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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.
Please note that anything marked with a ** is headcanon. I will use this sparingly and only when canon lacks details. Weaknesses are combined with Abilities for those looking to get a leg up. Also note that there isn't a lot of information about his backstory so I'll be drawing from the real mythos to fill in some blanks.
World: A notable difference from our world is that the number of ancient gods seems to be much smaller. The first generation is Zeus, king of gods and ruler of the skies. Poseidon is the middle god, ruler of the sea. Hades is the oldest and rules the underworld. Artemis and Apollo seem to be children of Zeus.
The following will also have mentions of self-harm.
Ganymede was born the youngest prince of Troy and was known far and wide for his beauty. Troy was a city thought to be loved by Apollo, god of the sun, and blessed by Zeus, king of the gods. Ganymede felt loved and blessed as well and enjoyed spending his days herding the sheep when he could. It was on the day his coming-of-age ceremony was being held that things changed. We see his mother helping him with the formal attire before Assaracus comes to get him and delivers news that their oldest brother would soon be getting married. He then gave Ganymede a dagger that had been gifted to him by their father for his achievements in combat, but he felt like Ganymede should have it instead. His beauty had been getting a lot of attention from the neighboring countries, and he claimed it's to help him fight off Zeus in case he catches the gods' attention, too. Based on the mythology in which others tried to kidnap Ganymede before Zeus, it's likely this was their way to prepare him should an enemy nation try to take him while still trying to protect his more innocent nature. Neither king nor prince would really think a simple dagger could protect against a god. However, Assaracus was more right than he knew as Apollo's voice breaks up their conversation. As he descended, Ganymede's brother attempted to save him but was killed instead. Unable to move, the young prince was carried into the heavens as Apollo told him to burn the memory into his mind since scorn can last for eternity.
It wasn't an obviously dark and evil fate that awaited him. Ganymede awoke not to the stereotypical bars, but to the sight of beautiful flowers. This was to be his prison, surrounded by beautiful things forever, himself yet another decoration, never to age, not to grow old or wrinkle. Apollo claims to have surrounded him with nothing but the most beautiful things, including pure-white flowers and twinkling stars that never go away, casting an endless night. His new home would be the Parthenon, Athena's old temple that's now housed inside of the garden. However, it's obviously seen better days.
Ganymede didn't originally believe this man to be Apollo for the people of his land thought highly of the god, and there's little reason to trust a man that killed your brother. When he argues that Apollo doesn't fit the right image at all, Apollo mocks him and even materializes a bow to taunt a common assumption: that he'd be a hunting god. Apollo even stabs him, which does indeed hurt, to show him he won't die. Later, he continued to fight him, both verbally and physically, something Apollo seemed to find charming. Ganymede tried many things to attack Apollo, including throwing his dagger and trying to roll large ruins closer together to reach him. However, there was little a human could do to a god, no matter how hard Ganymede tried. We see Ganymede trying to run from the Parthenon, only to end up right back at the building even if it seemed like he'd been surrounded by nothing but flowers only moments before. Eventually, he gets so bored and numb that he begins to intentionally hurt himself since pain is the one thing that does change, even if it fades.
And so, his life had changed forever. Days turned into years, which became decades, centuries, and finally, thousands of years had passed.
Eventually, Apollo answers a human's prayer; Heinz seems to have chosen the right words as he agrees to do something in return for having his wish granted. All he must do is take Ganymede to the cliffs at the end of the garden, and they can both go home. Sadly, the prince already knows that this is a ploy, another way for Apollo to pass the time. This causes quite the clash of personalities as Ganymede is annoyed at Heinz both for relying on Apollo and being used in his game. He's initially very rude and refuses to budge. Heinz, desperate and assuming that Ganymede is a pretty boy who can get anything he wants, is also annoyed but says he'll go to the end of the world and bring back proof. Without realizing Ganymede's situation, Heinz accuses him of being blessed with everything and claim he's not willing to do anything but look down at people, not realizing that Ganymede's tried for a very long time to find a way out.
Heinz's is determined. He walks away, promising to come back with proof. To fuel the illusion, Apollo allows Heinz to find a cliff and bring back dirt to prove a point to Ganymede. Knowing Apollo's tricks, he doesn't believe it, but Heinz is insistent and moves to drag Ganymede to the cliff. Still refusing to play his game, Ganymede cuts off his arm, which erupts into an explosion of flowers before growing back. Though this understandably shocks Heinz, he doesn't give up and eventually gets Ganymede to agree to come with him.
In truth, it was Apollo's greatest trick of all.
Heinz, it seems, is looking to excavate ruins, though the people of his world believe it's a silly dream. As they run toward the 'end', Ganymede's hopes finally revived, Heinz begins to describe the city he was excavating. It's between two rivers, one which flows cold from the mountains and the other is warm as it flows from the ocean. The sun rises from the mountain called Mount Ida at the place where the two rivers join. It was that ocean that the enemy used to cross and launch a coastal attack, killing the entire royal family.
He describes Troy.
The entire time, Ganymede has had hope that his home still stood. It was such a large, prosperous city that he had no doubt of its fate. While acknowledging that it might go under a different name, he'd been sure it was still around. To hear, so casually, about the fate of its people right when he started to have hope again enrages Ganymede and he sends Heinz away. Just as Apollo knew all along, there was no real way to get Ganymede out, even if he'd found the cliff. Heinz's dream was crushed.
Ganymede is left behind, now aware that his city fell to the enemies long ago.
This continues with Ganymede growing increasingly drained. Apollo visits to taunt and tease, but his purpose has yet to become clear. Finally, long after Troy falls, Zeus descends to the garden. His presence is so mighty that Ganymede's body shakes and consciousness begins to leave him before Apollo urges Zeus away. He does visit many times after, and Ganymede's body slowly starts to adjust. We learn that the classical story is involved within the manga. People tell stories on Earth, stories that are identical to those in our world: Zeus took Ganymede for his lover, and it happened when he was just a child. Ganymede finds the stories ridiculous, especially when he learns the rumors had him as a baby. A baby of unsurpassed beauty.
After many visits, as a reward for serving his purpose, he allows Ganymede to meet Hades. To him, Hades looks like a normal woman but the god looks different to everyone. It's Hades who reveals that Ganymede was actually a lure to have Zeus come to the garden. Their hope was that by luring Zeus to the garden by Ganymede's presence, they can make the uncaring god care. This in turn upsets the perfect balance needed to maintain the world, and the world could possibly end. They're bored enough to find this interesting. Ganymede is sane enough to that this idea.
The god also reveals that Ganymede can be freed of his cursed life by going mad and separating himself from his mind, essentially going insane. Telling him to step off the cliff when he's ready, the god disappears. However, he finds himself unable to do it. In the end, he decides to stay with Apollo and we see that eventually, Ganymede can even talk to Zeus. Ganymede's the first to see Zeus smile. His relationship with Apollo seems to be a bit more positive, too.
It's important to note that we only see current Ganymede in his prison, and his interactions are either Apollo or orchestrated by Apollo. 
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.
