An 11th Century Philosophers’ Game For Meditating On The Universe

Back in 2017, I wrote an article about Herman Hesse’s fascinating Glass Bead Game. The idea of two people at a table moving shiny glass beads around on a complex game board filled with mysterious glyphs, pondering incredible connections between disparate concepts still intrigues me terribly. I imagine a near-impossible breadth of knowledge needed to master this imaginary game, and its best players discovering hidden patterns behind reality and history as they ply their ingenious strategies.

Awesome.

Still, that’s fake. No such thing. Not really. But I wanted to bring something to your attention that has been around since the 11th century and that you can still buy on Etsy or whatever that isn’t fake at all. And if you squint real hard and just go with it, you’ll see something equally fascinating: an engine to tune your mind to the workings of the cosmos (sort of).

Anyway, I’m going deep right now into Medieval cosmology. Don’t ask. I don’t always pick these intellectual bunny trails; sometimes they pick me. Has to do with D&D’s Spelljammer, the Troika roleplaying game, and something I’m going to write up here in the future on Grailrunner. Will be great; I promise. Still cooking.

But this though:

That’s a vellum manuscript dating back to 1000AD, a copy of a work titled De Arithmetica by a philosopher named Anicius Boethius who actually wrote the work in the 6th century. He’s more famous for a conversation with philosophy in woman form called “On The Consolation Of Philosophy”, which is a bit of a mood piece about the fickle nature of fate and how you should deal with that. Not my topic today. Let’s talk about that book in the picture.

“Wait a second. You’re a blog about nerd stuff and science fiction. Why are you on about this right now?”

I hear you. Hold on to that. We bring you inspiration, and wonderful little nuggets that you can file away for your own creations. Edison said “All you need to invent is an imagination and a pile of junk.” And so we proceed…

“So what’s the big deal about Boethius?”

Boethius is important because he served as a bridge between ancient philosophy and the Middle Ages. He didn’t just translate Aristotle, but also commented on the works and added newer insights. He brought ideas from Neoplatonists like Porphyry into wider recognition and helped people make sense of them. In De Arithmetica, he translated De institutione arithmetica libri duo by Nichomachus of Gerasa, who was writing around 100 AD. You see what I mean about this guy being an important bridge of older thinkers, yes?

Philosophy is about pondering things, seeing the beautiful and intricate architecture behind things in flashes of insights and through establishing connections where others can’t see them. Boethius saw the foundation of philosophy as a bedrock he called “the quadrivium”, consisting of arithmetic, geometry, music, and astronomy. Fundamentally and at their innermost core, he might tell you, these four things merge.

“I thought we were talking about a game?”

Yes, we are. Here’s a wikipedia article about Rithmomachia, also called The Philosophers’ Game or The War Of The Numbers. The game is based on the study of numerical proportions and harmonies that Boethius studied and wrote about, much of which you could find perusing through that book up there. In fact, historian David Sepkowski said of Rithmomachia that between the twelfth and sixteenth centuries,

“Rithmomachia served as a practical exemplar for teaching the contemplative values of Boethian mathematical philosophy, which emphasized the natural harmony and perfection of number and proportion, that it was used both as a mnemonic drill for the study of Boethian number theory and, more importantly, as a vehicle for moral education, by reminding players of the mathematical harmony of creation”.

“What?”

Here, check this out to see what he’s talking talking about, then I’ll tell you what it’s like playing this game:

  • Arithmetical proportions: Say I give you the numbers 3 and 15. You can find the missing number between them that would form an arithmetical proportion by summing the first and last numbers of the sequence (3 and 15 for a sum of 18), then dividing by 2. So in this example: 3, 9, 15.
  • Geometrical proportions: Say instead I give you the numbers 2 and 72. You can find the missing number between these that would form a geometrical proportion by multiplying the first and last numbers of the sequence (2 x 72 = 144) then finding the square root of that. So in this example: 2, 12, 72.
  • Harmonic proportions: Say now, finally, I give you the numbers 12 and 20. You can find the missing number between these that would form the harmonic proportion by multiplying the first and last numbers of the sequence and also by 2 (12 x 20 x 2= 480) then dividing that by the sum of the same two numbers I gave you (12 + 20 = 32). So in this example, 480 / 32 = 15 and the sequence is 12, 15, 20.

To the minds of the Greeks, all the way up for centuries after Boethius wrote about this, number sequences like this have a magic to them, because they’re tuned to reality itself. Nature and the cosmos, the very music in the air, the movement of the moon and the stars, all tied in to these perfect, intellectually satisfying numerical relationships. Measure anything in the stars or on the water or in the music from a harp and you’ll find these sequences, they would tell you.

Make fun of that if you want, or look down on it as caveman thinking, but I felt the same kind of magic in school when I studied this little wonder:

That’s Einstein’s field equations, tying together everything that ever was. It’s one of the most verified things in Physics. Explains how the world goes round, why things fall, and the future of the universe. Gorgeous. Absolutely gorgeous. That’s the way the monks felt playing Rithmomachia, clashing their little game pieces together looking for ways to feel these proportions. Not to just learn them.

To feel them.

If you’re at all interested in learning more about this wonderful game, seeing its rules clearly delineated for you, and seeing some nice illustrations of game play maneuvers, then head to Amazon and read Rithmomachia by Seth Nemec.

He does an amazing job walking you through why the number sequences mattered to those to whom this game was more than a pastime and a learning mechanism, but rather a way of worshipping and meditating on the very fabric of the cosmos. If you have Kindle Unlimited, it’s free.

I’ve read it four times myself over the years just because he makes the game seem like an awful lot of fun, and somehow important. It makes me want to hop in the minds of those monks and feel the way they felt playing it, and to see that crazy board and its pieces on a big old oak table read to go.

In fact, it was the idea of a philosophers’ game with real-world implications that inspired a story collected in Kyot: The Storybook Puzzle Box. That one’s called The Berserker’s Game, and a far bit darker than Rithmomachia. Read it here if you like.

Overview of Rithmomachia

Quick summary of the game though, so I can tell you whether I beat my son on Christmas Eve or not (and some insights we had playing it):

Game pieces: Game pieces are either circles, triangles, or squares, all with numbers on them. The two game piece sets aren’t the same, nor are they symmetrical, though the White player’s pieces are based on even numbers and Black’s pieces on odd numbers. The numbers themselves, their placement in the starting setup, and the movement rulesets are all based on Boethius’s proportions. Precisely defined stacks, one stack per opponent and called ‘pyramids’ are provided for as well.

The board: The board is an 8 x 16 squares grid, basically two chess boards set end to end.

Moves: Everything can move orthogonally or diagonally, but circles move 1 space at a time, triangles 2 spaces at a time, and squares move three spaces at a time. Piece moves can’t come up short – you move exactly 1, 2, or 3 spaces when you move. Pyramids may move in the manner corresponding to their component parts, as long as the requisite shape is represented somewhere in the pyramid (meaning it can’t move a single space any longer if it’s lost all its circles, for example).

*

Attacks: Four basic attacks exist (but the attacking piece does NOT move into their victim’s space as it does in chess or checkers, you just call it and take the piece):

1. Siege is surrounding a target piece on four sides, either orthogonally or diagonally (board edge counts). Surround them and call it, taking the piece.

2. Encounter is when an attacking piece COULD legally move into the space where an opponent’s piece (of equal numerical value) is located. Just call it and take the piece.

3. Eruption is when you multiply the attacking piece’s number by the spaces between it and the target piece to obtain the target’s number. Say your 8 is 2 spaces from your opponent’s 16 (which in this game means side by side because the squares they’re on count in this calculation). Since 8 x 2 = 16, and that’s the target’s number, you call it and take the piece. Division okay too.

4. Deceit is when you surround a target pieces on 2 sides, and the two attacking pieces sum to the target piece’s number.

*

Victory conditions: A number of victory conditions are provided across two categories – those defined based on pieces captured and those defined based on numerical progressions formed with remaining pieces on the board. Simplest possible is Victory Of Goods, meaning pick an overall score (say 100) and the first player to capture pieces summing to that number wins.

“So you’ve played this? What’s that like?”

I built a Rithmomachia board based on Nemec’s book a few years ago when I first encountered the game, just to see how the rules played out, and what differences I experienced in game play between the opposing sides, given the asymmetry of their assigned numbers. It’s been in my closet a while now though. My son is in college, majoring in computer science and math, and I knew he’d be into this when he was back home for Christmas (2022 as I write this). It’s right down his alley now, and he’s devious and sly enough to uncover slick strategies in any new game.

And he’s not afraid to get mean when necessary.

Some interesting insights based on our game play:

  • Eruption is awesome. It’s just awesome. It was our signature move, because of the level of aggression and devastation you can wreak with it. Planning Eruption attacks feels like planning moves for Bishops, Rooks, and the Queen in chess, only slightly more difficult due to running all the permutations through your head.
  • I see now why the checkerboard needs to be as long as it is – Eruption needs spaces on the board to provide for more multiples and make the math behind the attack useful in going after larger numbers. If you’re only multiplying by 1 or 2 each time, that isn’t much to work with.
  • The rules allow you to take multiple pieces in one attack as long as conditions are met for the respective pieces, so we really focused on trying to make that happen. It felt a lot like chess in that respect, with long turns of staring at the board. (We had very little luck in this though.)
  • The fact that you don’t move the attacking piece into the captured piece’s position flavors the entire game very, very differently to chess or checkers. It’s much more cerebral, constantly checking different combinations and possibilities mentally. Since you can’t move and attack in the same turn, this forces you to spend some turns moving just to change up the board configuration.
  • We stuck to very basic attacks and lower numbers. Yet there are numbers on the board like 289 and 361. You’re dividing a lot, trying to seize one of these big pieces, but you can see pretty quickly that won’t be easy at all to just go for the one big kill shot, due to their placement in the startup configuration. We really should have moved more pieces versus the constant attacks, to change up the dynamics of the board

And the single biggest observation that became apparent within the first few moves was surprising to me. I hadn’t expected a game designed by monks for monks, engineered at its core for instruction and meditation on the harmony of the cosmos would be a poker face game of deceit.

“What do you mean?”

So many of the attacks work both ways. Since you can’t move and attack in the same turn, when you move into position for your planned attack, in many cases, the other guy can do it to you instead. That was especially true for us because of our fascination with the Eruption attack. It meant you had to keep a straight face, look elsewhere on the board, even say deceitful things to distract your opponent from what you’re scheming.

Our game deteriorated quickly into a broadsides shootout between our two pyramids and with a few surrounding pieces, blasting away with Eruption attacks since we kept getting confused about what was concealed in the stacks. It was a way of trying to surprise the other guy.

I just hadn’t expected a monk’s game to require so much deception and stealth. Crazy.

“Well, who won?”

I got a lucky strike in, which sent me over the goal for a win. Honestly, it’s just a lot to keep in your head with many, many possible sneak attacks. You start to feel a little paranoid about that.

But overall, I did start to get a feeling for the numerical patterns, the weight of the larger numbers, the reasoning behind their placement and the logic of the startup configuration. It’s a fascinating game, and easy to see why people who felt these patterns were the language of God would see wonder in the board and its pieces.

Anyway, that’s what I wanted to tell you about this week. Great game, and Nemec’s book is worth a read.

What do you think?

Till next time,

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