Symmetric Ternary (Short Count 2)

August 22, 2031, Rocket City, Liang Nation

Today is a special day, the day designated by the legendary Captain Mirov to open the Maya Notes envelope, exactly 52 years after 1979.

Irene, who had given the Maya Notes to Michael, had also been thinking about this matter. However, she had called too early, and Michael and Daphne were not yet awake.

"Irene, thank you for the reminder. We remember this day and will open the envelope today. Can you come over and watch with us?" Michael asked Irene.

"I can't. I'm in Country B, and the Atlantic Ocean is between us. I can't confirm my return trip yet; it depends on my husband Andek's health."

Daphne, listening to the phone call beside Michael, said eagerly, "How is Mr. Andek's condition? I wish him a speedy recovery."

Rocket City is located in the low-latitude region of Liang Nation. It is early autumn, and the temperature is still high. As soon as the sun peeked out, humid air rolled in from the sea to the southeast, sweeping across the vast tidal flats.

After finishing their morning routines and breakfast, Michael and Daphne, filled with curiosity and excitement, faced the old, worn, but tightly sealed large envelope. Both were eager to discover the secrets held within Captain Mirov's Maya Notes.

Daphne tiptoed to the bedroom door and peeked through the crack. She saw little Vito sleeping soundly in his crib.

When Daphne returned to the workbench, Michael had already opened the envelope with a utility knife. Inside was a red notebook and another small sealed envelope. The cover of the notebook had Russian writing on it, which Daphne recognized 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 under the scanner, flipping through the pages quickly. The scanned pages were automatically translated and archived, and then the couple began reading on the computer.

"Hello, friend from the future! I am Mirov. This year is 1979. When you read this, it should be 2031, 52 years later."

"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 leaps and bounds in a few decades. If that were the only reason, I could have set it for 50 or 53 years."

"52 years has special significance. It's taken from the Maya calendar; specifically, 52 periods of 365 days, totaling 18,980 days. You must find it hard to believe that the Maya marked each of these 18,980 days with a unique symbol, creating a distinct calendar they called the Short Count calendar."

Hearing this, Daphne said to Michael, "I know a little about the Maya's Short Count calendar. 365 days is called the Haab' calendar, and 260 days is called the Tzolk'in calendar. The least common multiple of these two numbers is 18,980 days, which is 52 times 365 and 73 times 260."

Michael said, "365 days clearly refers to Earth's orbital period. What period is 260 days?"

As if sensing Michael's question, the notes continued: "The calendars of almost all civilizations in the world are based on astronomical cycles, and the Maya are no exception. Their calendars are long and short, but all correspond to astronomical cycles."

"Besides 365 and 260 days, the Maya also used 584 days, which is the synodic period of Venus and Earth. This explains what 260 days means. The synodic period of Mars and Earth is 780 days, and 260 days is one-third of this synodic period."

"Earth's orbital period is 365 days, and the synodic periods of Venus and Mars with Earth are 584 and 780 days, respectively. The least common multiple of these three numbers is 113,880 days, which is 312 times 365 days. The Maya might have considered 312 years too long for a Short Count calendar, so they took half of the Venus synodic period and one-third of the Mars synodic period, resulting in 52 times 365 days, or 18,980 days."

Daphne said, "Yes, 18,980 multiplied by 1 is divisible by 365, and multiplied by 2 is divisible by 584, and multiplied by 3 is divisible by 780."

"I've noticed that the least common multiple of 18,980 days, the Earth year, and the synodic periods of the inner and outer planets seems special. It can be expressed in a symmetric mathematical form: 18,980 = 3^9 - 3^6 + 3^3 - 3^0."

Daphne exclaimed, "This is ternary! In ternary, 18,980 is represented as (22200222)₃. The 3 outside the parentheses indicates ternary. It's so symmetrical. The Maya used a vigesimal system, so why does ternary appear here?"

The notes seemed to answer: "Do the Maya find numbers with symmetry represented in ternary beautiful? I found confirmation of this at the Maya pyramid ruins in Chichen Itza, Mexico."

"This pyramid has four sides, each with 91 steps, for a total of 364 steps. In ternary, 364 = (111111)₃. Six 1s, truly symmetrical!"

"Each side of the pyramid steps has nine levels. 91 * 9 = 819. The Maya calendar is indeed magical; 819 days is also an important date in their calendar. 819 = 3^6 + 3^4 + 3^2 = (1010100)₃."

"The Maya's astronomical observations are astonishing. They measured Earth's orbital period to be 365.242 days. Although their observed synodic period of Mercury, 117 days, was off by a day and a half from the actual period, their measurements for the synodic periods of Venus, Mars, Jupiter, and Saturn with Earth were extremely accurate."

"As mentioned earlier, the synodic periods of the inner and outer planets with Earth, namely Venus and Mars, are 584 and 780 days, respectively. Their least common multiple with 365 days is 113,880 days. If we also consider the synodic period of Mercury, 117 days, and Saturn, 378 days, the least common multiple of these five numbers is no longer 113,880 days."

"Magically, the least common multiple of 819 days, contained within the pyramid, and 113,880 days is 7,174,440 days. This number is precisely the least common multiple of the five periods: Mercury (117 days), Venus (584 days), Earth (365 days), Mars (780 days), and Saturn (378 days)."

"Even more unbelievably, 7,174,440 in ternary can be written as twelve 1s and three 0s, i.e., 7,174,440 = (111111111111000)₃."

"If you find the previous numbers too large, the Maya actually derived two numbers from the synodic periods of Venus and Mars: 8 and 13. They discovered a coincidence: the greatest common divisor of Earth's year (365 days) and Venus's synodic period (584 days) is 73. 365 = 73 * 5, and 584 = 73 * 8. Therefore, they considered 8 to be Venus's contribution number and inscribed this coincidence on stone tablets."

"Mars's synodic period is 780 days. 780 = 20 * 3 * 13. The Maya used a vigesimal system. In both the Short Count and Long Count calendars, 13 was considered a magical number. They believed 13 to be Mars's contribution number."

"In ternary, 8 and 13 are also numbers with symmetric beauty. 8 = (22)₃, and 13 = (111)₃. Mathematically, in ternary, there are only three digits: 0, 1, and 2. If we represent 2 as -1 and use 1, 0, -1 to represent ternary, it's called symmetric ternary. Conventionally, -1 can also be represented by the letter T, so 8 = (10T)₃."

Michael was confused. Later, when he read about "1, 0, -1," he seemed to understand Mirov's intention and said, "Ternary computers. In 2024, researchers at Jinshengzhou University simplified the binary calculations of artificial large models using 1, 0, -1, which increased calculation speed by 5 times and significantly reduced energy consumption to 13 watts, approaching the human brain."

The content of the notes seemed to follow Michael's train of thought, immediately jumping to: "The study of the Maya calendar and the various numerical coincidences of planetary synodic periods made me realize the simplicity and beauty of ternary. Binary has only two judgments: 'yes' and 'no,' while ternary adds a state, meaning it includes three judgments: 'yes,' 'no,' and 'uncertain.' This clearly aligns better with the logic of the human brain."

"In 1955, various countries began developing computers. The hardware for computation was based on vacuum tubes in circuits. The mainstream design solution used diodes, with their conducting states representing 0 and 1 for 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 produced ternary computers and sold nearly a hundred units to Europe. They were fast and energy-efficient. Why did they abandon them later?"

The notes answered Daphne's question: "In 1979, the year I wrote this note, the Soviet ternary computer project was officially shelved. There were three reasons: first, the country didn't have enough money; second, the country lacked a complete industrial chain; and third, and most importantly, Liang Nation and Europe did not impose sanctions on us, so we could always buy binary-based computing equipment produced by them. Compared to starting from scratch and independently developing ternary computers, it was cheaper and more convenient."

"The reason I wrote this note and gave it to Irene, asking for it to be opened 52 years later, is because I believe that by then, binary-based computers will encounter a huge bottleneck with the geometric increase in computing speed. One bottleneck is the conflict with power supply, and the other is that when computers imitate the human brain, binary does not align with the human brain's thought patterns."

"I suspect that after seeing my notes, even if you realize that ternary computers have obvious advantages over binary, you might not be willing to abandon the accumulated hardware and software achievements and switch to ternary. In that case, please consider this note a well-intentioned reminder."

Michael Maxwell fell into deep thought. Although ternary computers were theoretically faster and more energy-efficient than binary, for the vested interests within the entire industrial chain of the Western world, led by Liang Nation, who would be willing to start over?

The clever Daphne smiled and said to Michael, "You big shots in the electronic information field will probably unite to resist ternary, won't you?"

"Jinshengzhou University only improved algorithms using symmetric ternary, significantly increasing computing speed and reducing energy consumption. What's more, if the underlying computing architecture were changed to ternary. If ternary computers represent a complete revolution, threatening vested interests would be unavoidable."

Michael also smiled and said, "I don't know what others 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 this note, I will accelerate the development of ternary computers and apply them to my Brain-Computer Fusion Project."

Daphne picked up a smaller, still unopened envelope. Scanning the text on it, it read "Jupiter and the Long Count Calendar" with a small line of text below: "If you think ternary computers are unnecessary, then you don't need to open this small envelope, because more complex positional numeral systems would be even more unnecessary for you."

Daphne was very curious and was about to open it when Michael stopped her. "We've already received a lot of information today. Let's digest it first and look at this later."

Daphne withdrew her hand but couldn't help but say curiously, "Jupiter? The Short Count didn't include Jupiter's synodic period with Earth, which is 399 days. Let's guess what kind of thoughts Mirov might have had about the number 399."

Michael analyzed, "The Maya used a vigesimal system. 399 = 19 * 21. In the vigesimal system, '21' is represented as 11, and '19' is represented as 1T."

Daphne seemed to understand and blurted out, "399 = 20² - 1 = (19.19)₂₀ = (10T)₂₀, which is exactly 1, 0, -1 in the vigesimal system."

Poem of Collected Verses from Chapters:

Short stele, deep carvings, Ming, Shen Zhou

Once here, proving my past life. Song, Shi Shu

Clever calendars cannot calculate, Song, Chao Gongsu

Taking books, I peek at the ancients. Song, Qiang Zhi