Gauss's Pythagorean right triangle proposal | Hacker News Reader

Gauss's Pythagorean right triangle proposal

en.wikipedia.org

44 pointsbenbreen27 comments

But seriously, what could we do to make ourselves evident in every direction to distant beings with telescopes at least as good as ours? Something that would never ever happen naturally.

I'm thinking something passive (i.e. reflective) and orbiting a Lagrange point... maybe one we don't need much, like Mars L4 or L5. Make a dozen reflective spheres, of mirror mylar inflated with a little bit of some big-molecule gas that won't leak out. Arrange them in a static icosahedral form at the Lagrange point.

From any angle except perhaps very normal to the ecliptic, there would be a distinctive changing pattern of reflected light. Maybe each sphere could have a different notch filter coating on it so there would be some signal on the spectrograph too.

consider the situation at home of already being evident to other species and its results

Aye. Our sensory apparatus, let alone our ability to process the totality of inputs from it, is pretty limited. WTF can comprehend even what fungi "feel"? And we're of the same origin.

Maybe in 100 million years? Chump change, of course, even in biological time; but then "we" won't be worrying about triangles.

I think parent meant that whenever another species notices you, it tries to kill you or hide.

More generally, a mutual lack of understanding or cooperation regardless.

Yeah that's not bad. Any way to work prime numbers into it? I'm thinking bright objects worked into an orbit, with the gap between each reflector in orbit being say 5, 7, 11, 13, 17 so you'd see these periodic increases in brightness in an unmistakable pattern?

It would be much easier to pick the nearest N stars and point lasers at them with a peculiar frequency that would stand out on any spectra.

This random quora answer[1] suggests lasers could potentially have a spread of ~0.1 arcseconds so that would be (10 * 60 * 60)^2 * 40k ~ 1e13 times less power required (per target) than an omnidirectional source.

[1] https://qr.ae/pvZZjd

It would have to be huge to be seen with telescopes as good as ours.

Looking at https://en.wikipedia.org/wiki/List_of_smallest_exoplanets:

- https://en.wikipedia.org/wiki/SDSS_J1228%2B1040_b has about 1/1000 the radius of Jupiter, but only could be detected because it’s heavy at 70% of the mass of the sun.

- https://en.wikipedia.org/wiki/Kepler-37b is larger than the moon.

So, it seems we can’t (¿yet?) build such a structure in a way that makes it visible enough.

If we let go of the idea that a proposal must be feasible with current tech, here’s an idea: split the sun in two halves, creating a binary star system. Then, merge the two again. Repeat. That should draw attention.

In order to have something visible in every direction is just impossible I think. It's best to just do something like flash lights or radio waves in such a way to spell out the first few prime numbers. EM waves spread out naturally from a small source, that seems much easier to me.

I'm not convinced that drawing a huge 2-dimensional representation on a non-flat surface (i.e. a globe) really demonstrates intelligence

But it might trigger some aliensplaining: they will come here to tell us about non-euclidean geometry and how to do it properly

Aliensplaining like the conquistadors? No thank you.

Much better is the idea to redirect asteroids into the sun’s orbit that encodes flashes of light in some unique pattern (pi?). The light sends the signal in potentially all directions and the signal would last for a lonnng time.

The ability to achieve that feat is in itself evidence of the beings intelligence and accomplishments.

“Intelligence was here”

Liu cixin would be proud. And horrified.

Aha, that's what Tunguska was: aliens telling us to not bother with this.

We might want to be quiet until we have a better idea of our surroundings.

hmmm.. what about a giant dickbutt, that way it would be much easier to draw.

Every civilization will discover the Pythagorean theorem; only a select few will discover dickbutt.

And the dickbutts of some civilizations will look like the Pythagorean theorem.

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

I don’t see how the sequence (-1, 1, -2, 2, -3, 3...) demonstrates having found Euler’s identity.

I looked for functions that generated the sequence in the music. The closest function that generated that sequence (which describes the scale notes in the violin piece) is:

y = (cosx(x)/2pi

...an increasing wave, which can be expressed alternatively as:

1/4 π e^(-i x) + 1/4 π e^(i x)

...which I have interpreted as requiring knowledge of the relationship between e, i, pi, 0, and 1.

...implying knowledge of Euler's identity.

It isn't a proof, but given it was procedural music, these kind of harmonious coincidences seemed consistent with expressing intelligent intent.

Arvo Pärt is still alive, I just wish there were a way to reach out an ask him if this interpretation is correct.

I’m pretty sure people were aware of trig functions before they became aware of Euler’s identity, and became aware of the general idea of “oscillation” before that.

Toward Gauss' goal of engraving a beacon into Siberia to signal our civilization to extraterrestrials, what is more representative of our current evolution and intelligent intent, an elementary theorem from 3,000 years ago written in megalithic land art with rocks and sticks, or, our most succinct representation of mathematical beauty, transmitted with complete consistency over all spectra of sound, radio waves, visible light, and even x-rays?

What's unique about this function is that it is an oscillation of reals, but between integers. It's the effect of representing the integers in terms of the real, imaginary, and transcendentals. That oscilating sequence is such that if they have discovered Euler's identity, they would be able to see that we have as well.

My point is that if another human person came up with this sequence themselves, this would not by itself convince me that that person is aware of Euler’s identity. There are many other ways to produce this sequence.

For example, ceiling(n/2) times (-1)^n . Or just “it flips between negative and positive, and increases in magnitude by 1 every other step (starting with -1)”.

Like, yes, it shows that they are familiar with the concept of something oscillating.

It does not demonstrate that they are familiar with any trig functions nor that they are familiar with complex numbers.

I'd agree we don't need complex numbers to produce this, because it's right there in the function (cosx(x)/2pi) without them. However, by way of intent, if someone says "hey check out these integers in order, but both positve and negative simultaneously, in the shape of a universal function over reals - as though they thought it was important to show they could produce those integers from the reals. Of what is that an expression?"

What other identity or function of reals yields all integers? I don't actually know, but I'm betting being wrong on the internet that the artist behind this song in this context would pick this specific one for both aesthetic and practical reasons.

However, let's say I'm wrong but wanted transmit Euler's Identity in a universal physical signal anyway, what function would we use?

In the piece, there is a piano accompanyment of triads I haven't done the analysis of to see how they are linked to functional melody notes - but given the composer states his music is an expression of geometric principles, there may be more information in that, and if I persist in my folly instead of posing the half baked version in a comment, I will become wise. Thank you for indulging, the friction refines the idea faster.

Maybe I should have included this small note previously, (I had figured it was a small typo or something) but, x cos(x) / (2pi) does not actually produce integers for integer values of x (other than for x=0) .

I imagine what you mean is more along the lines of x cos(pi x) ? However, this does not produce the sequence -1,1,-2,2,... either. Rather, it produces -1,2,-3,4,...

Which, again, can be produced with (-1)^x • x , not using any trig functions. I suppose one might object that (-1)^x doesn’t have real values for non-integer values of x, but I’m not sure why that should be super relevant to the “what does this sequence of numbers indicate”.

If you want a formula for -1,1,-2,2,-3,3,... then, well, there’s the one I posted earlier, using the floor or ceiling function, or if one wanted one could come up with something for it using multiple uses of trig functions, but it would be inelegant.

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