Where do you see instances of Ancient Greek culture within The Odyssey by Homer?
Probably the most outstanding thing that the Ancient Greeks had in common with The Odyssey (by Homer) is the presence of the belief in powerful gods who were responsible for things that happened in the world and to human beings. There was a story that explained the changing of the seasons, and as well as the passage of the sun through the sky during the day. The story of Prometheus, for example, explains how man was given fire.
There were also beliefs about the gods and what they were "responsible" for in the world of humans. Athena, for example is...
...the goddess of war, civilization, wisdom, strength, strategy, crafts, justice and skill...
A warrior would pray to Athena before entering into battle. Worship of the gods was a central part of the life of the Greek civilization.
In The Odyssey, Odysseus boasts and makes fun of Polyphemus, the Cyclops who killed his men.
In Book Nine, Odysseus yells across the water to Polyphemus:
O, Cyclops! Would you feast on my companions?
Puny, am I, in a Caveman's hands?
How do you like the beating that we gave you,
you damned cannibal? Eater of guests
under your roof! Zeus and the gods have paid you! (390-394)
Furious, the Cyclops calls on his father, Poseidon (a god who was equally as powerful as Zeus) to punish Odysseus. Homer explains that Odysseus' further delay is caused by Poseidon.
Odysseus' patron goddess is Athena (the daughter of Zeus), who ultimately intercedes on Odysseus' behalf that he might finally return to his wife and son. As Poseidon is not on Mount Olympus at the time, Zeus agrees.
In Book Twenty-One, Homer writes that Zeus shows his support of Odysseus when he returns home to a hoard of unwanted guests who want to take his home, his wife and all he owns. (Odysseus has been gone twenty years, so these interlopers feel it is their right to take what they want.)
As Odysseus (dressed as a beggar) confronts the men (who have broken sacred laws of hospitality—philoxenia—according to Greek custom), Zeus shows his approval of Odysseus:
Then Zeus thundered
overhead, one loud crack for a sign.
And Odysseus laughed within him that the son
of crooked-minded Cronus had flung that omen down. (207-210)
The thunder is credited in the story to Zeus, personifying him as a god who is giving aid to the human, Odysseus.
Whereas the Greek civilization believed that their lives and their world were affected—even controlled—by the gods, The Odyssey presents the same ideas. This epic poem attributes success, strength, temptation, and punishment to the gods—or their children...or minor gods and/or goddesses. In this way, the beliefts of Greek civilization are reflected in Homer's work.
[Outside the realm of literature, we also see a respect the Greeks had for warriors—in particular soldiers like Odysseus—who was gifted with his bow. Apollo was worshipped as the protector of archers, but in this instance, Zeus supports Odysseus.
Additionally, we see a respect for a guest, based on the Greek concept of philoxenia: kindness to strangers.
Both of these elements are also seen in The Odyssey.]
The Language of Literature. Evanston: McDougal-Littell Inc., 2006.