They form a nuclear family in the abandoned house. Jim plays the father figure. Judy the mother. Plato the son.
In this scenario, we have an Oedipus Complex. Jim and Judy would have had physical relations, had it not been for Plato. Plato is jealous of Jim (or is it Judy? Sal Mineo was gay, and his character seems that way too)--let's just say he's jealous of one of the parents--for taking away the other to another part of the house, thereby leaving him alone. So, he lashes out against one of the parents so that the other will give him affection.
Also, Plato works entirely from his id--his inner child. He curls up in the fetal position, cries, self abuses, pouts. He seems to have abandoned his superego altogether.
Jim and Judy are battling between their ids and superegos. That's their coming-of-age struggle--to cast off their ids and operate as adults would.
Look at the pool scene:
JIM (yelling) Quick! Fill the pool! JIM falls in. PLATO rushes to him. JIM Let's see how long we can stay under. PLATO Man, you're schizoid! JIM (in another outburst of laughing) I'm what? What? JUDY You can't talk underwater! JIM (gargling) I bet you hear everything I say! PLATO (gargling) Isn't he schizoid? JIM (gargling) Hey! How 'bout that! They laugh again. JIM swings up the ladder and goes to JUDY. PLATO follows. PLATO Haven't you noticed your personality splitting? JIM Not lately. They all sit on the blanket. JIM How do you know so much about this junk, Plato? PLATO I had to go to a head-shrinker. I only went twice though. My mother said it cost too much, so she went to Hawaii instead.
Concave objects, like swimming pools, represent the female, the bosom, breasts, womb. Notice the pool is empty--no water (no life, no amniotic fluid, no nurturing). The mother is empty, barren. That's the source of all of Plato's problems: "My mother said it cost too much..."