misinterpretation.
After that came the symbols of common arithmetic, with examples showing how to add, subtract, multiply, and divide. From there it was a short step to negative numbers, fractions, powers, and irrationals. Imaginaries you would introduce using fractional powers of negative numbers. Then on to series of powers, and the elementary transcendental functions such as sines, cosines, logarithms, and exponents. In every case you would give enough examples to be sure there was no confusion. After providing series expressions for the universal transcendentals such as ? or e, you would provide a check that all was being interpreted correctly by quoting one of mathematics’ enduring wonders, a formula that mysteriously links transcendental and imaginary numbers with the basic numerical building blocks of 1 and 0:
e i? + 1=0.
Mathematics was easy, the obvious way to start. After that, Milly would proceed to astronomy, physics, chemistry, and, finally and most difficult, language.
The trouble, of course, was that it was not up to Milly. She was not sending a message. She was receiving one. The difference, in terms of self-confidence, was the difference between being a doctor and being a patient.
The good news was that she was not working alone. People as smart as she, and probably a whole lot smarter, were her allies. The displays in front of her provided an overview of the whole signal in schematic form, subdivided into twenty-seven regions.
Using her console to control the rate at which she advanced, Milly set out to scan the entire length of the signal. The Puzzle Network team had worked cooperatively to attach their analyses to the appropriate regions. The result was like a gigantic snake, of which the string of digits of the signal itself formed a narrow backbone. Here and there, in places where something particularly interesting and significant had been discovered, the snake bulged out like a python that had swallowed a pig.
Milly backed the scan to study Section 7, the fourth bulge, which at first sight was bigger than all the others. The comments were offered in ordered bunches:
Attoboy: The structure here is odd. High entropy sequences of average length 106 digits are regularly interspersed with low entropy regions each of constant length 3.3554 X 107 digits. Any thoughts?
Sneak Attack: Yes. We could be seeing sections of “text” (variable but roughly equal lengths) that introduce or describe a “picture” (something in image format, with a constant array size). Maybe square arrays of black and white images, each about 6,000 X 6,000 elements?
Claudius: More likely, a gray scale image 4,096 X 4,096 (212 X 212 — that supports the notion of binary representations), with 2 bits (4 levels) for each pixel. That fits with the exact size of the low entropy regions, 33,554,432 bits.
Sneak Attack: Could just as well be 2,048 X 2,048, with 256 (8-bit) gray levels.
Claudius: Should be easy enough to find out which. If we assume a particular line length and do cross- correlations of successive lines, the correct line length should jump right out at us when we get to it, because the correlation will be a lot higher. Let me take a look.
That was all for that cluster of messages. Presumably Claudius did not yet have an answer, or at least not one that “jumped right out.” Milly moved on.
The seventh bulge along the signal’s spine, Section 12, contained remarks similar to the previous one, except for three added comments:
Megachirops: In this case the low entropy regions have a constant length of 4,194,304 bits, exactly one- eighth as long as in Section 7. Does anyone else find this somewhat surprising?
Ghost Boy: We would probably make them all the same sire. The difference may be part of the message, trying to tell us something.
Claudius: Or could these be line drawings? — binary images, black and white with no shades of gray.
The ninth bulge supported a hypothesis offered early in the history of SETI:
The Joker: My frequency analysis of this section suggests that we are dealing with base 4 arithmetic, rather than the base 2 binary we have seen elsewhere. The temptation to interpret this as a biological description in terms of strings of four nucleotides is strong.
Attoboy: Beware of anthropomorphism. But I agree, the temptation is strong. I will try to correlate this section with everything in the genome library.
Not surprisingly, Attoboy had not yet reported results from that effort. The task was a monstrous one. The library to be examined held complete genomes for more than two million species, everything from humans and oak trees and mushrooms to the smallest and simplest viruses. No one, no matter how optimistic, would hope for an exact match. It would be a miracle (and enormously relevant to the universal nature of life) if anything correlated at all with a living creature from Earth. But Attoboy was right, you couldn’t afford not to look.
Milly worked her way on through the signal, section by section. The exercise was giving her a strong inferiority complex. The results that she was seeing had been performed so quickly, and offered such powerful evidence of ingenuity — what could she possibly contribute? The team had already established the existence of unique start and stop sequences, each fourteen bits long. Numerical base and reading order were known beyond doubt: integers were base 2 and base 4, with the most significant digit to the right. Sequences of primes and squares and cubes had been discovered, more than long enough to be unambiguous.
When she came to the very end of the signal, with its termination as a repeated pattern of the fourteen-bit stop sequence, Milly at once went back to the beginning and started over. The easy part, following the trail that others had already marked out, was over. Now she had to do something to justify her own presence in the group. Sit, observe, learn, keep quiet; that was all very well — for the first half-day. After that, Milly hoped to bring her own special knowledge and experience to bear. She went to a section near the middle of the signal, where analysis and comments by members of the Puzzle Network were meager and felt tentative. This was a place with special significance for Milly. It was here where she first noticed the oddity that had evolved into the Wu-Beston anomaly, and she had studied it extensively.
Something she had brought with her from Argus Station, more important than clothing or personal effects, was her own suite of processing programs. She had no illusions that they were better than anyone else’s; what she was sure was that they were different. Also, they were hers, and she knew them inside-out.
She began her analysis. It was similar to what she had attempted months ago, with one crucial difference: she could now build on everything established or conjectured by the Puzzle Network group. The start-stop coding sequence was known. She was sure of the integers. Perhaps most important of all, she knew that what she was dealing with was a signal. Puzzles always become easier when you know that a solution exists.
The section that she clipped out for inspection was only a small section of the whole, roughly a hundred million digits out of twenty-one billion. You could eat that up very quickly with images, but she had deliberately avoided low-entropy data runs. What she hoped to find was “text” — whatever that term might mean to an alien mind. It was too early in the game to hope for keys to an actual language.
After the first few minutes, Milly entered the twilight zone. Her mind became a place where symbols took on their own life and formed their own relationships. The signal contained dozens of these, short and well-defined strings that had been identified by other workers as common repeat patterns, but were not yet understood as to meaning. Sometimes the unknown strings appeared close to known integers, sometimes they coupled only to other unknowns. At this stage of understanding, the “knowns” lay within a great quagmire of uncertainties. The trick — if a trick existed — was to stand on a firm starting-point, something you definitely understood, then discover a sequence that allowed you to scramble out along it to reach another point of understanding.
Milly worked on, oblivious to where she was or how long she had been sitting there, until she found her attention returning, again and again, to a sequence containing only a few tens of thousands of digits. She had culled it piece by piece from the sea of a hundred million bits, without knowing at a conscious level what she accepted and what she rejected.
What made these samples different from any random selections? They seemed like a meaningless mixture. The pattern, if it could be called a pattern, comprised sets of small integers, always separated from each other by a
