Something I've been hacking on that I don't think anyone has discovered yet is that around the time of the decision of what images and artifacts place on the Voyager spacecraft as evidence of human civilization, there were a number of candidates. Among one of the successful candidates was Laurie Spiegel[1], an early electronic musician (1970s) and programmer from Bell Labs, and her work "Music of the Spheres" introduces the section "Sounds of the Earth," instead of being included in the "Music of Earth" section on the golden record we sent in the hopes of it being discovered by some other spacefaring civilization.

There is a coincidence I'm running down to see if there is any substance to it, as there was another more prominent piece of music written at the same time by a more prominent composer that was not included in the space program, and it too used a procedural method, which could also be used to encode information to an advanced civilization in an unambiguous way, independent of what those aliens might look like, so long as they understood the same physics.

Obviously, it would make more sense to transmit a message via radio or laser or some other means that travelled at the speed of light, but the big question is, what message would you use, and how would you encode it so that it would be an unambiguous signal of our place in the universe?

Arvo Pärt is an Estonian, former Soviet subject and orthodox Christian, whose early work was largely inspired by gregorian chants, and early, religious music, and he's reputed to be a bit of a mystical hermit. Some writers have written about symmetries and "shapes" in the structure of his pieces, and what people examinging Arvo Pärt's music to interpret his "Tintinabuli" and geometric procedural composition style have overlooked is the time and context in which this specific piece was written. The piece was famous then as it is now, and it is called "Spiegel im Spiegel" or Mirror in Mirror by Arvo Pärt.[2]

Not only do I think that Pärt was inspired by Gauss's proposal, but what I think I have found that is new is that the violin melody from Spiegel im Spiegel isn't just made of triangles, as some academics have suggested, but rather a sequence of negative and positive integers, and specifically the integer sequence that is the result of iterating Euler's Identity e^ipi = -1. When you scratch the surface of what function could produce this undulating wave of -1, 1, -2, 2, -3, 3... it is a succinct demonstration that we have found e, pi, i, 0, 1.

This is what we should encode as a message in sound, or in light, to send out as evidence of the state of our civilization. That this encoding of Euler's identity in procedural quasi-religious music was written at the same time as our societies around the world were reflecting on who they were and what artifacts they would choose to represent themselves to a distant civilization - and it happened to be coincidentally named for one of the artists whose work was sent out - is perhaps a just-so story, but it has been quite a fun journey into who these people were and what they may have intended.

TL;DR, I think I discovered that Arvo Pärt's Spiegel im Spiegel is an encoded expression of Euler's identity e^ipi = -1, and since it can be looped infinitely at any scale, over any analog transission medium, just like two mirrors reflecting each other, it is a better candidate for Gauss's proposal than using Pythagoras. I've been trying to figure out what the piano accompanyment was produced by, but it's pretty likely that there is something similar encoded in it.

[1] https://en.wikipedia.org/wiki/Laurie_Spiegel
[2] https://www.youtube.com/watch?v=kVaEUBvppMI