The O.G. Birdman
Who would have thought Birdman—you know, the caped crusader of Hollywood fame—would be referencing ancient Greek mythology? But he does:
BIRDMAN: We have to end it on our own terms, with a grand gesture, flames, sacrifice, Icarus.
This is what Birdman says to Riggan as he ascends to the top of the building and prepares to jump off. Birdman is all about spectacle and wants Riggan's life to end with a bang, not a whimper. If Riggan slowly becomes washed up and irrelevant nobody will remember him. He will be that one dude who played the bird guy in those action flicks, and Birdman can't let that happen.
No, he needs to jump off a New York City building in broad daylight with onlookers shouting his name as he falls to his fiery demise, sacrificing his life to uphold his reputation and his celebrity.
Yeah. Birdman's a jerk.
But let's back up a sec. What are we talking about when we talk about Birdman? Well, in Greek mythology, Icarus is the son of Daedalus, the craftsman who built the labyrinth for King Minos to imprison his stepson the Minotaur. But when Daedalus helps the princess Ariadne make her way through the labyrinth, Daedalus himself, along with Icarus, are locked away in the labyrinth by Minos.
Daedalus, being a clever craftsman, builds two sets of wings made of wax and feathers so that he and his son can fly away and escape. Before taking off he warns Icarus of flying too high and having the sun melt the wax. Basically, he's metaphorically warning against becoming too big for your britches, and Icarus unfortunately doesn't heed his warning. He flies too high, melting his wings, and then falls into the sea.
Icarus' rise and fall parallels Riggan's pride. It's Riggan's inability to let go of the respect and fame he once had that's causing his career and his life as a whole to come cascading down upon him in a fiery mess.