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For context, Ganymede is taken from various canon points. In order to have him take part in some memes, such as truth memes, I assume forced compulsion to a certain degree since he's very tight-lipped otherwise.
Though slender, Ganymede seems to be in perfect health. While in the garden, he hasn't aged a day and has the same youth he did upon his abduction.
PERSONALITY
It's important to note that we only see current Ganymede in his prison, and his interactions are either Apollo or orchestrated by Apollo.
After millennia of captivity, Ganymede isn't the regal prince one might imagine. When he lived in Troy, we saw someone innocent but carefree to the point of almost lacking competence. Over time, he's become easily annoyed and quick-tempered with no fear of pain or death. All of these traits are shown quickly with the first interaction we have with him in the form of Heinz. Heinz is happy for the chance to have his wish granted and optimistic Apollo will keep his word. Ganymede is none of these things. His first comment after meeting Heinz is rudely insulting his name, followed by refusing to go with him, and eventually ignoring him. The same attitude is apparent when he speaks with Apollo where much of the conversation is filled with Ganymede insulting the god. It's likely a more complex anger, however. Not only is he mad, but he's also bored and depressed. He's had everything taken from him, seen his brother die, and has been isolated for a very long time. In addition to having little-to-no practice socializing, it's difficult to be nice to someone when you can trust nothing.
One of the reasons Ganymede's considered so interesting is his resourcefulness. His power level is far too below Apollo's to work, but he does try, and it's never a stumping of his feet. A good example comes from an early encounter when Apollo is mocking him from a high pillar. Ganymede throws the dagger, but it doesn't reach him. So, instead of just yelling from below, he actually tries to follow another pillow closer to try again. He can hardly be called a genius, but he's also not someone who curls into a ball and gives up because something is hard.
Thanks to the trauma he suffers and being locked away for so long even after many, many escape attempts, Ganymede's become quite the cynic and is slow to trust. However, time hasn't left Ganymede a husk of negative traits. Despite failing time and time again to escape, he still holds more hope than he'd like to admit. Though he tries to hold onto his doubt, Heinz is able to convince him to try and escape. Later, he's given a choice to finally end it all by giving into madness. While he does give it serious consideration, Ganymede decides to see it to the end, whatever the end may be. But it appears in smaller ways. Even when he refuses to go with Heinz, he doesn't tell him to give up on his dream. He tells him to get the girl even if it kills him, just not to involve him.
Once, the people of his time worshipped gods and depended on them for many things. After seeing behind the celestial curtain, Ganymede's a lot more cynical while also having a more independent attitude. As mentioned before, he urges Heinz to get the girl of his dreams without a god's help. In his mind, anyone who believes and relies on the gods is an idiot just as he'd once been an idiot. Likewise, after learning that the world believes the rumors about a baby Ganymede being kidnapped by Zeus to service him, he considers the mass acceptance of such a ridiculous story idiotic. In his mind, people should think for themselves and not just accept a story, no matter if it came from a god or other people.
On that note, he's always admired his brother's ability to do well in battle and their seemingly fearless approach to combat. Ganymede had hopes he too could be so brave and capable one day.
It's difficult to judge very many traits based on his personality before being kidnapped as time and pain have warped him, but it's likely he's kept a few traits that aren't as likely to fade. Even as a prince, Ganymede was willing to work the fields, and he doesn't seem to shy away from working hard to get back at Apollo. Likewise, he was loyal and loving to his family, and even in the garden, he does initially plan to kill Apollo in revenge before escaping. Perhaps the biggest show of loyalty comes in his anger at the possibility of the world ending even though he wants his life in the garden to end; he doesn't think the people of the world deserve such a fate.
As stated, all of these traits happen while he's still in a place that makes him miserable surrounded by a god who killed his brother and enjoys mocking him. We do see him being a bit less angry upon initially meeting Zeus and Hades which implies it's Apollo and anything Apollo-adjacent that earns his immediate ire. While it's impossible to say he'd be social even to everyone else, there's a good chance his rudeness will be at a milder level.
APPEARANCE
Ganymede is a small individual with a headcanon height of 5'2 to match his slender frame. Based on the myth of a boy that was kidnapped by the gods due to his beauty, Ganymede is meant to be a striking figure with long, golden hair that reaches past his waist and large blue eyes. He's described in canon as 'shining' due to his bright coloring.
Since taken, he's worn the same ceremonial outfit he'd been wearing when abducted. You can see it here. While in Troy, his clothing seemed to be more modest, especially for a prince, but he also herds sheep so it's not a royal family who's afraid to do work and his style probably reflected that with nicer clothing only being worn for special occasions.
I'll include some canon reference images. Clicking on a picture will take you to a larger version.
Ganymede seemed to enjoy herding sheep, and I imagine he was happiest when he was outside.
Assaracus gave him a dagger that their father had given him as a reward for his achievements in battle. This was his last gift before Apollo descended, killed his brother, and kidnapped Ganymede. Reference photos for the dagger can be seen here and here.
A reference picture for the garden can be seen here with picture of the Parthenon here and here.
Olimpos is modeled after the myth of Ganymede, who was kidnapped and taken to live with the gods because of his beauty. The story isn't a 1-to-1 recreation, but the differences between Olimpos and the real story are mocked in the series as being a rumor.
Not entirely related to Ganymede, but the gods seem to test what is possible and isn't possible based on their ability to say the words as gods can't lie.
WARNINGS
Dark content. Click to expand
KIDNAPPING/CAPTIVITY/ISOLATION - While it's not a dark canon, Ganymede was kidnapped from his home, saw his brother die while trying to protect him, and is kept separated from everyone who isn't Apollo or the occasional person Apollo decides to use to tease him. This has been going on for hundreds or thousands of years.
SELF-HARM Ganymede is trapped for thousands of years. The stars never move, and there's no sense of the moon. Nothing changes. He's largely alone. He's become so bored that he has killed and hurt himself several times though he does know it will never stick. This isn't played for laugh and shows just how damaging isolation can be.
If you wish to opt-out of interactions with Ganymede, please comment here. If you want to leave a note, please feel free, but an explanation isn't required. For those interested in tagging with him but would prefer to avoid the topics discussed above, I can certainly work around them.