Nicomachean History
The Years of the Squared Triangle
100: Nicomachus discovers a cosmic truth
Our story begins around the year 100 C.E. The philosopher Nicomachus of Gerasa was studying the mystical properties of numbers. Mathematics, he thought, was divine truth, deeper than any accident of matter. We consider 100 to be a round number because it’s 10*10, or “ten squared,” and we have ten fingers. But people on another world might work in base 12 instead, where 100 would refer to twelve dozen rather than ten tens.
Gerasa, a city in the Roman province named Syria, had a “Hellenistic” culture, meaning their beliefs and practices were modeled after those of ancient Greece. Nicomachus was specifically following the example of Pythagoras and his cult. Gerasa had been briefly conquered by Judea, so in his time it also had a significant Jewish community, and was a Jewish haven during Roman purges.
There are plenty of facts about ten that are the same in all possible worlds, and that’s what Nicomachus was looking for. For example, ten is what’s called a “triangular number,” because it’s one plus two plus three plus four, so you can lay out ten squares like this:
Since each row in the triangle is one larger than the one before it, the next triangular number is ten plus five, and the one after that is six more, and so on.
A hundred, then, is a square of a triangular number, which means you can lay out a hundred squares like this, where each side is 1+2+3+4 units long.
Nichomachus discovered something surprising about these numbers: they also describe the sums of cubes. You have to break some of them in half, but you can make a 10*10 square out of the layers of a 1x1 cube, a 2x2 cube, a 3x3 cube, and a 4x4 cube. Here’s an illustration using the next squared triangular number, 15 squared:
Both the cubes at the top and the square at the bottom have a total of 15*15=225 unit cubes, making 225 the next square triangular number.
225: A sun sets, the queen of the sea rises.
225 C.E. is notable as the year Sun Shao, a Chinese chancellor, died at age 63, and as the best-guess year that Lady Triệu, the Vietnamese “Beautiful Queen of the Sea,” was born. Sun Shao got unpersoned by his rivals after his death, so we don’t have a lot of biography on him, but there’s some evidence his goal was to avoid war, with some success. The year after his death, though, his king, Sun Quan, got it into his head to conquer Vietnam, and there was nobody around to talk him out of it. His troops stormed northern Vietnam and conducted mass executions of the nobility.
With the leaders of the previous generation murdered, it fell to an orphan of unknown parents, who’d fled to the mountains as a young woman after killing her abusive sister-in-law, to once again defy the occupiers. Vietnam has fought many wars of independence, but never led by someone as celebrated as Lady Triệu.
Chinese histories unpersoned her too, but Vietnamese accounts all give her this quote:
I'd like to ride storms, kill orcas in the open sea, drive out the aggressors, reconquer the country, undo the ties of serfdom, and never bend my back to any man.
Gerasa, by contrast, had enjoyed centuries of political stability, and was thriving, now a major city of 20,000 in the Roman province of Arabia. But with the rise of Christianity, that was soon to change.
441: The First Council of Orange
By 441, the Christians had taken over and the Roman Empire had splintered. The shard we call the Byzantine Empire ruled Gerasa, and it was not as tolerant of multiculturalism as its previous suzerains. The synagogues were torn down or converted into churches, and mystical mathematics became a dangerous hobby to have. Thanks to these disruptions, and to the loss of the pax romana, Gerasa began to decline.
Meanwhile, the other major shard, the Western Roman Empire, was also Christian, and was still trying to figure out what exactly that meant. In a city named Arausio, whose name would later be corrupted into “Orange,” seventeen bishops met to sort out the rules. The meeting was presided over by Bishop Hilarius, whom we now call Saint Hilary so that we can take him slightly more seriously. Among the thirty “canons” agreed to, by these seventeen men, was that women were not allowed to be priests. In later debates over the issue, that’s what everyone on the nay side always cited: the 26th Canon of the 441 Council of Orange.
784: The Mayan House of Orange gets a new ruler.
In 784, a man named Itzamnaaj Kʼawiil became the ruler of Wak Kab’nal, a city-state in modern-day Guatemala.
We don’t know why the place was named Wak Kab’nal, or really what it means. The standard “translation” of Wak Kab Nal is Six Earth Place, but what “six earth” refers to remains unknown, and even “place” is disputed. Probably “Six Earth” meant something valuable was there, because the area had already been settled and fought over for centuries before the coronation of Itzamnaaj Kʼawiil.
Spanish conquerors, eight hundred years later, translated the name to “Naranjo”—Orange Tree. The fruit didn’t show up in the Americas until after Columbus, so it can’t be the origin of the name. That suggests that if the translation was at all accurate, “Six-Earth” was the color orange. According to this paper, there were quite a few Mayan lords referred to with honorifics like Wak kab nal ajaw, which presumably translates roughly to Prince of Orange.
This wouldn’t be Outlandish Claims if I didn’t build a speculative castle in the air about why this place was named Orange. We know that the Mayans cared about color and used orange dyes in their art and pottery. Orange dye was difficult to produce, so any naturally orange clay would be valuable. So let’s check what Guatemala’s largest clay export is, and, what do you know, it’s bentonite, which depending on its iron content and how long it’s been exposed to air can be white, yellow, red, brown, or…
So I’m going to guess that “six-earth” is bentonite clay, and the name stuck even after they’d quarried it all out.
Back in Gerasa…no, you can’t go back to Gerasa. If you’ve got a date in Gerasa, she’ll be waiting in Jerash. The Byzantine Empire had been conquered by the Umayyad Caliphate, who renamed the capital to Istanbul, and Gerasa to Jerash. Under Muslim rule, the city began to thrive once again. Houses of worship of multiple faiths existed side by side, and the most common inscription on ceramics of this period is “Made in Jerash.” But its recovery was soon to be interrupted.
1296: Zhou Daguan spends a year in Cambodia.
1296 is the eighth square triangular number, and therefore also the number of rectangles on an 8 by 8 chessboard. By 1296, Jerash was a ghost town, either completely abandoned or with a double digit population. First a devastating earthquake, and then the Crusades, had all but destroyed the city.
In 1296, a Chinese official begin a year living in Angkor, the capital of the Khmer Empire. The book he published afterwards is a unique glimpse into this society, since the only Khmer writing that’s survived is religious. It’s part anthropology, part travel guide. They no longer revere Chinese people as divine, he warns. If someone prostrates himself before you, you are about to be scammed.
The society seems mostly patriarchal, but with the important exception that women are in charge of all economic activity. Zhou advises Chinese immigrants to get a wife or girlfriend immediately, partly for that reason. Also because the Khmer had vastly fewer prohibitions around sex than back home. One of the king’s duties, apparently, was to have sex every night with the nine-headed snake goddess who owns the land.
Zhou is critical of a lot of Khmer society, but he seems to take some of their mystical beliefs at face value. (Or he was being ironic and it’s gotten lost in translation.) In particular, he says the Khmer use trial by ordeal to resolve many disputes—spirits will curse the guilty and protect the innocent, so just put people in a dangerous situation and see what happens. This is great, Zhou writes. It’s a shame we don’t have spirits like that in China.
2025: Happy New Year!
2025 is the ninth square triangular number, meaning you can lay out 2025 squares like this:
The next such year will be exactly a thousand (ten cubed) years from now.
A lot went right in the last nine-cubed years.
Today, Vietnam has driven out the aggressors. Mayans have regained the numbers they had before the Spanish conquest, and their culture endures. Women can be priests in many denominations.
Cambodia has fared worse than almost anywhere else on earth. The United Nations classifies it among the “least developed” countries. And yet. A child born in Cambodia this year is expected to live to the age of 71. When Zhou Daguan was there, Chinese life expectancy was less than half of that, and he wrote that the Khmer seemed to age faster, so theirs was probably even less. Of the places and times I’ve written about so far today, Cambodia in 2025 would be the best to spend a year in.
I also hope, some day, to visit the city where this story began. With the help of friend of the blog Joe Jaeger, Jordan created a national park system that has restored the ancient ruins. It’s now divided into two parts—the preserved ancient city of Gerasa, and the modern city of Jerash, with a population of over 50,000. Every year, one of the biggest artistic events in the Middle East is held at the boundary of the old and the new. Here’s a scene from last year’s Jerash Festival, via a Chinese news outlet:
History isn’t a straight line from worse to better. It’s a chaotic mess that resists any attempts to organize it into neat little squares. But on the whole, so far, civilization has won.