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The Enterprise is involved in extremely delicate negotiations with the government of the planet Gideon. Gideon apparently wants to become a member of the Federation, but its government is extremely restrictive on allowing any non-natives access to the planet. The planet is shielded from sensor probes, but is reportedly a paradise.
Ambassador Hodin, who is the native leading the negotiations, finally agrees to allow only Kirk to beam down to the planet. Spock conducts Kirk's transport personally. We see Kirk beam in to what appears to be the Enterprise's transporter room. He doesn't know what has happened, but when he searches the ship, he cannot find any other member of the crew. He also has a bruise on his arm and determines that he lost about 10 minutes of time during the transport.
He eventually encounters a young woman named Odona. Odona has no idea how she got on the ship - she was somewhere in a crushing crowd, trying to breathe, and then suddenly was on the ship. Kirk believes she is from Gideon, but she claims she doesn't know the name of her planet. She revels in the empty space on the ship, saying that her planet is so overpopulated that every bit of space is crowded with people.
The initial mystery of Kirk arriving on an empty Enterprise is nice, but fortunately the episode doesn't try to drag this mystery on to long. We see that the rest of the crew is still on the real ship, which turns the mystery Kirk is involved in into a mystery concerning what Odona is really doing on the "ship".
Meanwhile, Hodin reports that Kirk didn't arrive on Gideon. Spock requests permission to beam down to Gideon to search. He and Hodin exchange a lot of very careful and polite words, but ultimately Hodin refuses Spock's request. Hodin promises that he will conduct a search. Spock sends a message to Starfleet requesting permission to beam down without Hodin's permission. His request is denied.
Kirk spends his time questioning (and kissing) Odona. Events become more eerie when we start getting visuals of crowds of faces outside the "ship". We also see that Hodin and his assistant are monitoring what Kirk and Odona are doing. Kirk becomes more and more suspicious of Odona, but then she faints. Kirk carries her toward Sickbay, but is intercepted by Hodin and his security men.
Hodin's plan has succeeded: he allowed Kirk to beam down, and sent him to a replica of the Enterprise, in order to draw some of Kirk's blood and inject it into Odona. Kirk had a rare virus, Vegan choriomeningitis, which he barely survived; now Odona has the disease and will die without immediate treatment. Odona was placed on the replica ship with Kirk in an attempt to gain Kirk's affection so that he will be happy to stay on Gideon and continue supplying his blood.
Why has Hodin concocted this plan to kill his daughter? He reveals that Gideon is horribly overpopulated, causing complete misery among the people. In the past, the environment of the planet was so benign and germ-free that the people developed extremely long lifespans and an overwhelming respect for life. Consequently, death is rare, but the population keeps growing. Odona's death is supposed to show the people that their traditions can change, and people can die young. Anyone who volunteers will be injected with the virus from Kirk.
Not surprisingly, Kirk is aghast at this plan. He offers a number of reasonable alternatives: the Federation can help them, possibly through sterilization or birth control. Hodin shrugs off these ideas. Odona's disease is progressing rapidly, and she asks for Kirk. We find out that she was a volunteer for Hodin's plan and isn't afraid to die, but has fallen in love with Kirk.
On the real Enterprise, Spock has decided to violate orders and beam down to where Kirk did. He arrives on the replica ship, but since he knows something fishy is going on, and he has a tricorder, he sees through the ruse immediately. His tricorder leads him to Kirk, Odona, and Hodin. Kirk orders Spock to have the two of them and Odona beamed up to the Enterprise.
On the ship, McCoy cures Odona. Although she had been prepared to die, she's just as glad to live. Now she will be the source of the virus, since her blood carries it, just like Kirk's.
This episode has some interesting ideas, but also some serious flaws. Some aspects of the writing and directing were very effective. The visuals of the "faces in the window" were pretty spooky. The later visuals of the crowds of people shuffling around the replica's stage did a good job of conveying the situation on the planet.
Some of the dialog was also very nice. When Odona is describing the overpopulation on her planet, she says that any one of them would be willing to kill or die in order to be alone. Kirk then says, "Odona, who are you here to kill? Odona, are you here to die?" This is some nice foreshadowing of Odona's actual plan.
I also appreciated the diplomatic scenes between Spock and Hodin. I thought they did a good job of showing how careful negotiators must be in touchy situations. Spock is an excellent negotiator, since he cannot be offended or lose his temper. One thing that Spock said was silly, though. He told Sulu to begin scanning all 360 degrees of space to search for Kirk. 360 degrees is only a two-dimensional circle - what about looking "up" or "down" for Kirk? Guess they won't find him if he's there.
The main problem with the episode is that it was slooow. While the scenes between Kirk and Odona had some interesting aspects, it was mainly filler to kill time until Odona got sick. The negotiation scenes were also intellectually interesting, but they were slow and a little too staid.
The overpopulation plot was a mixed bag. I realize that the episode is trying to take the problem of overpopulation to an extreme, but in forcing the plot to be as it was, there were some logical problems.
If the planet really was so incredibly crowded, it's hard to believe that none of the natives are organizing any efforts to help. Surely there might be some grass-roots movements for birth control, or abstinence, or suicide. If it's so crowded, one would think that tempers would be hot, and there would be violence over small things that would begin killing people. We don't hear about anything like this. Alien societies in TOS are notoriously monolithic, and so all of the people revere life to the extreme.
Hodin's plan is that once Odona dies, young people of her generation will volunteer to also get the disease and die. Why couldn't the same type of plan be implemented with birth control? Or voluntary, painless suicide? I suppose a disease is more "natural", but not when you have to import it from off-world and then purposely inject it into people.
Hodin is also very short-sighted: why not join the Federation and then use the Federation's help to colonize other worlds? We can only assume that the natives' isolationist views don't let them think of this possibility. The extremity and idiocy of Hodin's plan are a big detraction from the episode. It feels like the writers really wanted to set up the disease plot, and so they forced the situation.
More practically, how could Hodin have implemented his plan? How did he get Kirk's medical records? How did he gain the information to construct a perfect replica of the Enterprise? How did he get the space to construct the replica, if the planet is so crowded?
Other issues: what kind of forcefield protects the planet from sensors, but allows transporter beams? The difference in coordinates between Kirk's beam-down and Hodin's assistant's beam-up was obvious - how did Hodin think he could get away with it? Hodin had to realize that eventually Kirk's abduction would be revealed - did he think the Federation would be happy about it?