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Character error
When Spock sits next to Alexander and begins laughing, actor Michael Dunn breaks character and begins laughing with him. (You can see he's trying unsuccessfully to restrain himself.)
Parmen gives a large shield to Kirk. During the ensuing conversation, shots including the entire landing party show Kirk holding the shield in front of his body, covering his insignia. In the close-ups of Kirk, his insignia is present and there's no sign that he is holding a shield.
McCoy goes to knock out Parmen with a hypo when Parmen throws furniture around. McCoy readies the hypo near his shoulder. When there's a close-up of Parmen, the hypo and McCoy vanish. McCoy reappears when the wide angle returns.
Chess came from a game called chaturanga, which originated in India in the 7th Century, CE. A group that visited Earth at the time of Plato could not know about it.
Philana says her people stop aging, yet Liam Sullivan (Parmen) was 45 years of age, and he clearly looks even older than that.
Leonard Nimoy's double is just visible as Mr. Spock loudly dances "flamenco" dangerously beside Captain Kirk's face. When the dance finishes, it is obvious that the dancer's boots have higher heels than the ones Spock is wearing.
Spock's ears are quite obviously fake prosthetic ears when he and Kirk wear the laurel wreaths on their heads towards the end of this episode. His left ear in particular has been bent to prevent the wreath slipping down any further and because of this it looks very abnormal and fake.
Parmen refers to the "pièce de resistance," a French term which did not exist when the Platonians visited Earth.
The Platonians repeatedly mention "Utopia". It is impossible for them to know the term, as it was coined by Thomas More in 1518 and they left Greece centuries before he was born.
If the Platonians left Earth at the time of Plato, they could not know about "Tweedledee" and "Tweedledum".
In his Captain's Log, Kirk states that the Platonians' "planet went nova." Planets don't go nova, only stars.
During the performance by Uhura, Nurse Chapel, Kirk and Spock, one of the Platonians refers to "Cupid". Cupid is a Roman deity; in Greek mythology the name of the deity is "Eros".
When Spock sits next to Alexander and begins laughing, actor Michael Dunn breaks character and begins laughing with him. (You can see he's trying unsuccessfully to restrain himself.)