There’s this wall of pure Azbantium. It’s 400 times harder than diamond and 20 feet thick. Once every few days*, the Doctor comes and punches it a few times. And when the Doctor punches his way all the way through the 20-foot-thick Azbantium wall…he’ll escape to Gallifrey.
According to Doctor Who episodes “Heaven Sent” and “Hell Bent”, the Doctor spends four and a half billion years punching his way through that wall in one of the most epic scenes in Doctor Who history. But how long would it really take to do that? Is it even possible to punch through a wall 400 times harder than diamond? Just how many seconds are there in eternity?
First off, let’s establish that a soft material can, in fact, damage a harder one. A diamond is about 20 times harder than steel, but if you hit a diamond hard enough with a steel blade, it will crack. Place it in a heavy enough steel press, and it will shatter. Water can’t be said to have any hardness at all, and yet, given enough time, it can wear away rock.
There’s another factor at work here, too. After the Doctor has worn away a little bit of the wall, the Azbantium has to go somewhere. After a while, he probably gets microscopic bits of Azbantium dust on his fist just by being in the room, which will help him chip it away faster. Also, the wall doesn’t look like a single crystal, so he might be putting stress on crystal interfaces, which are much weaker than the crystal itself. And perhaps the oils of his skin react with it chemically over billions of years, wearing it away.
Now, how much Azbantium can the Doctor remove from the wall per punch? Honestly, I have no idea. Even if it were just ordinary diamond, I would have no idea. I have no reference from which to guess it. I wouldn’t even know where to begin to look for how diamonds react to being punched. (Also, the Doctor is implied to be stronger than a normal human.)
But let’s say each punch removes one layer of atoms where his fist strikes, just to give us a number to work with. His fist strikes with an area of about 25 square centimeters, and the tunnel he carves out has a cross section of about two square meters. That means it takes 800 punches to remove one layer of atoms from the tunnel. The tunnel is 20 feet long, or about 6 meters. We don’t know what Azbantium is made of, but the bond length between carbon atoms in a diamond is 154 picometers. That means the tunnel is about 25 billion atoms long, requiring 20 trillion punches to make.
The doctor is trapped in the castle for 4.5 billion years, or about 1.6 trillion days. This would suggest that he needs to deliver an average of 12 punches per day, quite a bit more than he appears to do in the show.** However, he wouldn’t be able to just punch his way through. After all, the wall extends all the way down to the floor. He would need to kick out the bottom part, which he could do much faster since he’s wearing shoes and wouldn’t need to break his toes.
So while the 4.5 billion years number was probably made up…it’s actually surprisingly plausible for the closest attempt we can make at even trying to figure out how long it would really take to get through that wall. Go figure.
(Also, I can definitely recommend the Christmas special, “The Husbands of River Song”, which is a delightful romp that offers excellent closure for a beloved character. 4.5 out of 5.)
* We see two nights take place in the episode, and the Doctor says it takes him a day and a half more to climb back to the top of the castle, so he couldn’t have reached the Azbantium wall any more often than once every four days. It could have been much longer, possibly as long as a hundred years, as in the story he tells, but that seems unlikely.
** Edit, December 26, 2018. On rewatching the episode, in the early cycles where he doesn’t have the extra few feet of the tunnel, the Doctor punches the wall exactly six times. Toward the end, he appears to get eleven punches in. He also strongly implies that he remembers all of the previous cycles, like Rory after “The Pandorica Opens”.