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I have a strange problem that you may be able to help me with. I was recently hired by company that manages several communications satellites, and I've gotten stuck with the overnight shift. Earlier this week I was getting some weird interference on the C-band receiver during a pass sometime after midnight. I didn't think too much of it initially, but it reappeared again Friday night, at around the same time. This time I managed to record it:

Time-domain plot

enter image description here

Frequency-domain plot (Spectrogram)

enter image description here

Zooming in on the first few seconds:

enter image description here

Download the signal as audio

Unfortunately it hasn't reappeared since. I want to ask one of my co-workers to scan the source of the signal with the big dish, but I've forgotten which satellite the antenna was pointed at when I recorded the signal! Can you help me figure out where in the sky the signal originated?

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    $\begingroup$ @2012rcampion The links are all dead. The Youtube video is still being processed and the links to the sound files return "404 Not found" errors. $\endgroup$ – jarnbjo Jun 29 '15 at 9:23
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    $\begingroup$ @Beta I don't see any prime numbers, but one of the interns pointed out to me that the frequency of the message was the same as the signal depicted in the movie: pi times the frequency of the hyperfine transition of the hydrogen atom's ground state. That's why I think it might be aliens. $\endgroup$ – 2012rcampion Jun 29 '15 at 16:37
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    $\begingroup$ I find it interesting that this is tagged math... $\endgroup$ – Bobson Jun 30 '15 at 17:34
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    $\begingroup$ I am really impressed by this puzzle. Way above my head, but impressively deep. (To use two aspects of the same metaphor) $\endgroup$ – Bobson Jul 6 '15 at 18:54
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    $\begingroup$ This was an outstanding puzzle! $\endgroup$ – Stephen Donecker Jul 16 '15 at 22:02
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+350
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Solution

Gliese 785

  • Right ascension 20 15 17

  • Declination -27 01 58

Alien Message Translation

enter image description here click to enlarge

Explanation

Page 1 The purpose of this page is to teach us the alien number system. They use a base 3 number system that we call balanced ternary with the '+' symbol defined as +1, the '-' symbol as 0, and the 'x' symbol as -1. In addition they define the '>' symbol as the radix point. In the examples they show us how to represent the numbers 1 through 9.

Page 2 Here they move on to teach us the concepts of addition/subtraction, multiplication/division, and equivalency. In each of the examples the symbols that represent the various operations are defined and can be seen clearly in the above image.

Page 3 Now they demonstrate the properties of the their number system and how it represents decimals. They begin by showing that shift operations in balance ternary are achieved by multiplying by 3 to shift left and dividing by 3 to shift right. They then demonstrate a new symbol 'O' when drawn around any set of digits defines a repeating group identical to bar notation here. They also introduce a 'o' symbol when placed at the end of an equation it means continue the series infinitely.

Page 4 They move on from basic number representation and operations to define the concept of and the symbol representing Pi. They introduce a new symbol '|' that was previously introduced in page two as '>|' to mean equals. Or rather in the previous use it was customary to write equals without a space between other symbols. At the top of the page they appear to be introducing additional equation concepts that utilize perpendicular calculations and back to back operators. I assume that some sort of order of operations and/or the equivalent of parenthesis is implied. Clearly this looks like some sort of series expansion. We see the familiar 1, -1/3, +1/5, -1/7 ... expansion used to calculate Pi.

Page 5 Similarly to the previous page, they derive and define the constant 'e'. They also define the concept and syntax of exponents with two examples.

Page 6 Here they give models representing Carbon, Oxygen, Hydrogen, and Helium atoms followed by a number in alien units. We define this unit as the Alien Mass Unit (AMU). By referencing the various properties of these atoms you can quickly see that they are defining the mass of each atom in terms of the mass of an electron (me).

1 AMU = 9.10938291E-31 kg

Page 7 In this page they show a model of a water molecule with distinguishing characteristics such as the number of protons and the angle between the two hydrogen atoms. The aliens also define an length unit in terms of the bond distance between the oxygen and hydrogen atoms. We define this unit as the Alien Length Unit (ALU). Referencing various materials we can see that the radius of the hydrogen atom is well defined as the Bohr Radius which is roughly 53 pm, and that the O-H bond distance is 95.7854 pm.

1 ALU = 52.9177211 pm

Page 8 Now they show a model of the alien radio transmitter. We translated the antenna diameter, and converted the ALU units to meters and determined that the diameter of the antenna is 2197 m. The number on the lower right of the diagram represents the transmission frequency in alien units and the number in the center of the diagram represents the transmission power also in alien units. We define these units as the Alien Frequency Unit (AFU) and Alien Power Unit (APU) respectively. Since our receive frequency was 4460MHz we can calculate AFU.

1 AFU = 1.170960 Hz

Given that the receive antenna is 10m with a receiver sensitivity of -130dBm we can calculate a reasonable maximum transmission distance. A 10m dish antenna with an efficiency of 0.7 will have a maximum of 50dB gain and the alien dish antenna will have a maximum of 99dB gain at 4460MHz. So the maximum free space propagation can be determined given these quantities and the signal to noise ratio (SNR) of the received signal in addition to the alien transmit power. However the power is currently unknown. But we can determine a reasonable estimate somewhere between 60 and 90dBm. Using these values we can safely estimate a maximum transmission distance of ~200 ly.

Page 9 In this page they depict the aliens planetary system with numbers that represent the distances from the center of their star. The distance units will be determined in later pages.

Page 10 Now they specify the mass and radius of planets 1, 2, 4, and 5 as well as the mass of each moon and distance to the related planet. The mass and radius are in currently unknown alien units which we will define as Alien Planetary Mass Unit (APMU) and Alien Planetary Radius Unit (APRU) respectively. The distance units will be determined in later pages.

Page 11 In this page they continue from page ten with the third planet and its associated moon. The planet and moons are specified in units of APMU, APRU. However when looking at the numbers for the fourth moon of the third planet we see that they have defined an equation. The two equations define the APMU and APRU units.

1 APMU = 1.454E+53 AMU

1 APRU = 4.880E+16 ALU

Again the distance units will be determined in later pages.

Page 12 In this final page of the alien message they show us something unexpected. We see that the aliens have specified a relative distance from their system to another star with additional orbiting planets. Looking at this and the last few pages we conclude that the unit of distance between the moons and the associated planets, as well as, the unit of distance between the planets and the associated stars is in units of APRU. Finally we notice that the aliens use a different unit for specifying the distance between the two stars. As is the case here on Earth we define our unit of distance AU to be the distance between Earth and the Sun. Here we notice that the aliens chose to define their unit to represent the distance between their moon and it's associated planet. We define this unit as the Alien Astronomical Unit (AAU).

1 AAU = 301 APRU

Calculations

star radius (sun)

7.7191e-01, 1.0020e+00

star mass (sun)

7.3138e-01, 1.0027e+00

system distance (au)

1.8251e+06

system 1 planet distance (au)

1.7197e-01, 3.1877e-01, 1.1828e+00, 2.4915e+00, 4.6432e+00

system 1 planet mass (earth)

5.5857e-02, 1.7033e+01, 2.4707e+01, 9.0710e+00, 4.0661e-01

system 1 planet radius (earth)

3.8028e-01, 7.0252e+00, 4.3683e+00, 3.2424e+00, 7.6557e-01

system 1 planet 2 moon distance (au)

2.1058e-03

system 1 planet 3 moon distance (au)

4.4303e-04, 1.1047e-03, 3.1587e-03, 5.1955e-03, 9.6315e-03

system 1 planet 4 moon distance (au)

6.7317e-04, 9.3208e-04

system 1 planet 5 moon distance (au)

3.9700e-04

system 1 planet 2 moon mass (earth)

7.4030e-04

system 1 planet 3 moon mass (earth)

9.3203e-07, 3.1099e-04, 3.8536e-04, 2.2179e-02, 1.8780e-05

system 1 planet 4 moon mass (earth)

3.7184e-05, 1.4874e-04

system 1 planet 5 moon mass (earth)

9.9157e-05

system 2 planet distance (au)

5.2094e+00, 9.5129e+00

system 2 planet mass (earth)

3.1797e+02, 9.5213e+01

Search

Analyzing the above calculations, we recognize that the planets and star in the second system match Jupiter, Saturn, and our Sun exactly. Now searching through the various databases for systems around 28.86 ly away, I found Gliese 785 matches the specifications of the alien system star, planet 2, and planet 3 exactly. Therefore we can conclude that this is the origin of the alien signal transmission. Clearly we are currently unaware of the alien systems other plants and moons. I foresee future space missions.

Here are two of the more helpful online exoplanet databases.

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  • $\begingroup$ Even if we assume their transmitter power is something crazy like 1 GW, 3000 ly seems a little bit far for a 2200 m antenna... they'd need an AU-sized phased array. $\endgroup$ – 2012rcampion Jul 13 '15 at 0:59
  • $\begingroup$ I will have to go back and check my numbers tomorrow but I recall getting GW and GHz numbers for the transmitter. $\endgroup$ – Stephen Donecker Jul 13 '15 at 1:41
  • $\begingroup$ Can you provide the receive antenna size/gain, and noise figure of the reciever? $\endgroup$ – Stephen Donecker Jul 13 '15 at 1:44
  • $\begingroup$ I do remember which antenna I used... let me double-check that it's OK for me to share. $\endgroup$ – 2012rcampion Jul 13 '15 at 2:23
  • $\begingroup$ I recorded the signal on our 10-meter antenna, whose receiver has a -130 dBm noise floor. $\endgroup$ – 2012rcampion Jul 13 '15 at 2:33
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Update:

The aliens are using a rather nifty number system that is similar to base 3, but with digits corresponding to zero and ±1. Here's a bit of Python that will convert them to decimal form:

def a3(s):
  r,p,dp = 0,0,0
  for c in s:
    if c not in 'x-+>o':
      raise LookupError("Illegal symbol ('" + c + "')")
    if c in 'x-+':
      r = r * 3 + 'x-+'.find(c) - 1
      p = p + dp
    if c == '>':
      dp = 1
  if p == 0:
    return r
  return r * 1.0 / (3**p)

Examples: a3('++-x>') = 35, a3('+>+--x+') = 1.325

The numbers that intersect at a triangle enclosed in a circle are a form of scientific notation. From the example on page 5: a3('+>xx+--x') * 3**a3('xx>') = a3('>---+xx+--x') = 0.0073. Recurring digits are enclosed in rounded rectangles as shown in page 3, along with the identity +>(x) = >(+) (both are equal to 0.11111… in base 3, or 0.5 in decimal).

Pages 4 and 5 describe the mathematical constants $\pi$ and $e$. Page 6 shows the atomic structure of C, O, H and He, and their respective atomic masses relative to the mass of an electron. Page 7 shows a water molecule, including the bend angle of 104.45° (expressed in radians), and the distance between the H and O nuclei (in units of the Bohr radius).

Page 8 seems to be describing the transmitter that sent the signal. The number on the left is equal to $\alpha^{-1}$ (the reciprocal of the fine structure constant). The number at the top is perhaps the size of the transmitter in Bohr radius units.

Page 9 appears to show the planets orbiting a star. I assume the numbers at the bottom are the mean orbital distances of the planets, but I have no idea what distance units are being used here. Apparently our alien friend inhabits one of the moons of a ringed gas giant.

Annotated message, pages 1-9
Click to embiggen

The last three pages may have more information about the planets in this system, but I haven't made much progress there yet.


Original answer:

Except for the "synchronization pulses", each part of the signal contains two superimposed sine wave signals at different frequencies. If these two frequencies are used as the left and right inputs to an Etch-A-Sketch, then they trace out some rather interesting pictures, with the synchronization pulses marking the boundaries between them. The only trouble is that a 1024-element FFT results in pictures of rather low resolution:

A little green message from a little green man?

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  • $\begingroup$ I know there exist methods that allow you to interpolate between samples in an FFT with an appropriate windowing function (see for example this paper, in particular (21)); and a PLL will give you the best tracking, but they're hard to develop. Let me pass your idea on to one of our analysts, she should be able to produce a pretty clean picture. $\endgroup$ – 2012rcampion Jul 2 '15 at 0:50
  • $\begingroup$ Probably not related to the Voyager probes though, since they've barely left the solar system proper. Great animation by the way! $\endgroup$ – 2012rcampion Jul 2 '15 at 0:51
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    $\begingroup$ <out-of-character> Looks like I made a typo... 16 == 5 should be 5 == 5. I sort of expected one to slip through since I copied all the values by hand, but how did I mess up that one?! I double-checked again and all the other values look OK. </out-of-character> $\endgroup$ – 2012rcampion Jul 2 '15 at 18:43
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    $\begingroup$ The alien numerical system is known as en.wikipedia.org/wiki/Balanced_ternary $\endgroup$ – March Ho Jul 2 '15 at 21:42
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    $\begingroup$ Another interpretation of page 9 is that the circled planet is where they're from, and they're transmitting from one of several moons. Minor difference, though. Page 11 may be a more detailed description of that particular planet & moons, judging from the double-ringed shape, with four in-line dots and one out-of-line, which would correspond to the 4+1 moons. $\endgroup$ – Bobson Jul 6 '15 at 19:03
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This is also very much a work in progress, but ...

Do these figures make any sense to you?

enter image description here

.

enter image description here

.

enter image description here

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  • $\begingroup$ <out-of-character> I think you're on the right track =) </out-of-character> $\endgroup$ – 2012rcampion Jul 1 '15 at 14:27
  • $\begingroup$ Is it just a coincidence that the signal decodes to images looking like stylistic animals (especially the last two) when using an otherwise obviously too low resolution in the time domain? I used a plain FFT transform for these images and obviously lost a great deal of detail. $\endgroup$ – jarnbjo Jul 2 '15 at 23:56
  • $\begingroup$ Not a coincidence: by essentially creating structured, semi-random images you stimulate the brain to recognize familiar shapes. <ooc> The resemblance is not intentional though. </ooc> $\endgroup$ – 2012rcampion Jul 3 '15 at 0:23
  • $\begingroup$ My grandfather is Navajo, and I recognize some of those shapes from painted tribal designs. In the ancient stories, they represent a tale of a warrior hunter named Gliese 785. $\endgroup$ – user1717828 Sep 23 '15 at 1:06
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Work in progress.

Based off the work of Quark:

Added labels, powers of 3, what might be the solar system, and H2O. Full size image enter image description here

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Partial Answer

Going off of OP's newest hint:

The second-to-last 'page' seems to contain two nonsensical equations:

= (1) (1.45e53)

= (1) (4.88e16)

The first one corresponds to the mass of the universe in kg

The second when googled (4.88e16), find very few results, the only coherent one (aside from this post) being this link, a new population of planetary nebula discovered which seems to be more than a coincidence. Perhaps the signal came from there.

But if that was true, the entire puzzle would be a build up to a disappointment so I'll chalk that up to coincidence.

So, what we have so far is a measurement in kg, meaning either the 1.45e53 is by chance and not referring to the (average of 3 arbitrary measurements) mass calculation of the universe, or the signal came from Earth, particularly someone who referenced that Wikipedia page. I'll go with Earth for now (since we know who actually created the signal).


Here's a start on decoding the image from the audio signal:

enter image description here


https://en.wikipedia.org/wiki/Wow!_signal

This wiki page has almost too many things in common with this puzzle to be a coincidence, haven't been able to progress further however so posting to give others some ideas.

Edit 1: Image removed to avoid confusion, refer to images posted by OP

Edit 2: Added (probably) useful info source

Edit 3: Added work in progress for image decoding

Edit 4: Added interpretation of OP's hint

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    $\begingroup$ Wow. Now I can totally tell what the signal is. :D $\endgroup$ – COTO Jun 29 '15 at 4:07
  • $\begingroup$ @COTO Heh well at least you can see the (probably not random) spikes in the audio waveform and the dual frequency pattern. I suspect there may be some graphical significance from the frequency of the notes. $\endgroup$ – Quark Jun 29 '15 at 4:11
  • $\begingroup$ I'm just venting my frustration. The YouTube video claims "We're processing this video. Check back later." The embedded files are only accessible via creating a Google account. I'll chew glass before I give those data sucking sons of guns any browsing data I don't have to, hence I was hoping to be able to solve the problem based on your post. But if there's a pattern in there somewhere, I can't see it. ;) I'll be interested to see what the solution is. The signal must have significance as audio. There's no logical reason why a C-band RF signal would be presented as audio otherwise. $\endgroup$ – COTO Jun 29 '15 at 4:25
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    $\begingroup$ @Quark You would have to answer the same questions thy were posed to me for my answer. $\endgroup$ – LeppyR64 Jul 9 '15 at 17:29
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    $\begingroup$ Don't know if it is important, but 1.45e53 electron mass is 1.32e23 kg, about 10 times the mass of Pluto. $\endgroup$ – mmking Jul 10 '15 at 19:56
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This is a "canonical" answer meant to move the hints and partial results out of the question, as well as present a unified description of the solution and explain some of my thoughts about the puzzle.

Warning, unmarked spoilers ahead!

Squeamish ossifrage Found a method that demodulates the signal into images in his answer. Using his technique we can extract a series of 12 'pages,' which are described below. Click on any of the pages to see an "annotated" version with the numbers converted into familiar units.

Page 1 — Counting

This page introduces us to the number system that the message uses, a form of balanced ternary. "Ternary" means it's base-3; instead of ones, tens, and hundreds places, we have ones, threes, and nines. "Ordinary" base-3 uses the digits $0$, $1$, and $2$, but balanced ternary instead uses the digits $-1$, $0$ and $+1$. $\def\T{\mathtt T}\def\Z{\mathtt 0}\def\E{\mathtt 1}$

In the message these digits are represented by "×," "−," and "+," respectively, but in this explanation I'll use the more standard $\T$, $\Z$, and $\E$ instead, to avoid confusion with mathematical operators. The examples given:

$$ \begin{array}{rcrcr} \E &=& {}+1 &=&1 \\ \E\T &=& {}+3-1 &=& 2 \\ \E\Z &=& {}+3\phantom{{}+0} &=& 3 \\ \E\E &=& {}+3+1 &=& 4 \\ \E\T\T &=& {}+9-3-1 &=& 5 \\ \E\T\Z &=& {}+9-3\phantom{{}+0} &=& 6 \\ \E\T\E &=& {}+9-3+1 &=& 7 \\ \E\Z\T &=& {}+9\phantom{{}+0}-1 &=& 8 \\ \E\Z\Z &=& {}+9\phantom{{}+0+0} &=& 9 \\ \end{array} $$

Notice that counting is similar to ordinary base-3; each digit cycles from $\Z$ to $\E$ to $\T$ and back to $\Z$, and when we "roll over" from $\E$ to $\T$, the next digit increments as well.

I chose balanced ternary not just because it's a little unusual, but because it also has a number of cool properties (no pun intended). For example:

  • You can represent negative numbers without a minus sign; take the positive form of the number and replace all $\E$s with $\T$s and vice versa.
  • Addition and subtraction are easy. When adding, a $\E$ and $\T$ cancel, and a pair of the same digit turn into the opposite, with a same-signed carry: $$\begin{array}{r|rrr}+&\T&\Z&\E\\\hline \T&\T\E&\T&\Z\\\Z&\T&\Z&\E\\\E&\Z&\E&\E\T\end{array}$$
  • Multiplication is similarly easy. multiplication by $\E$ leaves the multiplicand unchanged, by $\Z$ yields $0$ of course, and by $\T$ flips the sign of the multiplicand: $$\begin{array}{r|rrr}\times&\T&\Z&\E\\\hline \T&\E&\Z&\T\\\Z&\Z&\Z&\Z\\\E&\T&\Z&\E\end{array}$$ This means that when multiplying two numbers, you can skip writing down the multiplicands and go straight to writing the partial products. For example to multiply $\E\T\Z$ ($6$) by $\E\T\E$ ($7$) we would write: $$\begin{array}{r}\E\phantom{\Z\Z\Z}\\\E\T\Z\\\T\E\Z\phantom\Z\\{}+\E\T\Z\phantom{\Z\Z}\\\hline\E\T\T\T\Z\end{array}$$ Which is, of course, $81-27-9-3=42$.

Page 2 — Arithmetic

This page demonstrates the basic arithmetic operators (equality, addition/subtraction, and multiplication/division) and their notation.

Expressions are represented in prefix notation, instead of the typical infix notation. In infix notation the operator is placed in between its operands, for example $1+1$. In prefix notation, the operator comes before its operands. In prefix notation, $1+1$ might be written ${}+1\ 1$.

The reason I chose prefix notation is that it doesn't require parenthesis, making the number of symbols I had to invent and explain smaller; similar to how I eliminated the need for a minus sign by choosing balanced ternary. However, I do use parenthesis in the annotated drawings to make things a little easier to read.

Expressions are read in the direction of the triangles. They serve as a graphic indicator of which end of the numbers are most and least significant. This direction indicator allows expressions to be written in any direction (or in multiple directions, as we will see later). I'll call the most significant end (flat end of the triangle) the upstream end, and the least significant end (point of the triangle) the downstream end. Note that the prefix operators are upstream of their operands.

Operands with no operator between them are separated by a small vertical bar, which acts a sort of comma. If the vertical bar comes next to the direction marker, the two are combined together into a single symbol that looks sort of like a "fast-forward" or "skip" button.

The operators themselves are based on diagrams of the operations they represent: comparing two things side-by-side for equality, placing two things end-to-end for addition, and forming an area from two perpendicular lengths for multiplication.

The block-like depiction of the example operations are probably inspired by the manipulatives we used in elementary school math.

Addition and multiplication both have three parts: the two addends (or augend and addend) and the sum, and the multiplier, multiplicand, and product. However, their operators only have two arguments or inputs. Which roles the arguments play are determined by the two parts of the operator that are shown; the missing third part is the value of the whole expression. For example:

  • Addition is represented by the two addends: two short bars placed end-to-end.
  • Subtraction is represented by a sum and addend (minuend and subtrahend): one long bar next two one short bar.
  • Multiplication is represented by the multiplier and multiplicand: two bars at right angles.
  • Division is represented by the product and multiplicand (dividend and divisor): a rectangle with a bar on its side.
  • Equality has no "output" value, so it is always represented by both its left-hand and right-hand sides: two bars of equal length side-by-side.

The example for equality has the first mistake of this puzzle: the example equation reads $16=5$ instead of $5=5$ due to an extra $\E$.

One final detail: the order of the parts in the operator corresponds to their order in the expression. $a-b$ is represented by 'long bar, short bar, $a$, $b$.' But if we want to switch the order around and do $b$-$a$, we could represent it without changing the order of the operands as 'short bar, long bar, $a$, $b$.'

Page 3 — Decimals

Here we see that the triangle functions not only as a direction indicator, but also as a decimal point. We also introduce the circle as a symbol for repetition. A small circle at the end of a number it functions like ellipsis, while a circle or oval around a set of digits denotes a repeating decimal (like a vinculum or overbar).

Probably the most interesting detail on this page is that one number is shown to have two repeating decimal expansions! This is the balanced-ternary equivalent to the expression $0.999\ldots=1$.

Page 4 — $\pi$

This page shows the geometric meaning of $\pi$, its approximate value, and one continued fraction expansion for it. The expression is two-dimensional, with the dividends branching off at the division operators. In (linear) prefix notation the expression is:

= π + 3 ÷ × 1 1 + 6 ÷ × 3 3 + 6 ÷ × 5 5 + 6 ÷ × 7 7 + 6 ...

This is equivalent in standard notation to:

$$ \pi = 3 + \cfrac{1\times 1}{6+\cfrac{3\times 3}{6+\cfrac{5\times 5}{6+\cfrac{7\times 7}{6+\cdots}}}} $$

Note the explicit multiplication, since we haven't introduced a notation for exponentiation (squaring).

Looking back, I wish I had used $\tau$ (tau) instead of $\pi$ here.

Page 5 — $e$ and Scientific Notation

Like the previous page, this one shows a geometric definition of $e$, its approximate value, and one continued fraction expansion for it. In prefix notation the expression is:

= e + 2 ÷ 1 + 1 ÷ 1 + 2 ÷ 2 + 3 ÷ 3 + 4 ÷ 4 ...

This is equivalent in standard notation to:

$$ e = 2 + \cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4+\cfrac{4}{\cdots}}}}} $$

We also see a form of scientific notation. Repeated zeros are "pulled" into the decimal point; to denote this, the decimal point is circled and labeled with the number of decimal placed moved (the exponent). Remember these are base-three decimal places, so increasing the exponent by one increases the value by three times, not ten!

Page 6 — Atoms

Here four atoms are depicted, along with one lone electron. The electron is labeled $1$. The atom with one nucleon and one orbiting electron, hydrogen, is labeled $1837$. This is the ratio in mass between a proton and electron, so the labels are the masses of the atoms in electron mass units. Note that the mass of the carbon atom is slightly different from the value you find in the periodic table: that value is a weighted average of the different isotopes of carbon that occur on Earth, while the value given on this page is the weight for pure carbon-12 (the isotope depicted). The values for hydrogen, helium, and oxygen will be the same as in the periodic table because they are all dominated by one isotope on Earth.

I chose these specific elements because they represent the main stages in the stellar evolution of a Sun-like star: hydrogen burning on the main sequence produces helium, helium burning forms carbon as the star transitions into its red giant phase, and the carbon fuses into oxygen as the star contracts into a white dwarf.

There is also a line next to the hydrogen atom equal in length to the radius of the electron's circle; this line is also labeled $1$. This indicates that the distance unit is the Bohr radius $a_0$, in some sense the "radius" of that electron's orbital.

If you look carefully you will notice that the electrons are labeled with $\E$ and the protons with $\T$, a sign convention opposite to ours. This probably makes electronics a little easier for these aliens!

Page 7 — Water

This page shows a water molecule. The atoms, hydrogen and oxygen, are labeled by their number of electrons. The atomic weight is given in electron mass units; the bond length is given in Bohr radii; and the bond angle is given in radians.

Labeling by the number of protons probably would have made more sense, since the electrons move around when atoms are bonded; but since the electrons are represented as positive, the aliens probably think of them as the more important part of the atom.
I initially considered putting an ammonia molecule here, but decided to make it a little more obvious for the puzzle-solvers.

Page 8 — Transmitter

A parabolic reflector is depicted on this page. The diameter of the dish and the wavelength are labeled in Bohr radii. The wavelength, 21.16 cm, is the wavelength of the hydrogen line, equivalent to a frequency of 1420 MHz, theorized to be an important frequency for SETI.

I thought about making the frequency "pi times hydrogen," or 4462 MHz as a reference to Contact, but eventually decided against it.

The number to the right along the transmitted beam is the speed of light, given in atomic units. In atomic units, $c$ is dimensionless and equal to my favorite number: $1/\alpha$, the inverse of the fine-structure constant.

Finally, the number at the focus of the transmitter is the transmitting power (again, in atomic units).

Page 9 — The Solar System

This page gives an overview of the solar system the aliens are transmitting from. From left to right, we have:

  • The star;
  • A small Mercury-like planet;
  • Two hot, Neptune-like gas giants:
    • One with one moon,
    • and another with a ring system and five moons;
  • A cool, rocky super-Earth with two moons;
  • And a cold, rocky planet with one moon.

The distances from the star are given in some as-yet unspecified planetary distance units. The large fourth moon of the second gas giant is shown transmitting; it's (presumably) the home of the aliens who sent the signal.

Designing the solar system was probably the most fun part of this puzzle. Other than the star and the two gas giants, everything is made up (including the rings).

Page 10 — Uninhabited Planets

The characteristics (mass, radius, and orbital distance) of the first two and last two planets and their moons are described on this page. They are given in the same unspecified distance and mass units.

The second mistake in this puzzle is an extra $\T$ in the orbital radius of the gas giant's moon. The correct value is given in the annotated image.

Page 11 — Inhabited Planetary System

The characteristics of the ringed gas giant and its system of moons are described on this page. Note that the first two distances near the planet are supposed to be the inner and outer radii of its rings. One of the moons (the large one shown transmitting on page 9) has its characteristics described by two equations:

$$ 1 = 1.45\times 10^{53} \\ 1 = 4.88\times 10^{16} $$

These give the values of the planetary mass and radius units in terms of the electron mass units and Bohr radius units used in the earlier part of the message.

Page 12 — Stars

The final page describes two stars, one of which has two planets depicted. The stars' and planets' characteristics are given in the same planetary units as before. Converting to our units, we can see that the second system (mass and radius of the star, and the mass and orbital radii of the planets) exactly match the Sun, Jupiter and Saturn.

The third error in this puzzle is that the masses for Jupiter and Saturn are swapped. The corrected values are in the annotated image.

The distance between the stars is also given; but it is not in the planetary distance units: it is in units of the orbital radius of the aliens' moon!

This distance was originally supposed to be in terms of the radius of the aliens' star. I thought that this had a nice symmetry: distances between objects are measured in terms of a "standard" object of that type. However, I mistakenly used the wrong unit in my calculations and ended up with what you see in the puzzle.

Conclusion

Using the information from the message, the only star that matches is Gliese 785.

$\endgroup$

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