Quantomaton is a ‘zero-player game’ based on Conway’s Game of Life and a small collection of cyclic music pieces. You can think of it as a music album with a visualiser attached, or as a cellular automaton with a soundtrack. Quantomaton is an app for iOS and Mac OS, coming soon to the Apple App Store. You can already download the public beta through Apple TestFlight.
Note: Quantomaton is a continuation of my earlier ‘A Game of Quantum Life’ project. For more details about the maths and physics behind my implementation of the Game of Life, please read my write-up on that project.
About the game
Conway’s Game of Life is probably the most famous example of a cellular automaton – a grid of cells that evolves through different generations based on a few simple rules. It was invented (or discovered, depending on how you look at it) by mathematician John Conway. Over the course of the game, the cells form various patterns determined by the initial configuration of the grid. Quantomaton takes the Game of Life as its starting point, and adds a quantum-mechanical twist to it.
Classical rules
- All cells can be alive or dead
- Live cells stay alive if they have either two or three live neighbours
- Dead cells turn alive if they have exactly three live neighbours
Additional quantum rules
- Cells can also be in a superposition of alive and dead
- Superposed cells with two or more live neighbours will be measured and turn either alive or dead, each with probability 1/2
- Dead cells with fewer than two live neighbours may go into superposition with probability 1/8192 (1/2¹³, or about 0.012%)
- Dead cells with exactly two live neighbours may turn alive with probability 1/8192 (1/2¹³, or about 0.012%)
- Superposed cells with fewer than two live neighbours may get measured by the environment and turn either alive or dead, each with probability 1/16384 (1/2¹⁴, or about 0.006%)
Quantomaton has two selectable modes: Classical and Quantum. In Classical mode, only the first three rules are applied. In Quantum mode, the other rules are applied as well, adding an element of chance to the game.
The neighbourhood of a cell consists of the eight cells directly around it. The playing field is toroidal, meaning that the edges of the grid wrap around so that it is continuous in all directions and every cell on the field has exactly eight neighbours. Each time a new game is started, the playing field is seeded with a random configuration, following the percentages selected in the ‘seed’ setting.
From there on, the game evolves automatically – the rules are applied to the initial configuration, then to the resulting new configuration, and so on for each subsequent generation. This is the ‘zero-player’ part – after you’ve hit the ‘start game’ button, all you need to do is sit back and enjoy the show. In Quantomaton, the game plays out at a set rate of three generations per second.
About the music
There are three pieces of music to choose from. All three are cyclic in nature, meaning that they end up back where they started to form an infinite loop that keeps going for as long as you care to watch and listen. Although the game and the music don’t influence each other, they run on the same clock and are always in step. When you select a new piece of music, the game’s options are set to values that I felt best matched the mood of the piece, but you can change them to suit your preferences.
‘Meridian Crossings’ is my most recent cyclic composition. It follows a cycle of 28 just-intonation pentatonic scales. This cycle uses the gentlest of harmonic motions: each of the 28 segments is purely pentatonic, and differs from the previous segment by just a single note. Still, the piece as a whole passes through seven different tonal centres and employs 15 unique notes per octave. It also incorporates a long-form polymeter of the first five primes (2, 3, 5, 7 and 11). One full cycle takes 4620 generations (25 minutes and 40 seconds) to complete and loop back to the starting point. In Game of Life parlance, you could say that the piece as a whole oscillates with period 4620.
‘Polychrome’ was originally written for solo acoustic guitar, though I added a second part for this version. This was the first cyclic piece I ever wrote. Rhythmically it is very simple, but harmonically it covers a lot of ground. As the piece progresses, its theme modulates through all 24 major and minor keys of 12-tone equal temperament. Once it has passed through every key, after 768 generations (4 minutes and 16 seconds), it comes back to its starting key and repeats.
‘A Game of Quantum Life’ is a slow-moving sequence in four-part harmony. It uses a single diatonic scale (F sharp Lydian) in just-intonation tuning. Each of the four parts has its own pace, and together they form a polymeter of the first four primes (2, 3, 5 and 7). After 2520 generations (14 minutes), the chord sequence and the polymeter come full circle and the piece repeats. I originally wrote this piece to accompany my first quantum version of the Game of Life, built on the PICO-8 platform.
All sounds are synthesised in real time by the Arithmophone sound engine I developed for ‘Pentatone’ – a music app at the opposite end of the spectrum from this one, in that you actually play it yourself. Coincidentally, the 28-scale cycle behind ‘Meridian Crossings’ comes from that project as well – I came across it while mapping the relations between different pentatonic note sets.
About the game, once more
In Classical mode, Quantomaton works exactly like the regular Game of Life. Even though its rules are very simple and the game evolution is completely deterministic, the Game of Life famously produces highly complex behaviour that tends to stay varied and unpredictable for many generations. In Quantum mode, the behaviour is no longer deterministic and superposed cells interfere with the evolution. This reduces the stability of classical patterns, typically keeping the game’s evolution dynamic for many additional generations.
On a grid of the scale used in Quantomaton (a few thousand cells, the exact number depending on the device size and settings), the Classical Game of Life typically takes between 500 and 2500 generations, or roughly 3 to 15 minutes, to reach a steady state. This final state usually includes a number of permanently live cells as well as some patterns that oscillate with period 2 (i.e. they repeat every two generations).
In Quantum mode, the game can easily keep going for ten to a hundred times as long as in Classical mode. Eventually it will settle into a ‘vacuum’ state where isolated live cells die immediately and only the ‘quantum fluctuations’ of dead and superposed cells persist. However, this vacuum state is not final: a new epoch of life can arise spontaneously at any time. The probability of that happening is quite small though, so you may need to stock up on snacks if you want to witness it – I calculated the median wait to be around 18 days when the game is run on a 64 by 92 grid (iPad in fill mode) and around 45 days on a 48 by 48 grid (iPhone in square mode).
Disclaimer
This app is not in any way a rigorous representation of the mathematics of quantum theory. I am a composer/designer, not a physicist or mathematician. Serious research into quantum-mechanical cellular automata does exist, and is probably very much worth checking out if your maths skills are up to the challenge, but this app is not that. Quantomaton is primarily intended to provide an interesting audiovisual experience – it may lead you to contemplate the nature of reality, if you’re so inclined, but it takes generous artistic licence with the actual maths and physics of quantum mechanics.