Maths in the Movies: Indiana Jones and the Dial of Destiny

The 5th movie in the Indiana Jones series was released in the UK on 28th June 2023. Amongst the fans in my circle of friends it has been well-received, certainly more so than the 4th instalment (Kingdom of the Crystal Skull) was. This isn't a general review^[1], but a brief discussion of some of the mathematical themes that make an appearance.

I'll do my best to make the following spoiler-free, but as I'm discussing things that happen in the movie I can't guarantee that I'll manage. With that in mind, if you haven't yet seen the movie the safest course of action would be to lock this post in a snake-ridden vault and grab your hat & whip and head out to the cinema before reading any further.

1. An Enigma Machine

Absolutely 0 spoiler danger in this bit (unless you're not already aware that Indy's adventures invariably relate to key real-world events of the 1940s).

I'm always excited when a certain piece of World War II era encryption tech makes an appearance in a movie, so I was glad to see a fleeting glimpse of an Enigma machine. We see a small portion of the machine for a fraction of a second. Blink and you'll miss it, but it's there. It has absolutely no bearing on the plot, so the fact that they bothered to source one for this fraction of a fraction of a scene made for a lovely, surprising little easter egg, seemingly just for me.

2. A Misunderstanding of Polybius

This bit is potentially a little spoilery, pertaining to some plot points, but I don't think there are any major spoilers here. 

Some of the information required by our protagonists and also sought by their antagonists is hidden using a form of the cipher made famous by ancient Greek scholar Polybius. It works by arranging the letters of the alphabet in a 5 × 5 square grid (hence the cipher's common name "Polybius square")^[2]. The columns in the grid are labelled 1 - 5, as are the rows, and pairs of these labels are used much like coordinates to describe the position of each letter in the table. Using the table below, for example, the position of the letter B would be described by "12" and X's location is 53:

	1	2	3	4	5
1	A	B	C	D	E
2	F	G	H	I/J	K
3	L	M	N	O	P
4	Q	R	S	T	U
5	V	W	X	Y	Z

A message is encoded by substituting each letter with the pair of numbers describing its position in the table. For instance, if we were to encrypt "INDY" using the table above we'd get "24 33 14 54".

At this stage, this is really just a code: memorise the grid and you can read anything encoded with it. For it to be useful as a cipher we'd need to be able to change it somehow so that it's hard for anyone to read the message even if they know how the Polybius square cipher works. A decent way to do this would be to change the order of the letters in the grid. For someone else to read our message (and when encrypting a message we always want someone to be able to read it, or why bother writing it down?) we need to be able to share that new arrangement with them. A convenient way to do this is to use a keyword that describes the new arrangement. Filling in the keyword "DESTINY" and then writing the remaining letters of the alphabet in order would give us the following arrangement:

	1	2	3	4	5
1	D	E	S	T	I/J
2	N	Y	A	B	C
3	F	G	H	K	L
4	M	O	P	Q	R
5	U	V	W	X	Z

Encrypting "INDY" with this table would give us the ciphertext "15 21 11 22". Decrypting that requires knowledge of the key in order to accurately recreate the grid^[3].

The final piece of the puzzle is, of course, that in the context of the movie this would have been (literally) all Greek to them. Ancient Greek. A bit of a hole in this part of the plot is that Helena Shaw (Phoebe Waller-Bridge's character) is able to read the message just by looking at it. I can accept that she is capable of deciphering ancient Greek by sight-reading, but not that she can determine the correct configuration out of all 800 septillion (I think...) possible arrangements of the grid on the fly. Also, the quick view of the message that we saw on screen looked, to me, like it might have been an actual Polybius square which would mean that she was reading the key rather than the message. You might argue that if this is the thing that I find most unrealistic about an Indiana Jones movie that I should probably engage with the real world a little more closely.

3. Wax on... wax off

This bit might count as an irritating mini-spoiler: it doesn't give away any major plot points but could remove a surprise that's key to a scene or two, but forms the third in a trilogy of things relating to the history of cryptography (which is, of course, a subset of the history of maths).

Point 2 above is revealed to be moot as the Polybius square is a red herring: It's scratched into the surface of a wax tablet, and melting the wax reveals the real information hidden underneath. This, like so much Indiana Jones lore, is inspired by actual events. Wax tablets - literally a piece of wood covered in wax in which written information would be scratched with a stylus - have been discovered in archaeological finds from as far back as the 14th Century. Greek historian Herodotus recorded that in the 4th or 5th Century BCE, exiled Spartan King Demaratus noticed a build-up of forces in his new home of Susa and wanted to warn his former home of an impending attack. Fearing that any obvious message might be intercepted and read, he scraped the wax off a tablet, scratched details into the wood, and then re-covered it with wax. What appeared to be a perfectly ordinary, blank table reached its intended recipient and (by some luck as far as I can make out) the decision was made to scratch off the wax, revealing the message beneath.

4. An Enigma of a Machine

This is potentially quite a big spoiler, so be wary all ye who continue beyond this point, for ye have been warned.

The main non-human character of the story is something called the Antikythera mechanism. I was so excited when this was revealed as it's a truly interesting device shrouded in mystery. I wrote about it some time ago (here: blogstronomy.blogspot.com/2011/05/who-created-mechanical-model-of-solar.html) but in short, it's a hand-powered mechanical orrery discovered in 1901 in a shipwreck from the 1st or 2nd Century BCE. It was initially assumed to have originated from a time much later than the shipwreck (possibly finding itself there by pure coincidence) because analysis has revealed a device of significantly greater technical detail than was thought possible of that time, capable of predicting eclipses with great accuracy, and even modelling irregularities of the Moon's orbit around the Earth. It has since been confirmed to originate from the same time as other objects retrieved from the shipwreck but still represents technology far in advance of anything else discovered from the same period. It is little wonder that this very real device has found itself the subject of fantastical fiction, but in real life it highlights that some pretty impressive mathematics was being done at the time of its creation.

Have you come across any mathematics in the movies? Tell me about it! Either recommend the movie to me or write about it yourself: I'm happy to host well-written guest posts!

Footnotes

1. I did something like that here: letterboxd.com/teakayb/film/indiana-jones-and-the-dial-of-destiny/. [back]
2. As there are 26 letters in the modern Western alphabet and this isn't a square number, we tend to double up I & J in ciphers like these. [back]
3. Ideally someone would know the key because we'd told them what it was, but of course there are cryptographic techniques that an enemy might use to rebuild the table without us telling them: the Polybius square cipher is not very secure by modern standards. [back]