The characteristics of Gilgamesh and Odysseus that were rejected by their cultures were boastfulness and overconfidence.
Gilgamesh was a demigod and set out in search of immortality. On this journey, he encountered many dangers which he faced with courage, strength, and bravery. All this highlighted a very respectable masculinity in the society he was a part of. However, Gilgamesh proved to be arrogant at times, which caused him to make terrible mistakes that caused him to suffer very serious losses.
Odysseus showed all the characteristics expected and valued in Greek men. He was smart, strategic, strong, brave, and had a lot of bravery. However, he had strong overconfidence and this was not valued in society, for this reason, he is punished, as he becomes responsible for the death of most of the crew that traveled with him.
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