Logopolis was the seventh and final story of Season 18 of Doctor Who. It was Tom Baker’s last story as the Fourth Doctor and introduced Peter Davison in the role as the Fifth Doctor, at the very end of the story. It also introduced an interim version of the Doctor known as the Watcher, whose sudden presence foreshadowed the regeneration of the Doctor. What triggered the Watcher’s creation, however, was not fully explained.
Logopolis also marked the first appearance of Janet Fielding as new companion, Tegan Jovanka. After being introduced in the previous serial, Nyssa, played by Sarah Sutton, began her travels with the Doctor here. It additionally introduced a recurring element in the TARDIS’ Cloister Bell. This was also Anthony Ainley’s first full story as the Master.
A policeman is talking from the telephone of a police box, as a TARDIS materializes around it in disguise. Suddenly, the phone goes dead, a hand drags him inside and there is an evil chuckle. Meanwhile, the Fourth Doctor is pacing around the TARDIS Cloister Room, pondering decay and entropy. As he and Adric prepare to leave, the large bell in the center of the room begins to ring. This worries the Doctor. The sound of the Cloister Bell is a sign of impending universal catastrophe.
To divert himself, the Doctor decides to repair the TARDIS’ chameleon circuit, which has frozen it into the shape of a police box. To do this, he intends to materialize the TARDIS around a real police box, and then obtain its precise measurements in twenty-seven dimensions. With these measurements, he will have the inhabitants of the planet Logopolis produce a mathematical calculation — a Block Transfer Computation — to reset the circuit. However, the “police box” he materializes around is actually the TARDIS of the Master, who has survived their encounter on the planet Traken. When the Doctor materializes around the Master’s TARDIS, a recursive loop of TARDISes within TARDISes is formed.
Meanwhile, an airline stewardess, Tegan Jovanka, is being driven to the airport by her Aunt Vanessa. The car breaks down and Tegan decides to go to the “police box” for help, but finds herself lost in the TARDIS instead. The Doctor and Adric enter another police box in a duplicate TARDIS. The Doctor, telling Adric to wait behind, finds himself outside the box. He meets policemen, who find the shrunken, dead bodies of Tegan’s aunt and the earlier policeman. The Doctor realizes that the Master has escaped from the planet Traken and must be somewhere nearby.
The police think the Doctor has caused the incident, but Adric creates a distraction. This allows the Doctor to escape. In the distance, a mysterious, white-clad stranger watches the proceedings. Realizing that the shrunken bodies are the trademark of the Master, the Doctor decides to materialize the TARDIS underwater, to literally flush him out. The Doctor misses the River Thames, however, and lands on a boat instead. The mysterious stranger appears here too. He beckons to the Doctor, telling him to go to Logopolis.
As the TARDIS arrives at Logopolis, Tegan finds her way to the control room, annoyed. She asks where her aunt is. The Doctor, realizing that Tegan’s aunt was the dead woman in the car, evades the question. Once they exit the TARDIS, the Doctor asks the Logopolitan leader, the Monitor, for his help. The Logopolitans are able to model reality by pure mathematics and whatever they calculate can take physical form. Since block transfer computations cannot be calculated by machines or computers, the Logopolitans speak aloud a line of calculations and pass the results on.
Unknown to the group, the Master has arrived on Logopolis and killed several Logopolitans. This disrupts the calculations for the TARDIS. When the Logopolitans produce the requested computation, the Doctor tries it on the TARDIS. It shrinks to half its normal size and causes strange effects inside the ship.
The Logopolitans try to stabilize the TARDIS. They use sonic projectors to produce a stasis field while the Monitor and Adric attempt to uncover the fault. Meanwhile, Nyssa has been brought from Traken by the Watcher, the mysterious white figure the Doctor spoke with. She is searching for her father. The Monitor and Adric work through the city and discover the shrunken bodies of three Logopolitans. Fixing the error this has caused, they bring the new computation to the TARDIS. Tegan holds the notes up to the TARDIS so the Doctor can read them through the scanner and correct the fault. The Doctor emerges from the restored TARDIS and tells Tegan that her aunt is dead. Meanwhile, Nyssa finds the Master, whom she believes is her father as he is inhabiting Tremas’s body. The Master gives her a bracelet; it is actually a device which will allow him to control her actions.
The Master attaches a device to the sonic projectors and sets up a counterwave that brings silence to the Central Registry, preventing the Registers from making their calculations. He goes to the Registry’s control room (a replica of the Pharos Project on Earth, a radio telescope tasked to seek out signs of extraterrestrial life). He demands the Monitor tell him the true purpose of Logopolis. The Doctor arrives with Adric and Nyssa. Adric deactivates the Master’s device, only for the Master to have Nyssa attempt to throttle him. Tegan restores the device and the Master repeats his demand. The Monitor warns the Master that bringing Logopolis to a halt will cause universal disaster, but the Master replies that it is only a temporary effect. He attempts to demonstrate this assertion by deactivating the suppression device.
The silence persists. The calculations do not resume. They go outside, and find all the Logopolitans dead, crumbling to dust, and the city collapsing. The Master thinks this is a trick and tries to have Nyssa strangle the Monitor, but the control device ceases to function. He tries to increase the device’s power, but it falls apart as local decay increases. The Monitor explains the situation: the universe has long ago passed the point of heat death. To stave off final collapse, the Logopolitans have been modelling temporary Charged Vacuum Emboitments, like the one through which the TARDIS was previously transported into E-Space. The excess entropy generated by the universe had been passing through the CVEs to other universes. The Master’s interference has closed the CVEs and the universe is now dying at last. The Doctor realizes he has no choice. To save the universe, he has to work with the Master. He orders his companions into the TARDIS. When they argue about him working with the Master, the Doctor points out that he never chose to travel with any of them; Adric came aboard as a stowaway, Tegan’s curiosity brought her into the Doctor’s life and Nyssa came to him asking for help finding her father. With that, the three return to the TARDIS. The Master holds out his hand to the Doctor on their agreement to work together. “One last hope,” says the Doctor and they shake hands.
Adric, Nyssa and Tegan enter the TARDIS. The Doctor has the Watcher take it out of space/time. However, Tegan refuses to cooperate and follows the Doctor, Master and Monitor back to the Logopolis control room. The Monitor reveals that they had been completing a program to make the CVEs permanent. He prepares to use it on one of the surviving CVEs, but entropy takes hold of him. He disintegrates before their eyes. The Doctor dismantles the computer and realizes the program is stored in bubble memory that they can use with the real Pharos Project. The Doctor, Master and Tegan escape from Logopolis in the Master’s TARDIS.
Adric and Nyssa watch helplessly in the Doctor’s TARDIS as a portion of the universe is wiped out by encroaching entropy — including Traken. On Earth, the two Time Lords reconfigure the Logopolitan program and feed it into the Project’s computers, but the Master points out that the transmitter is pointed away from the last surviving CVE. After speaking with the Watcher, Adric brings the Doctor’s TARDIS to Earth as the Doctor and the Master run to realign the dish. The Doctor’s companions distract the guards and the two Time Lords go to the dish’s control room, hooking up a light speed overdrive from the Master’s TARDIS to ensure the signal gets to the CVE in time. On transmission of the program, the CVE begins stabilizing.
The Master’s co-operation with the Doctor has been a ploy, however. Holding the Doctor at gunpoint with his Tissue Compression Eliminator, he transmits a message to the peoples of the universe, saying that if they do not acknowledge his rule, he will send a signal to close the CVE and restart the collapse. The Doctor climbs onto the radio telescope’s gantry to disconnect the power cable and the Master tries to prevent him by tilting the dish. The Doctor disconnects the cable, but falls off the tilted gantry. As he hangs onto the disconnected cable, visions of old enemies mock him: the decaying Master, a Dalek, the Captain, the Cyber-Leader, Davros, a Sontaran, a Zygon and the Black Guardian. Losing his grip, the Doctor plunges to the ground. The Master enters his own TARDIS, de-materializing before the Pharos Project guards reach the control room.
The Doctor’s companions run to where he has fallen. Dying, the Doctor sees visions of the companions that have accompanied his current incarnation on his travels: Sarah Jane Smith, Harry Sullivan, Brigadier Alistair Gordon Lethbridge-Stewart, Leela, K9 and Romana’s first and second incarnations. Smiling, he looks up at his companions who has gathered at his side and says, “It’s the end… but the moment has been prepared for.” They turn to see the Watcher approach. Nyssa realizes that the Watcher was the Doctor all the time. As the companions look on, the Watcher merges with the dying Doctor, triggering his fourth regeneration. The Fifth Doctor lies where the Fourth Doctor once was, beaming with a delighted grin.