The planet where a small group of long-lived humanoids live by the ideals of ancient Greece, their powers derived from a mineral.
Platonius is the planet where the Platonians settled after their home star, Sahndara, went nova. They had once visited Earth during the time of ancient Greece and adopted its culture, naming themselves after Plato.
In Plato's Stepchildren, the Enterprise answers a call for medical help when Parmen is injured. The Platonians' psychokinetic powers come from kironide in the planet's food. When Kirk and Spock gain the same powers, the Platonians lose their hold over the crew.