Heaven Sent is an instant classic, a clockwork puzzle of a Doctor Who episode which may leave your head spinning. We’re going to try to explain what was going on, but it might take us a while. In the meantime, do you want us to tell you a story?

It’s about the holes in Winston Churchill’s shoe. 

Churchill died in 1965, but he still haunts the House of Commons. The door between the Member’s Lobby and the entrance to the chamber is guarded by a statue of the man, sculpted by Oscar Nemon and unveiled in 1970. A political tradition soon emerged. MPs are a nervous lot, so before a big speech or vote they would rub the late PM’s foot for luck as they passed through the door. (Presumably kissing it was beneath their dignity.)

This tiny action, repeated over and over again by hundreds of MPs throughout the years, eventually caused the dull bronze to shine gold. Churchill’s toes were glowing. (The statue of Molly Malone in Dublin is similarly polished, albeit in different areas.)

Eventually MPs were banned from touching the great man, due to the appearance of “small holes and fractures” in the metal shoe. They put up a sign and everyone lived happily ever after.

The moral? The smallest actions we can imagine – a gust of wind, a wave on the beach, walking through a door – shape the world when repeated enough times. Time breaks down everything in the end. Rivers carve out valleys, the sea eats away the land, you’re just not around long enough to notice it. 

Unless you’re the Doctor, that is. 

In Heaven Sent, the Doctor wakes up in an elaborate torture chamber that follows certain rules. We’re only really interested in two here:

  1. Rooms reset after a certain period of time.
  2. To escape, he either has to reveal his secrets or get through a very thick wall “400 times harder than diamond.”

The Doctor uses the rules against the prison. Realising that resetting the teleporter means another version of himself is stored in its memory, waiting to emerge, he starts playing a long, long game: running the maze over and over again just to punch that final wall once or twice, before dying and starting the process again.

Like the waves on the beach or MPs filing into the chamber, he becomes erosion incarnate, wearing down something impossibly hard with hundreds of tiny taps, a one man force of nature. But that does inspire one major question...

Can you actually punch through a wall of diamond?

Diamond is well hard. On the Mohs scale of mineral hardness, which ranks materials based on what can scratch what, it’s number ten, the tiptop. (Talc, as in talcum powder, is at the bottom.)

As well as sealing marriages and inspiring Beatles songs, diamonds are used to tip the drills on oil rigs. Would punching 'Azbantium', which is apparently 400 times harder than diamond, make any difference whatsoever? Even if you did it over and over again?

Here’s another question: is water harder than rock?

Answer: no, it’s just persistent, and there’s a lot of it. 

“Could you punch with your fists through a wall of diamond? I guess it depends what the wall of diamond was like,” says Professor David Dobson, an expert on high pressure experiments at University College London. “If it was a big single crystal, I would say no. If it was polycrystalline, and assuming that you have an infinite number of fists to use…possibly.”

Possibly?

“In principle, particularly when you have very hard materials that are very brittle, if you can get them to flex, then yes. As you punch with your fist, you cause a tiny amount of strain in the aggregate, so it bends slightly. What you would end up doing in that case would be causing the grain boundaries between the individual diamonds to actually break down.”

“The same rules apply to something 400 times tougher than diamond, only more stringently.”

If you were very, very clever (and perhaps were wearing some snazzy sci-fi sunglasses) you could even target weak points in the structure of the wall. Another tip? Stop using your fists, you bampot.

Why doesn’t the diamond wall reset? 

If every room in the castle resets to its original condition after a while, why doesn’t the diamond wall heal itself between versions of the Doctor? The simple in story explanation is that only certain rooms are designed to reset. The Doctor was only supposed to be in the final ‘trap’ room once. His captors intended that once he saw the wall he would realise it was impossible to escape, give up and reveal the identity of the hybrid. 

“Once you broke off a piece of the wall you could use that as a tool. Rather than punching the wall directly, if you made a nice diamond tipped wedge that could place the stress on a single point, that would significantly increase your chances of doing it in a finite length of time.” 

Nevertheless, it’s a long way from the ‘multi-anvil presses’ and ‘diamond cells’ Dobson uses in the lab, which can exert pressures up to 3 million times the Earth's atmosphere – the same as at the planet’s core. 

“That is somewhat harder than you can punch,” Dobson explains, “even if you put all your weight behind it, and you’ve been working out at lunchtime.” 

So, punching through a 20 foot thick wall of material 400 times stronger than diamond is possible, but how long would it take?

“I really wouldn’t like to say. I think possibly longer than the universe has existed, but maybe not longer than it will exist. Maybe a good 14 billion years.”

To put that in perspective: in one billion years the sun will burn Earth to a cinder, in 4 billion years the Milky Way will collide with its neighbour Andromeda, and in 8 billion years Sherlock series four will be released.

Or, to put it another way...

How old is the Doctor now? 

Unsurprisingly, it’s complicated. Some version of the Doctor was trapped in what we assume is his own Confession Dial for billions and billions of years, but that doesn’t mean the Doctor is billions and billions of years old.

Essentially, the Doctor was cloning himself again and again. Each time, the Doctor would emerge from the capsule, give the same monologue, make the same daring leap from the castle, get to the diamond wall, come up with his escape plan yet again, then throw a few punches before getting mortally wounded, dragging himself to the top of the tower and creating a new Doctor to begin the cycle over.

It wasn’t the Doctor who broke through that wall, it was millions of Doctors.

It’s important to note that this wasn’t a Bootstrap Paradox, as we encountered earlier in the series. It all makes logical, linear sense. There was a first Doctor who emerged from that teleporter and worked out the plan – leaving ‘bird’ as a clue for himself (a reminder of the Grimms' fairytale above) – and there was a last who finally broke through the wall. The story isn’t a loop but a straight line. This wasn’t time travel, it was a relay race with himself.

Is the Doctor no longer telepathic?

Faced with a locked (wooden!) door in tonight’s episode, Peter Capaldi lamented that he could have opened it when he was “young and telepathic.” David Tennant and Matt Smith's Time Lords occasionally used these abilities, including wiping Donna’s memories in Journey’s End and sharing memories with James Corden in The Lodger – so why did Capaldi’s older Doctor lose the power? He’s around 1,100 years older than Smith’s incarnation was when he last used telepathy, of course, so that could have something to do with it. Or maybe Capaldi’s Doctor is just a little less proficient with the skill – the door did open in the end, after all. HF

From the perspective of the Doctor who escaped, he spent a long time in the prison, but not billions of years: every clone of the Doctor comes to realise he is one of thousands, yet he can only remember from the moment he was created in the teleporter to the moment he died or escaped. 

Here’s the real brainteaser: is he still really the Doctor? Put another way, is the Doctor now much, much younger, almost a newborn?

Was the Doctor in his own Confession Dial?

When the Doctor first emerges from the transport pod, he comments that the equipment is “consistent with a modified long range teleporter”, meaning that (as referenced in previous episode Face the Raven) he is within one light year of his last location (Earth). However, he spends his time staring up at an alien sky, and later walks out of the castle onto the plains of Gallifrey. So what gives?

Best guess: the Doctor was teleported inside the Confession Dial Ashildr confiscated from him, then that was moved to Gallifrey. Which means he had been carrying his own prison in his pocket, and the Time Lords have been planning this for a long, long time...

He makes clear that the teleporter is ‘a fancy word for a 3D printer’. It doesn’t really transport the original, but copies it out of new materials and destroys the original. The original Doctor was ‘dead’ the moment he was teleported – a new man made of shiny new atoms stepped out.

(Presumably the only reason he can’t create an entire army of Doctors is a lack of energy. He only has one Doctor to burn.)

This isn’t a new problem. Philosophers have been tossing around versions of it for hundreds of years when debating the nature of identity, and it’s come to be called the ‘teletransportation paradox’. For obvious reasons it’s a favourite of sci-fi writers, but here’s another way of thinking about it. Let’s talk about the Sugababes.

Siobhán Donaghy, Mutya Buena and Keisha Buchanan first topped the charts in the late 1990s and early 2000s with bangers like Freak Like Me. 

But all was not well with the Sugababes, and one by one all of the trio left and were replaced. First Siobhán was swapped with a former Atomic Kitten, then Mutya went, followed by Keisha. By 2007’s ‘Get Sexy’, there were no original Sugababes in the Sugababes.

Ask yourself: was this band calling themselves the Sugababes still the Sugababes? If not, when did they stop being the Sugababes?

(Here’s the real mind-screw. Mutya, Keisha and Siobhán later reformed as the band ‘Mutya Keisha Siobhán’, and there’s talk of them trying to reclaim the Sugababes name.) 

Before the Sugababes existed, this problem was known as the ‘Ship of Theseus’. A version of it appeared in Only Fools and Horses, in a joke the Doctor referenced last series in Deep Breath.

So what are we to make of all of these girl bands, brooms and replacement Time Lords? Well, to paraphrase the Doctor, the key is to slow the problem down. Imagine if instead of an instantaneous transport, the Doctor was replaced piece by piece –limb by limb or cell by cell– over an hour or so. When does he stop being the Doctor?

What about you? Over the course of seven years, every cell in your skeleton dies and is replaced – when do you stop being you?

Also, bear in mind that the Doctor is frequently replaced by a new version of himself. After regeneration, he emerges with new cells, a new face and a new dress sense, but is still deemed to be the same character. Whether teleported or regenerated, the Doctor remains several thousand years old, despite his shiny new body.

The point is that whether you’re a normal person or an immortal Time Lord, and even when you’re trapped in a prison for billions of years, change is the only constant.