The one with the cardboard cutout Master. And Four had a big fall.
The Doctor and Adric make a pit stop in 1981 Britain, so that the Doctor can materialize right on top of a real police box and make hyper-precise multi-dimensional measurements of it, which he needs to take to the mathematics nerds of Logopolis so they can fix the chameleon circuit. (Just nod along, OK?) The Master, anticipating this, materializes his spare TARDIS (which he nicked from Goth that one time) around that exact same police box moments before. Which means that the Doctor's TARDIS now contains the Master's TARDIS containing a real police box. (Still following us?) Time and space get a bit wonky, so the Doctor and Adric investigate the string of TARDISes in the console room — all identical except for being progressively less well-lit.
Meanwhile, a plucky young Aussie named Tegan Jovanka is off to her first day of work as a flight attendant, when The Alleged Car breaks down on the way to Heathrow. Good thing there's a handy police box nearby, so she can call for help. But she hadn't planned on getting lost inside. She completely fails to be impressed by the fact that it's obviously an alien spaceship, and instead decides to go yell at the captain of the thing. Meanwhile, her aunt, who was driving her to the airport, has been murdered by the Master for no real reason.
The sonorous bong of the TARDIS's cloister bell (its Oh, Crap! signal) makes itself heard, so the Doctor and Adric eventually find their way through the nesting TARDISes back to the outside world. The Doctor suspects that the multiple TARDISes thing is yet another devious trick of the Master, and dematerializes at once, bound for Logopolis. Tegan, by now extremely confused, finds her way back to the control room and demands to know what the hell is going on and to be put back in 1981 right now, please — rather angrily, true, but then again "Sorry I'm late, I accidentally stepped into a spacetimeship that looked exactly like a police box and ended up traveling the universe in the company of an alien with an enormous scarf and a swotty maths geek from another dimension" is, as excuses go, pretty lousy. Especially for one's first day on the job.
And while all that is happening, a white figure quietly watches. The Doctor goes to have a word with it eventually, and comes back in an extremely gloomy mood, telling Adric to prepare for the worst.
The TARDIS soon materializes on the dusty, cave-pocked planet of Logopolis, home of the maths monks. It is, we learn, an enormous analogue of a computer. A task is broken down into a series of calculations; each monk performs a single calculation and passes the result on to the next monk over. Operations are overseen by the Monitor (get it?) in a sort of large workroom. After reuniting with Nyssa, who is deposited on the planet by the mysterious white Watcher, the TARDIS crew finds the Monitor. He explains that the universe is actually well past its sell-by date, and is kept from flying apart into chaos only by the power of their mighty minds, as they use their "block transfer computations" to keep the universe together.
Of course, it's time for the Master to show up and start mucking about with things. He starts small, with such pranks as shrinking the TARDIS (with the Doctor inside) and throwing a spanner into the endless work of Logopolis. Unfortunately, Logopolis can't be restarted once it's stopped, resulting in a rather large problem for those who still want to, you know, exist. The Master hadn't really counted on accidentally deleting much of the universe, and while Tegan and Nyssa are both horrified at even having to be near him (Nyssa, of course, especially because he nicked her father's body), the Doctor reluctantly agrees to try and help the Master fix the spreading entropy.
The Doctor tells Adric, Nyssa, and Tegan to enter the TARDIS, which he has arranged for the Watcher to pilot out of time and space so they will be safe. Tegan, however, still thinks the Doctor is the best bet for getting home and decides to disobey his order and follows him and the Master as they go to speak to the Monitor. The Monitor reveals to the two that there is a backup plan that might be able to save everything. The people had been completing a program that could automatize its operation, but before the Monitor can boot up the program, him and the computer equipment is swallowed by the entropy field. For a moment all seems lost, but the Doctor, however, realizes that they still have a chance; Logopolis' computer equipment was an exact copy of the Pharos Project, a giant radio telescope on Earth. The Doctor, the Master, and Tegan then escape from the disintegrating Logopolis in the Master's TARDIS, setting a course for Earth.
Adric and Nyssa, meanwhile, can only watch helplessly, as the encroaching entropy wipes out a potion of the universe. Much to Nyssa's despair it includes Traken.
Landing inside Pharos Project, the Master and the Doctor eventually manage in cooperation to install the backup system and restore balance to the universe. But true to form, the Master decides to double-cross the Doctor; he uses the radio telescope broadcast to a message to all intelligent life in the universe: they must submit to him or else he will knock out the backup system and unleash the entropy field upon everything again.
After a chat with the Watcher inside the deepest parts of the TARDIS, Adric is able to follow the Doctor, and the ducklings cause enough of a distraction to allow the Doctor to muck up the Master's plans. Fortunately, this leaves the larger part of the universe intact. Unfortunately, the Master and the Doctor break out into a fight on top of the Pharos Tower. The Doctor sees visions of his worst enemies attempting to kill him. Clashing on the gantry, the two Time Lords wrestle to the floor. The Master gets up first and bolts to the controls of the telescope. He tilts the radar dish until the gantry completely flips over. Grabbing a cable and desperately trying to hang on to it, the Doctor sees several visions of his various enemies taunting him. Gradually, his strength begins to fail him, and finally his grasp slips, causing him to fall. The cowardly Master runs away sniggering in his TARDIS after this cheap act of murder.
Dying, the Doctor has several soothing flashes back to his previous companions in that incarnation. The next thing he knows, he's surrounded by the trio of current companions who are quite worried about him, snapping him out of the daze. The Fourth Doctor simply gives them a serene glance and tells the group with an assuring smile, "It's the end... but the moment has been prepared for."
The Watcher approaches and merges into his dead body, and turns out to have been a time/space-transcending echo of his next incarnation all along. The Doctor regenerates into a young blonde gentleman, who greets everyone with a smile. The Fifth Doctor starts his first day almost drowning in the sheer amount of coat and scarf he's wearing, not to mention chased by some very angry security guards.
- A Million Is a Statistic: The story involves the destruction of one quarter of the entire universe; by even the most conservative estimates that's a single-episode death toll expressible only in standard notation. And not only does nobody seem to care (including one character whose home intergalactic supercluster was destroyed note ), the villain responsible gets the full Draco in Leather Pants treatment despite being responsible for one of the most horrifying massacres in the series. (Though in all fairness, it was an accident this time.)
- Artistic License – Physics: Though the description of how the heat death of the universe would work is basically correct, the way it's depicted bears no resemblance whatsoever to the actual theory, and in fact is more similar to the "Big Rip" theory.
- According to the expanded verse story (The Invasion of E-Space), CVE is (in that verse) the rarest of space-time events and almost undetectable, entities containing layers of highly charged quantum particles suspended in pockets of super-vacuums. They occupied the space between realities (possibly the void) and were so violent in their physics that when they formed they punched a hole between universes, pushing space aside to insinuate themselves into reality, releasing massive waves of energy in the process. (And since heat death means no more energy, the way they do so brings forth new energies to both universes.)
- "Blackmail" Is Such an Ugly Word: As the Master announces his threat to destroy the universe unless its people submits to him:The Doctor (outraged) Blackmail!
The Master: No, Doctor, I'm merely reporting the state of affairs.
- It's revealed that the CVE the TARDIS fell through earlier in the season was actually one of many created by Logopolis to let the entropy drain away and prevent the universe from collapsing.
- The Watcher's role in events may be a Call-Back to Cho Je from "Planet of the Spiders", thus linking the Fourth Doctor's origin-story to his finale.
- The Doctor reaffirms that he hates farewells.
- The Doctor and the Master confront each other on a radio telescope.
- Climbing Climax: The Fourth Doctor is defeated while trying to overpower the Master inside a radio telescope.
- Continuity Cavalcade: Before the Fourth Doctor regenerates, he sees images of his enemies and former companions, all shown by clips from previous stories - the Master, a Dalek, the Cyber-Leader, the Captain, Davros, a Sontaran, a Zygon, the Black Guardian, Sarah Jane Smith, Harry Sullivan, the Brigadier, Leela, K9 and Romana.
- Continuity Nod:
- The school uniform which Romana wore in "City of Death" can be seen in her room.
- The Doctor and Adric discuss Romana's recent departure.
- The Doctor mentions Totter's Yard.
- The Doctor and the Master's TARDISes become intertwined.
- Cosmic Keystone: The titular planet is itself a universal keystone. The people are bringing energy into the universe as a way to fight off entropy. When the Master decides to stop activity there for a little while (admittedly, not really understanding what they were doing) large chunks of reality get destroyed before the problem can be fixed. Even then, there's no Reset Button.
- Creepy Good: The Watcher. He looks unnatural, but he guides the Doctor to where he needs to go and merges back into him during his regeneration.
- Death by Origin Story: Tegan's Aunt Vanessa is murdered by the Master, plus Nyssa's home planet is wiped out.
- Didn't See That Coming: Oops. The Master didn't actually intend to destroy so much of the universe... including Nyssa's homeworld.
- Don't Fear the Reaper: A variant. The Watcher is a sort of representation of the Doctor's upcoming regeneration. He breaths mystery and uneasiness, from his look to his theme music. But in the end, he is helpful, telling the Doctor where to go and merging with him at the end.
- Dropped Him Off A Radio Tower: What causes the Doctor's regeneration.
- Early Installment Weirdness: Creator/Peter Davison appears with his natural brown hair, which was quickly changed to blonde to better contrast his appearance with Tom Baker's. Luckily, his role here is so brief that it's barely noticeable.
- End of an Era: Tom Baker's seven year run comes to an end.
- Enemy Mine: The Doctor and the Master must work together to save all of existence.
- Establishing Character Moment: When Tegan gets lost inside the TARDIS, instead of being astonished by it, she goes and looks for someone to yell at.
- Eyes Always Shut: The Watcher has some rather weird statue-like features.
- Evil Hand: Nyssa's bracelet.
- Face Death with Dignity: The Fourth Doctor is fully prepared for his death and accepts it with a smile.
- Famous Last Words: "It's the end... but the moment has been prepared for..."
- Femme Fatalons: The Watcher sports pointy claws for some reason.
- Finale Credits: Since Part 4 is Tom Baker's swan song, his face is visible during the Closing Credits for only the briefest of moments.
- Future Me Scares Me: The Fourth Doctor is visibly disturbed by the Watcher's presence (which means impending regeneration) hence the quote.
- Good with Numbers: Logopolis is this trope in city form.
- Go Out with a Smile: We get one last glimpse of Tom Baker's big toothy smile before the regeneration.
- Grand Finale: For Tom Baker's era as The Doctor.
- Floating Advice Reminder: Happens twice in Part 4. The first time, while hanging on the Pharos Project's power cable, the Doctor is taunted by visions of the (decayed) Master, a Dalek, the Pirate Captain, the Cyberleader, Davros, a Sontaran, a Zygon, and the Black Guardian. Then after falling, the Doctor sees visions of his past companions from Sarah Jane to Romana II.
- Happy Ending Override: In the previous story, the Doctor successfully saved the planet Traken. In this story, it is destroyed by the entropy field.
- Heroic Sacrifice: The Doctor. Actually a realistic one, as his death isn't a direct cause of saving the day, but because he pulls off an incredibly risky stunt that he doesn't recover from.
- Kill 'em All: Every single member of the eponymous planet and the Doctor die (he gets better, of course). Plus, approximately a quarter of the universe gets destroyed. By accident. Special points for this explicitly including all of the surviving characters from the previous story, except Nyssa. The only survivors are the companions (Nyssa's entire family dies in this story and Tegan's only on-screen family member gets murdered) and the Master. And, well, the Fifth Doctor is created at the end of this story.
- Leitmotif: Ominous theme always plays whenever the Watcher appears
- Man in White: Even the Watcher's face is white.
- The Master
- Mike Nelson, Destroyer of Worlds: The Master inadvertently manages to unleash great destruction upon about a quarter of the Universe, simply because he didn't fully understand the danger of interfering with Logopolis' operations. It is without a doubt the greatest atrocity he has ever committed on the show, if not the single greatest atrocity ever depicted on the show, and he basically did it by pure accident.
- My Brain Is Big: The inhabitants of Logopolis.
- Mysterious Watcher: The Watcher.
- Nightmare Face: His face is somehow unformed and rather clay-looking.
- No Name Given: The Watcher is just "the friend of the Doctor". "The Watcher" is a nickname given by Adric.
- Offscreen Teleportation: How exactly the Watcher is following the Doctor and transported Nyssa from Traken to Logopolis is never revealed.
- Off-the-Shelf FX: The TARDIS Chameleon Circuit control panel is obviously an old carpet sweeper turned upside-down.
- Oh, Crap!: The Master, when he realises the Doctor is going to disconnect the cable at the radio telescope.
- Planet of Hats: Their hat is genius, universe-maintaining mathematicians.
- Reality Warping: The Logopolitans are seemingly capable of arbitrary reality warping by meditating on the equations that physically describe the universe, or objects within it.
- Shoot the Shaggy Dog: A significant proportion of the entire universe is destroyed by entropy. Although insignificant on the scale of such an unbelievably cataclysmic event, the region destroyed includes the Traken Union, thereby almost immediately rendering all the events, people and struggles in The Keeper of Traken (the previous connected story) dramatically null, void and pointless. (The fact that this is only obvious when you stop and consider it suggests that this was a side-effect rather than dramatic intent on the part of the writers).
- Shown Their Work: Writer Christopher H. Bidmead seems to have done more reading up on computers than on entropy. The monks of Logopolis work and communicate in hexadecimal (base 16), just like real computers. At one point, the Doctor mentions "bubble memory"—which is real, and was the subject of a lot of research in 1981.
- Spanner in the Works: The Master doesn't actually plan on wiping out part of the universe or destroying Logopolis. See also: Gone Horribly Right.
- The Stars Are Going Out: the entropy spreads and consumes part of the universe.
- Stupid Evil: The Master not only accidentally kills octillions of people, he draws attention to himself by randomly killing a policeman and Tegan's aunt.
- The Watcher: Played straight with The Watcher, who watched the Doctor and his companions throughout the serial, and was revealed to be a manifestation of the Fourth Doctor, which Foreshadows his regeneration into the Fifth Doctor.
- Theme Song Reveal: The Watcher's ominous theme during his appearances. This is before the audience is let in on what he really is (although the Doctor obviously knows but isn't telling). However, in the final scene of the story, when the Watcher's true nature is revealed to everyone, the full course of the piece plays out with the last three notes being the first three notes of the Doctor Who theme song.
- Unwitting Pawn: The Master coerces the Doctor into an alliance. If the whole universe disintegrates, there will never be a galaxy to rule, right? Not so fast: The Master plans to bargain the healing technobabble for the unquestioned allegiance of the universe's inhabitants.Doctor: Blackmail.
Master: No, Doctor, I'm merely reporting the state of affairs.
- The Voiceless: Other characters refer to the Watcher speaking but even when he's having a dialogue scene with someone it's always shot from a distance and inaudible.
- Wetware CPU
- Wham Episode: Tom Baker was the most popular Doctor the classic series ever had, so his departure after seven seasons into the (then) youngest Doctor actor ever was a Wham.
- Xanatos Speed Chess: The Master's magnificent comeback from accidentally destroying about a quarter of the universe to holding the rest to ransom
- You're Insane!: Before their final confrontation, as The Master reveals his endgame, the Doctor utters with dread:Doctor: You're mad...you're utterly mad!