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Never guess. Unless you have to. There's enough uncertainty in the universe as it is.—The Doctor
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 math nerds of Logopolis so they can fix the chameleon circuit. (nod along, OK?) Oddly enough, though, when they try to materialise, a police box shows up in the console room! A police box that is a TARDIS! With its own TARDIS in the control room! Doctor and Adric investigate the string of TARDISes, 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--wouldn't you know?--the 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.
The sonorous bong of the TARDIS's cloister bell (its Oh Crap signal) makes itself heard, so the Doctor and Adric retrace their steps through the nesting TARDISes back to the outermost one. 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 shrilly, true, but then again "Sorry I'm late, I accidentally stepped into a spacetimeship that looked exactly like a police box and ended up at either MIT or a Buddhist monastery or possibly both at once, 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.
Anyway. 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 here for no very good reason, 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 do... something mathematical... 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.
Like any good computer installation, Logopolis has a backup system: it's on Earth, at a giant radio telescope called the Pharos Project. So back to Earth we go, with the TARDIS landing perfectly for once. But it turns out the Master has double-crossed the Doctor (which has never happened before) and is holding the universe hostage to his demands by threatening to knock out the backup system. To show he's serious, he lets a fair chunk of the universe dissolve into chaos... including Nyssa's home world, Traken. While she watches. No class, Master.
And during all of this, there's been a mysterious figure in white following the Doctor around. No one seems to recognize him, save the Doctor... who keeps mentioning things like "the moment has been prepared for."
But nothing bad's going to come from that, of course!
Watch it here.
- Didn't See That One Coming: Oops. The Master didn't actually intend to destroy so much of the universe... including Nyssa's homeworld.
- Dropped Him Off A Radio Tower: What causes the Doctor's regeneration.
- Evil Hand: Nyssa's bracelet
- 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.
- 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.
- 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.
- The Master
- My Brain Is Big
- Planet of Hats: Their hat is genius, universe-maintaining mathematicians.
- 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.
- Wetware CPU
- 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!