A young policeman stops at one of the last remaining police boxes in England to telephone in to his station. But his connection is abruptly cut off as the police box seems to shimmer with an odd vworping sound. The door then opens, and something chuckles as it drags the terrified policeman inside...

Adric finds the Doctor pacing about the TARDIS cloisters, thinking dark thoughts about entropy, age and the inevitable slow disintegration of the TARDIS. The Doctor has decided not to return to Gallifrey, as there are bound to be recriminations over Romana's decision to remain in E-Space; instead, the time has finally come to repair the chameleon circuit, and that means travelling to Earth, measuring a police box and taking those measurements to the planet Logopolis. But as the Doctor and Adric set off they are interrupted by the resonant sound of the cloister bell -- a warning of imminent disaster, and Universal catastrophe.

Meanwhile, in London, young flight attendant Tegan Jovanka is setting off for her first day at work. Her aunt Vanessa has offered her a lift to Heathrow, but unfortunately Vanessa's sports car is poorly maintained, and Tegan is forced to pull off the road due to a flat tire. Tegan sets about repairing the flat herself while Vanessa waits hopefully for a passing motorist to offer assistance. Neither of them notice a translucent figure in white watching them from a distance, or the abandoned policeman's bicycle still leaning against the nearby police box...

The cloister bell stops ringing as the Doctor approaches the console room, and he dismisses it as a side-effect of entropy eating away at the TARDIS systems. On the way back, he passes by Romana's old room, and realizes with some surprise that he's going to miss her. Once back in the console room he demonstrates the chameleon circuit to Adric, by programming the exterior of the TARDIS to change shape into a pyramid -- although when he implements the change it instantly reverts to the default form of a police box. He's no longer satisfied with such a distinctive form -- especially since the Master was able to use a functioning chameleon circuit to hide from him on Traken.

The TARDIS materializes around the police box on the Barnet bypass, and the Doctor and Adric begin to measure the box in every aspect, ratio, dimension and particular. These measurements will be taken to Logopolis, where the natives have perfected the art of Block Transfer Computation -- a means of modelling and generating real space/time events and structures through pure mathematical calculation. The Logopolitans can generate a duplicate of the TARDIS exterior and overlay it upon the real thing, and the interference will shake the image free and enable the Doctor to change the mathematical model of the exterior once again.

But as the Doctor and Adric complete their measurements, the TARDIS suffers an instrument failure caused by a localised gravity bubble. The Doctor pops his head out of the TARDIS to investigate and is oddly disturbed by his sight of the translucent white figure in the distance. When he returns to the console room he finds that Adric has picked the lock of the police box -- and when he and Adric enter they find another TARDIS console room, with another police box inside. And when Adric picks the lock of that one they find inside yet another console room with yet another police box. The cloister bell begins to sound again, as the Doctor realizes that they're caught in a dimensional anomaly -- and if it's an infinite regression they will be trapped here forever...

Tegan finds that Vanessa's spare tire is punctured as well, and reluctantly sets off in search of a garage. But on the way, she spots the police box, and decides to call for assistance. Instead, the doors open inwards and she stumbles into the TARDIS console room just as the police box inside dematerializes. The doors shut behind her, trapping her inside, and as she tries to make some sense of the console the cloister bell begins to ring. Tegan ventures into the corridors, seeking the crew of this weird ship, but soon becomes hopelessly lost. Meanwhile, her aunt Vanessa also finds and enters the TARDIS -- but this time the chuckling figure that accosted the policeman is waiting inside...

The detective inspector forcibly invites the Doctor to come to the police station to answer a few questions, but the Doctor can't waste time on this and loudly suggests that a diversion is needed. Adric uses the abandoned bicycle to pretend he's been in an accident, and when the constables try to help him, the Doctor is able to escape and rush back into the TARDIS with Adric. As he tries to dematerialize, he finds the gravity bubble holding him back, and is forced to jettison Romana's room to provide thrust for take-off. The TARDIS leaves the real police box behind, and the police manage to open it only to find it empty.

The Doctor sends Adric out of the console room, ostensibly to answer the cloister bell, but as soon as Adric has gone the Doctor answers the call himself. When Adric returns the Doctor reveals that the communication was from Traken; Nyssa's father has disappeared, and the Doctor can guess what has happened to him. The telepathic link shared by Time Lords must have allowed the Master to guess the Doctor's intentions, and he came to Earth to lay in wait for him. Now the Master is hidden somewhere within the structure of the TARDIS, and the Doctor decides to literally flush him out by materializing underwater and opening the doors.

Tegan, hopelessly lost in the corridors of the TARDIS, stumbles into the cloister room and is shocked when another police box materializes in front of her. In the console room, the Doctor materializes and drops into the Thames, and Tegan is then thrown off her feet by the jolt. Furious and frightened, she is then terrified when the police box door creaks open and evil laughter echoes about the cloisters. She flees back into the TARDIS corridors, only to get turned around and end up back in the cloisters once more. Despairing, frustrated and increasingly angry, she makes a run for it again -- while behind her, the police box transforms into a potted tree.

The Doctor and Adric are surprised by the strength of the jolt as they land in the Thames, and even more surprised when no water rushes in through the open doors. This is explained when they emerge to find that they've landed on a barge by error. The Doctor notices the figure in white watching them from a bridge over the river; the figure beckons to him, and, unnerved, the Doctor orders Adric to stay put while they speak. Adric is convinced that the waiting figure is the Master, but the Doctor refuses to discuss what happened; he seems unsettled, and claims to have seen the future. He sets the co-ordinates for Logopolis, telling Adric that they will have to part company; a chain of circumstances is about to fragment the very law which holds the Universe together...

Tegan finally finds her way back to the console room, where she furiously demands to speak to the ship's pilot. The Doctor, already surprised by her arrival, is appalled when she mentions that her aunt is waiting for her outside, and doesn't have the heart to tell her what's happened. She accompanies him and Adric outside into the city of Logopolis, where endless rows of elderly grey-haired mathematicians sit in cubicles chanting the numbers of their calculations. The leader of Logopolis, the Monitor, leads the Doctor, Adric and Tegan to the Central Registry, which the Doctor finds to be a very familiar-looking computer room with a radio telescope antenna mounted outside. There, the Monitor converts the Doctor's measurements of the police box exterior into Block Transfer Computations; as Block Transfer cannot be run on a mere mechanical device, the Logopolitans must chant the calculations themselves.

But the "tree" from the TARDIS cloisters has materialized in the streets outside, transformed into a stone pillar and appeared inside one of the Logopolitan's cubicles. The Logopolitan inside, occupied entirely with his calculations, doesn't realize that anything is amiss until it's too late; soon another shrunken doll-like figure has joined the list of the Master's victims...

The Monitor is appalled; there has never been a fault in the Logopolitans' calculations before, and now their honour and the Doctor's life are both at stake. The Logopolitans carry the shrinking TARDIS back to the Central Registry and use sonic projectors to generate a stasis field around the TARDIS, stabilising it while the Monitor and Adric hunt for the error in the streets outside. They soon locate the error -- in a street where three Logopolitan cubicles are now occupied only by shrunken doll-like figures. Adric is surprised when the Monitor refers to this not as murder, but as sabotage -- and as the most dangerous crime in the Universe. The figure in white is standing at a nearby intersection, watching...

The Doctor is unable to correct the fault from inside the TARDIS; he must stand by helplessly waiting for word from the outside. When Adric and the Monitor arrive with their corrected calculations, Tegan holds the notes up to the TARDIS so the Doctor can see them on the scanner. Adric sets off into the streets to find the Master, followed by Nyssa, who wants to know what has happened to her father. Adric notices the white figure in the distance and sets off in pursuit, but Nyssa is brought up short by the sound of a familiar voice -- and sees the man she still believes to be her father. He's changed, not merely younger but colder in attitude, but he claims that she can help him to learn the great secret of Logopolis. He gives her a bracelet which clamps tightly shut about her wrist, telling her not to let anyone know that she has seen him here.

Tegan confronts the Monitor about the humourless Logopolitans she's seen, convinced that they're being exploited somehow, but he skirts around the issue and claims only that they're devoted to their work. The Doctor is able to restore the TARDIS to normal, but once he emerges he's forced to admit to Tegan that her aunt has been murdered. He sets off to find Adric and Nyssa, vowing to stop the Master if it's the last thing he ever does. Meanwhile, two Logopolitans remove the sonic projectors to return them to storage, but the Master is waiting outside. He murders them and connects a device of his own to the projectors, and the other Logopolitans are too busy with their calculations to pay him any attention as he sets up the projectors behind them. When he activates the projectors they generate a sound-cancelling wave, and the Registers fall silent.

Adric, having failed to find the figure in white, finds Nyssa trying to remove the too-tight bracelet. As he examines it there is a discharge of energy, and for a moment it appears as though Nyssa is reaching out for his throat -- but at that moment the Doctor arrives, interrupting them. The figure in white is standing in the distance again, and Adric is surprised to learn that it's the figure who brought Nyssa here from Traken. The Doctor seems almost frightened of him, as if his mere presence means the worst is bound to happen. And then it does; Logopolis itself falls silent, and as the hush spreads over the city the Doctor realizes that he's been wrong all along. The Master isn't after him... he's after Logopolis.

The Master enters the Central Registry, having brought silence to Logopolis so he can discuss its future with the Monitor. He refuses to restore the calculations until the Monitor reveals Logopolis' great secret and the purpose of the Pharos Project's replica. The Monitor begs the Master to shut down his machine but can't bring himself to reveal the great secret. The Doctor, Adric and Nyssa arrive, and Nyssa is appalled to realize that the Master has murdered her father and taken his body. Adric tries to shut down the projectors himself, but the Master uses the bracelet on Nyssa's arm to force her to throttle Adric. Tegan is forced to move the projectors back into place, and the Master allows Nyssa to release her friend. The Monitor and the Doctor try to convince the Master that he has unleashed a universal catastrophe, and he contemptuously shuts down the projectors to demonstrate that Logopolis is still functioning. But it isn't.

The Doctor and the Master follow the Monitor back to the Central Registry to collect the project notes of the advanced research division, who were calculating a series of data statements which would stabilise the CVEs of their own accord. Tegan follows them, determined not to let the Doctor out of her sight; Adric and Nyssa remain in the TARDIS, which the Watcher takes out of time and space altogether to sit out the catastrophe. Tegan arrives at the Central Registry as the Monitor prepares to realign the project aerial with the nearest CVE -- but before he can make the final alterations he unravels into nothingness. The Master, horrified, flees for his life, only to be trapped beneath a slab of rubble as the streets crumble about him. The Doctor and Tegan pull apart the computer to find that the final project is still stored within its bubble memory; and as this room is an exact duplicate of the same room in the Pharos Project, that means they can run the Logopolitan programme from Earth. They head out into the streets, rescue the Master from the rubble and order him to take them to the Pharos Project.

Adric and Nyssa, waiting out events in the TARDIS cloisters, see the Watcher beckoning to Adric. Adric speaks with the Watcher but afterwards is unable to describe the conversation specifically; all he knows is that it seemed that the Watcher knows what is going to happen next. He returns to the console room with Nyssa and tries to set the co-ordinates to materialize at the Pharos Project. As he does so, he and Nyssa see the entire Universe on the scanner, and the dark field of entropy slowly blotting it out; and Nyssa watches in horror as her homeworld, Traken, falls to the blight. The man who killed her stepmother and father is now responsible for the destruction of her entire world.

The Master's TARDIS materializes in the Pharos Project computer room, but the night-shift technician is listening to classical music on his Walkman and doesn't even notice. The Master prepares to shoot the technician but the Doctor flings him out of the way, accidentally knocking him unconscious as he does so. The Doctor and Master work through the night to programme the Logopolitan project into the computers, and finally succeed as dawn breaks and the security guards come out on patrol. Now they need only install the light speed overdrive from the Master's TARDIS in the antenna control room, so the data will reach the CVE on time -- but as the TARDIS won't work without the overdrive unit in place, they must cross the project grounds on foot. The Doctor's TARDIS has materialized in the grounds now, and as the Doctor heads for the antenna he sees the Watcher standing in the TARDIS doorway, waiting...

The Doctor, the Master and Tegan are spotted when the Doctor tries to stop the Master from shooting the security guards, but Adric and Nyssa arrive and distract the guards, and Tegan rushes over to help. The Doctor heads for the antenna control room, but the Master returns to the computer room and steals the unconscious technician's Walkman. By the time the Doctor reaches the control room, high atop the antenna dish, the Master is already there; the light speed overdrive he gave the Doctor was a fake to make the Doctor trust him. He hands the real light speed overdrive to the Doctor, but as the Doctor installs it and begins transmitting data to the CVE the Master surreptitiously records a message for the peoples of the Universe on his Walkman. The CVE stabilises -- and then the Master holds back the Doctor at gunpoint and broadcasts his blackmail message to the Universe. Unless the peoples of the Universe submit to his will, he will send an energy pulse down the transmitter and close the CVE forever.

The Doctor rushes out to disconnect the transmitter cable, overpowering the Master and knocking the TCE out of his hand to the ground far below. The Master returns to the control room and begins to move the antenna, causing the Doctor's catwalk to tilt alarmingly. Nevertheless, the Doctor crawls across to the cable and yanks it out of its socket -- losing his balance as he does so and falling from the catwalk. As he dangles from the cable the faces of his past enemies seem to come back to taunt him... and when he lunges for the antenna's support struts, he loses his grip, and falls. The Master, thwarted, returns to his TARDIS and dematerializes.