The film starts as David Clemens (Keir Dullea) is brought to a residential treatment center by his apparently caring mother. He becomes very upset when one of the inmates brushes his hand, as he believes touches can kill him. Cold and distant, he mainly concentrates on his studies, especially that of clocks, which he appears to be obsessed with. We later learn that he has a recurring dream in which he murders people by means of a giant clock.
He meets Lisa Brandt (Janet Margolin), a girl who has two personalities: one of them, Lisa, can only speak in rhymes, while the other, Muriel, cannot speak, but only write. David befriends her by talking to her in rhymes. Following an argument with his mother when she comes to visit him, his parents decide he should leave the place. After staying at their house for a short time, David runs away and goes back to the residential treatment center, where he is allowed to stay. He has a small argument with Lisa, and she takes the train to the city, unnoticed by anyone.
In the final scene, David, who realizes she would go back to a museum in which she had embraced a statue before, finds her. Lisa appears to be cured and doesn't need to rhyme anymore, and David allows her to hold his hand on the way back.