαι μετὰ οἷς ἑτάροισιν, / τοὺς δʼ ἄλλους πρόσθεν· τὸ δέ τοι ξεινήιον ἔσται.(9.369-370), I will eat Nobody last, after his companions; the others before. This will be your guest-gift.
ὦ φίλοι, Οὖτίς με κτείνει δόλῳ οὐδὲ βίηφιν.(9.408), friends, Nobody is killing me by trickery and not by force.
Blind, he sits before the door to catch whoever tries to leave, and he feels the backs of the rams without noticing the men tied beneath their bellies. He speaks to the ram that goes out last, imagining it sad for its master, and says that if it could tell where Nobody hides, he would smash his brain against the ground. When Odysseus shouts his real name, Polyphemus recalls an old prophecy from the seer Telemus, that he would lose his sight at the hands of Odysseus, and that he had expected a large, strong man, not a small and weak one who overpowered him with wine. He calls Odysseus back, offering gifts and Poseidon's escort, and is refused.
ἦ σύ γʼ ἄνακτος / ὀφθαλμὸν ποθέεις, τὸν ἀνὴρ κακὸς ἐξαλάωσε / σὺν λυγροῖς ἑτάροισι δαμασσάμενος φρένας οἴνῳ, / Οὖτις, ὃν οὔ πώ φημι πεφυγμένον εἶναι ὄλεθρον.(9.452-455), perhaps you miss the eye of your lord, which a wicked man blinded, with his wretched companions, after taming my mind with wine: Nobody, whom I do not say has yet escaped destruction?
ὅς μοι ἔφη τάδε πάντα τελευτήσεσθαι ὀπίσσω, / χειρῶν ἐξ Ὀδυσῆος ἁμαρτήσεσθαι ὀπωπῆς. / ἀλλʼ αἰεί τινα φῶτα μέγαν καὶ καλὸν ἐδέγμην / ἐνθάδʼ ἐλεύσεσθαι, μεγάλην ἐπιειμένον ἀλκήν· / νῦν δέ μʼ ἐὼν ὀλίγος τε καὶ οὐτιδανὸς καὶ ἄκικυς / ὀφθαλμοῦ ἀλάωσεν, ἐπεί μʼ ἐδαμάσσατο οἴνῳ.(9.511-516), he told me that all this would be fulfilled later, that I would lose my sight at the hands of Odysseus. But I always expected that some large and handsome man would come here, clothed in great strength; now, being small, insignificant, and weak, he has blinded my eye, after taming me with wine.
Then he prays to Poseidon, repeating the name, lineage, and home he heard: if he is the god's son, let Odysseus not reach home, and if fate wills that he reach it, let him arrive late and wretched, without companions, in a stranger's ship, and find misfortune in his house. Poseidon hears him, and Polyphemus hurls a stone larger than the first.
εἰ ἐτεόν γε σός εἰμι, πατὴρ δʼ ἐμὸς εὔχεαι εἶναι, / δὸς μὴ Ὀδυσσῆα πτολιπόρθιον οἴκαδʼ ἱκέσθαι / υἱὸν Λαέρτεω, Ἰθάκῃ ἔνι οἰκίʼ ἔχοντα.(9.529-531), if I am truly yours, and you claim to be my father, grant that Odysseus, sacker of cities, son of Laertes, who has his house in Ithaca, not reach his home.
ἀλλʼ εἴ οἱ μοῖρʼ ἐστὶ φίλους τʼ ἰδέειν καὶ ἱκέσθαι / οἶκον ἐυκτίμενον καὶ ἑὴν ἐς πατρίδα γαῖαν, / ὀψὲ κακῶς ἔλθοι, ὀλέσας ἄπο πάντας ἑταίρους, / νηὸς ἐπʼ ἀλλοτρίης, εὕροι δʼ ἐν πήματα οἴκῳ.(9.532-535), but if it is his fate to see his own people and reach his well-built house and his native land, let him come late and wretched, having lost all his companions, in another's ship, and find misfortunes in his house.
In Book X, Odysseus continues the account after Polyphemus, and later Eurylochus uses the cave as an argument to accuse Odysseus of having caused the death of the companions. In Book XI, Tiresias tells Odysseus that the rancor of Poseidon comes from the blinded son.
In Book XX, on the eve of the slaughter, Odysseus bids his heart endure, reminding himself that he has endured something worse: the day the Cyclops devoured his companions, and he held on until his plan got him out of the cave.
τέτλαθι δή, κραδίη· καὶ κύντερον ἄλλο ποτʼ ἔτλης. / ἤματι τῷ ὅτε μοι μένος ἄσχετος ἤσθιε Κύκλωψ / ἰφθίμους ἑτάρους· σὺ δʼ ἐτόλμας, ὄφρα σε μῆτις / ἐξάγαγʼ ἐξ ἄντροιο ὀϊόμενον θανέεσθαι.(20.18-21), endure, heart! You have endured something even more vile, on the day when the Cyclops, of irresistible strength, devoured my valiant companions; you held on until the plan brought you out of the cave, when you thought you would die.
Themes Embodied
Polyphemus embodies the violation of hospitality, which the poem treats as the measure of a people, and with it Truth and lies: the name Nobody is the trick that defeats him, and the real name, shouted by Odysseus, is what gives him someone to curse. He also embodies the Revenge of a son who prays to a father, and Home, exile, and return, because the curse he hurls makes Odysseus's return long and costly.
Polyphemus is described by the lack of assemblies and laws: each one dictates the law to his children and wives, and his violence is a consequence of that lack. But the poem also gives him a moment of guest-language, when he calls Odysseus back offering gifts. One can read the offer as sincere and late, or as a trap, and Odysseus does not even discuss it: he answers with a wish for his death.