Today was a special day: the day Captain Mirov, a man of legendary renown, had designated for opening the envelope containing the Maya Notes. Exactly 52 years had passed since 1979.
Irene, who had given the Maya Notes to Michael, had been thinking about it too. But she called so early that Michael and Daphne were still in bed.
"Irene, thanks for the reminder. We've marked the date, and we'll open the envelope today. Can you come over and look at it with us?" Michael asked.
"I can't. I'm in Country B, with the Atlantic Ocean between us. I don't know when I'll be able to return yet. It depends on my husband Andek's condition."
Daphne, listening beside Michael, asked urgently, "How is Mr. Andek doing? I hope he recovers soon."
Rocket City lay in a low-latitude region of Bright Nation. It was early autumn, but the temperature was still high. The sun had barely risen before hot, humid air came rolling in from the southeast, sweeping across the endless tidal flats from the sea.
After washing up and finishing breakfast, Michael and Daphne faced the large, dusty, battered envelope, its seal still intact. Curious and excited, they were both eager to learn what secrets Captain Mirov's Maya Notes contained.
Daphne tiptoed to the bedroom door. Through the crack, she saw little Vito sleeping peacefully in his crib.
When Daphne returned to the worktable, Michael had already opened the envelope with a paper knife. Inside were a red notebook and another, smaller sealed envelope. The notebook's cover was written in Russian, and Daphne recognized the words as meaning Maya Notes.
Michael fetched an external device and connected it to the computer. It looked like a desk lamp, but was actually a high-speed overhead scanner.
Daphne opened the notebook and placed it beneath the scanner, quickly turning the pages one by one. The scans were automatically translated and archived. Then the two of them began reading on the computer.
"Hello, friend from the future! I'm Mirov. This is 1979. By the time you read this, it should be 2031—52 years from now."
"You must be wondering why I asked for this envelope to be opened 52 years later. It's not just because I believe human technology will make tremendous leaps over the next few decades. If that were the only reason, I could have said 50 or 53 years."
"Fifty-two years has a special meaning. It comes from the Maya Calendar—or, more precisely, 52 periods of 365 days, totaling 18,980 days. You may find this hard to believe, but the Maya marked every one of those 18,980 days with a different symbol, creating a unique calendar they called the Short Calendar."
At that, Daphne said to Michael, "I know a little about the Maya Short Calendar. The 365-day cycle is called the Haab' calendar, and the 260-day cycle is called the Tzolk'in calendar. Their least common multiple is 18,980 days: 52 times 365, or 73 times 260."
"Three hundred and sixty-five days is clearly the time Earth takes to orbit the sun," Michael said. "What cycle does 260 days represent?"
As if answering Michael's question, the notebook continued: "Almost every civilization in the world has based its calendar on astronomical cycles, and the Maya were no exception. Their calendars varied in length, but each corresponded to an astronomical cycle."
"Besides 365 and 260 days, the Maya also used 584 days, the synodic period of Venus and Earth. That makes it easy to explain what 260 days means: Mars's synodic period with Earth is 780 days, and 260 is one-third of that."
"Earth's orbital period is 365 days, while the synodic periods of Venus and Mars with Earth are 584 and 780 days. The least common multiple of these three numbers is 113,880 days, or 312 periods of 365 days. The Maya may have thought that 312 years was too long for a Short Calendar, so they took half of Venus's synodic period and a third of Mars's, arriving at 52 periods of 365 days, or 18,980 days."
"Right," Daphne said. "18,980 is divisible by 365; twice that is divisible by 584, and three times that is divisible by 780."
"I've noticed that 18,980 days—the least common multiple of an Earth year and the synodic periods of the two inner and outer planets—seems special. It can be written in a symmetrical mathematical form: 18,980 = 3^9 − 3^6 + 3^3 − 3^0."
"That's ternary," Daphne said. "In ternary, 18,980 is written as (22200222)₃; the 3 outside the parentheses indicates base three. That's so symmetrical! The Maya used base twenty, so where did ternary come from?"
The notebook seemed to answer: "Did the Maya consider numbers with symmetrical ternary representations beautiful? I found evidence of this at the Maya pyramid ruins in the ancient city of Chichen Itza, Mexico."
"The pyramid has four sides, each with 91 steps, for a total of 364. In ternary, 364 = (111111)₃. Six ones—now that's symmetry!"
"There are nine tiers on each side of the pyramid's steps, and 91 × 9 = 819. The Maya Calendar is truly fascinating: 819 days was also an important date in their calendar. 819 = 3^6 + 3^4 + 3^2 = (1010100)₃."
"The Maya's astronomical observations are astonishing. They calculated Earth's orbital period as 365.242 days. Their estimate of Mercury's synodic period was 117 days, a day and a half off the actual value, but their calculations for the synodic periods of Venus, Mars, Jupiter, and Saturn with Earth were extremely accurate."
"As I said earlier, the synodic periods of the two planets inside and outside Earth's orbit—Venus and Mars—are 584 and 780 days. Their least common multiple with 365 days is 113,880. But if you also include Mercury's 117-day synodic period and Saturn's 378-day period, the least common multiple of the five numbers is no longer 113,880."
"Amazingly, the least common multiple of the pyramid's 819 days and 113,880 days is 7,174,440 days. That number happens to be the least common multiple of Mercury's 117 days, Venus's 584, Earth's 365, Mars's 780, and Saturn's 378."
"Even more incredibly, 7,174,440 in ternary can be written as twelve ones followed by three zeros: 7,174,440 = (111111111111000)₃."
"If you think the numbers so far are too large, the Maya actually derived two numbers from the synodic periods of Venus and Mars: 8 and 13. They noticed a coincidence: 365, the number of days in an Earth year, and 584, Venus's synodic period, have a common divisor of 73. Since 365 = 73 × 5 and 584 = 73 × 8, they considered 8 Venus's contribution and carved this coincidence into stone."
"Mars's synodic period is 780 days, and 780 = 20 × 3 × 13. The Maya used base twenty, and in both the Short Calendar and Long Calendar, 13 was one of the numbers they considered magical. They believed 13 was Mars's contribution."
"In ternary, 8 and 13 also happen to be numbers with symmetrical beauty: 8 = (22)₃, and 13 = (111)₃. Ternary has only three digits, 0, 1, and 2. If you write 2 as −1 and represent ternary using 1, 0, and −1, it's called Symmetric Ternary. By convention, you can also write −1 as the letter T, making 8 = (10T)₃."
Michael was utterly bewildered. But when he read "1, 0, −1," he seemed to understand Mirov's point. "Ternary computers. In 2024, researchers at Jinshengzhou University used 1, 0, and −1 to simplify binary calculations in large AI models, increasing computing speed fivefold and reducing power consumption to just 13 watts—close to the human brain's."
The notebook seemed to follow Michael's train of thought, continuing: "Studying the Maya Calendar and the many numerical coincidences in planetary synodic periods made me realize how simple and elegant ternary is. Binary has only two states—'yes' and 'no'—while ternary adds a third: 'yes,' 'no,' and 'uncertain.' That is clearly more in line with how the human brain thinks."
"In 1955, countries began developing computers. Their hardware relied on vacuum tubes in circuits, and the dominant design used the on and off states of diodes to represent 0 and 1, performing binary calculations. At my suggestion, Soviet scientists used transistors, with positive charge representing 1, negative charge representing −1, and no charge representing 0, to develop ternary computers."
Daphne exclaimed, "I know! The Former Soviet Union made ternary computers and sold nearly a hundred to Europe. They were fast and energy-efficient. Why did they give up?"
The notebook answered Daphne's question: "In 1979, the year I'm writing these notes, the Soviet ternary computer project was officially canceled. There were three reasons. First, the country didn't have enough money. Second, it didn't have a complete industrial supply chain. Third—and most importantly—Bright Nation and Europe weren't imposing sanctions on us, so we could always spend money to buy their binary-based computing equipment. That was cheaper and easier than starting from scratch and independently developing ternary computers."
"I'm writing these notes, giving them to Irene, and asking for them to be opened 52 years later because I believe that by then, binary-based computers will have hit major bottlenecks as their computing speeds grow exponentially. One bottleneck will be the conflict with the power supply. The other will be that binary doesn't fit the way the human brain thinks when computers try to imitate it."
"I suspect that even after reading my notes and realizing that ternary computers have clear advantages over binary, you may not be willing to abandon the hardware and software achievements you've accumulated over the years and start over with ternary. So please take these notes as a well-intentioned warning."
Michael Max sank into deep thought. Though ternary computers would theoretically be faster and more energy-efficient than binary ones, who among the vested interests across the Western world's industrial supply chains, led by Bright Nation, would be willing to start over?
Clever Daphne smiled at Michael. "The bigwigs in electronic information technology will probably band together to resist ternary, won't they?"
"Jinshengzhou University has significantly improved computing speeds and reduced energy consumption just by using Symmetric Ternary to improve algorithms. Imagine what it would mean to switch the underlying computing architecture to ternary. If ternary computers are a total revolution, then threatening vested interests will be unavoidable."
Michael smiled too. "I don't know what other people think, but I won't resist any advanced technology. In fact, as far as I know, many companies, including ours, have been researching ternary components and algorithms for a long time. After reading these notes, I'll speed up the development of ternary computers and apply them to my Brain-Computer Fusion Project."
Daphne picked up the smaller, still-sealed envelope. She scanned the words on it: "Jupiter and the Long Calendar," followed by a line of small text: "If you think ternary computers are unnecessary, then you needn't open this little envelope. A more complex number system will be even less necessary for you."
Daphne was curious and was about to open it when Michael stopped her. "That's enough information for one day. Let's digest it first and look at this another time."
Daphne stopped, but couldn't help asking, "Jupiter? The Short Calendar didn't include Jupiter's synodic period with Earth. That period is 399 days. Let's guess: what kind of thoughts will the number 399 inspire in Mirov?"
"The Maya used base twenty," Michael said. "399 = 19 × 21. In base twenty, '21' is written as 11, while '19' is written as 1T."
Daphne seemed to catch on and blurted out, "399 = 20² − 1 = (19.19)₂₀ = (10T)₂₀, which is exactly 1, 0, and −1 in base twenty!"
Poem composed from collected lines:
Deeply carved upon a short stele, bright. — Shen Zhou, Ming dynasty
Here once I found proof of my former self. — Shi Shu, Song dynasty
Even clever reckoning cannot compute it. — Chao Gong-su, Song dynasty
I take up a book and glimpse the ancients. — Qiang Zhi, Song dynasty
Before you continue