Ghost stories are as American as apple pie. Washington Irving, Herman Melville—yes, Moby Dick is a ghost story—Edgar Allen Poe, Mark Twain, many of the greatest American writers wrote ghost stories. Shakespeare wrote ghost stories. Two of the greatest Shakespearian tragedies, Hamlet and Macbeth prominently feature ghosts who drive plot elements.
Our Puritan forebears slipped into a dark age when ghost stories began to drive the actual plotlines of their politics. The trials at Salem depended on the introduction of “spectral evidence” which the zealous judges of the Massachusetts Bay Colony candidly admitted had no place in a normal trial. Allowances had to be made to trap the practitioners of black magic.
And not much has changed. Ghosts still drive the plotlines of American politics. What is Russian collusion and Russian disinformation but spectral evidence?
The zeal with which these wraiths have been pursued exceeds that of our Puritan forebears. Special Counsel Robert Mueller, his agent Andrew Weissman, and the inimitably nefarious Representative Adam Schiff (D-Calif.) should all be depicted in sugarloaf hats and white collars, representing the superstitious judges of 17th century America who murdered the innocent to slake their taste for things invisible offered as an explanation for their own political failings.
Schiff’s use of the Sensitive Compartmented Information Facility in the impeachment drama turned the House Intelligence Committee into a vestige of the barbarism of Salem. The damage done by Schiff’s obsession with political sorcery will take years, if not decades to repair—if it can be repaired at all.
In contrast, what you see is what you get with President Trump. Trump clearly believes in a world of the living and the dead. He often speaks of persons who have left us “looking down” and judging. They participate in our world. Trump, however, has his feet on the ground. There’s a partition for Trump. They are up there. We are down here.
This is in contradistinction to his political tormentors like Schiff and House Speaker Nancy Pelosi (D-Calif.), for whom phantasms and reality are all mixed together, in the haunted mansion of their minds, where spooks from Muscovy hide among the cobwebs, mists, and fluttering curtains, spreading their chaos.
This brings us to our tale below, delivered to you in the spirit of the holiday and the American tradition of ghost stories. Our spectral yarn explores the order of chaos in an examination of numbers. It is based on actual events. Which portions are embellishments, I will leave it to the reader to guess.
But before we descend into superstitious musing, allow me to suggest there is an order to the chaos of Joe Biden’s campaign. There is no other way to describe the candidacy of a late septuagenarian defined by failing memory, an unpopular and inexperienced de facto top-of-the-ticket running mate, fear of an invisible pathogen, and a supporting cast of Marxist and anarchist rioters rising in a milky fog of tear gas. That order, they say, is found in the text number of the campaign—30330—which is of course the quotient of 2020 divided by the Number of the Beast.
Happy Halloween, my deplorable friends.
The Order of Chaos
If you like numbers, you probably have some favorites. Take Pi, for example. It is an irrational number, meaning it cannot be expressed as a fraction of two integers. Instead, it is a geometric ratio of the diameter of a circle to its circumference. As an irrational number, Pi is expressed in the familiar base 10 system as 3.14582 . . . The decimals extend out, never repeating exactly the same pattern, infinitely. Pi is essential to many practical and theoretical calculations, because circles are found everywhere in nature.
I have another favorite number, which is also an irrational number derived from a geometric ratio. It is the number that results when you divide a line into two parts so that the ratio of the whole line to the larger part is the same as the ratio of the larger part to the smaller part. It can be expressed (a+b)/a=a/b. Solving this expression resolves into the irrational number 1.618033 . . . Again, the decimals extend out, never repeating the same pattern, infinitely. This ratio is sometimes referred to as the Divine Ratio or the Golden Ratio.
Now here is another interesting fact. If you create a sequence of numbers by adding the last two numbers to get the next 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 . . . . The ratio of any two adjacent numbers in the sequence soon approximate, and then converge on, but never reach, the Golden Ratio. They never reach it, of course, because, as an irrational number, it cannot be expressed as a ratio of two integers. This sequence is known as the Fibonacci sequence.
The Fibonacci sequence can be expressed visually as a spiral, which is similar to the spiral of a nautilus shell, like this:
The Fibonacci sequence and the Golden Ratio, like the circle, appear in many natural objects. For example, the spirals of a pinecone follow Fibonacci numbers. More often than not, the number of petals on a flower does too, as do the leaves on the stem of a plant. The structure of insect wings can be derived from Fibonacci numbers, and the patterns of crystal fractals follow Fibonacci numbers. The breeding of rabbits, too, tends to follow Fibonacci numbers. The general explanation for this is the Fibonacci numbers are ratios that correspond to an efficient growth pattern. So things that grow—especially things that live—tend to reflect the Fibonacci sequence.
When we were first married, my wife and I bought a piano, a Steinway upright, lacquered in black. The piano was older, second-hand, but a beautiful instrument. It was the first big asset thing we owned, and for a time our only substantial piece of furniture. We bought the piano before anything else because we love music—my wife plays the piano quite well. To us, a piano was a symbol of our aspirations to build a certain kind of household.
I don’t play but I have always liked the piano because I know the piano to be a special instrument. When the piano emerged it changed the world of music because it has a range of seven octaves and can be played soft or loud by modulating the force with which the keys are struck. This feature makes the piano more expressive than other lead keyboard instruments, the organ and the harpsichord, which can only increase or decrease in volume in quantal amounts by engaging more pipes or harps. But I digress.
Music is mathematical. And a tuned piano controls harmonies which are the auditory expression of an arithmetic that is orderly and beautiful. An out-of-tune piano is a disaster, an assault on the ear. As pianos over time will do, our piano became out of tune, at first only slightly, so only my wife could hear it, but soon it became unusually, perversely, hideously out of tune. After seeking out several recommendations, we called a piano tuner to our apartment.
An eccentric-looking Eastern European man, the tuner came one Saturday and worked on the piano for an hour, then two, then three, turning pins, tapping keys. This relatively simple exercise consumed the whole of an afternoon. I could hear him muttering to himself as he worked. He seemed frustrated, confused. When he finished, he came to speak to me in my study. The afternoon light was failing, and the early evening gloom painted the scene. He spoke softly with a heavy accent.
“Where did you get piano?” he asked.
“We got it second hand, from a dealer in Queens,” I said. “Why?”
“Someone has . . . something is. I have tuned it but I suggest you get another. Something is wrong with piano.”
“I am not following. What is wrong? Is it tuned?”
“Yes. I did it. But it was not easy! Someone maybe has done something to this piano. Come!” he ordered.
We walked over to the piano. He lifted the cover.
“See these numbers?” he said pointing to the pins in the steel harp, “Someone has written this here.”
Indeed, above each pin someone had in pencil written a number: 0, 0, 0, 2, 5, 5, 8, 9, 11, 14, 15, 19, 21, 19, 17, 21, 23, 27, 26, 25, 29, 29, 29, 33, 29 . . .
“So what?” I said.
“This is not normal. There is no reason why anyone would put these numbers here. This piano was . . . I don’t know.”
“I don’t understand,” I said. “Someone jotted down some notes when they were working with the piano. Frankly, it doesn’t seem like it means anything. Maybe they were put there when it was manufactured.”
The man signed heavily and ran a hand through his curly grey hair. He looked up. His eyes were bloodshot.
“When you put together a piano,” he continued, “it is a mathematical undertaking. Someone was thinking about this when they fiddled with your piano. I don’t know why but this . . . ”
He paused. “Are you familiar with Fibonacci Sequence?”
“Yes,” I said.
“Yes,” he said, “ancient scholars very interested in numbers. They thought Fibonacci sequence and Golden Ratio express something about beauty and life.”
I nodded.
“Some of them,” he said, “they had other ideas. They wanted counter sequence, negative sequence. You know, Italian Renaissance full of people with ideas. Some of them good. Galileo. Some of them bad. Machiavelli.”
I nodded again.
“You know what prime number is, right?” he asked.
“Yes, of course. A number whose only factors are one and itself,” I said. “There are an infinite number of them, theoretically.”
“Yes. Yes,” he said. “Do you know what ugly number is?”
“Yes,” I said proudly. “It is a number whose prime factors are two, three or five. It’s a silly sequence. It doesn’t amount to anything.”
“No!” he said. “Not silly. Not silly at all. You see, if pattern out sequence of ugly numbers you get orderly progression. Not beautiful spiral, but orderly.”
“If pattern out sequence of prime numbers also you get order,” he said, “But if you subtract ugly numbers from prime number in sequence . . .” He arched his eyebrows dramatically and stared ahead.
I leaned forward.
“. . . you get something else. Completely different!” he almost shouted. Then he said slowly, “Kay-aahh-teek.”
“Chaotic?” I said.
“Yes,” he said. “My English is no good.” He straightened and softened his tone, “Chaotic.”
He took out a piece of paper and began to write out a series of numbers. Then he began to sketch something. When he finished, he said:
“Look. Look. Here is Fibonacci and prime and ugly.” He had made a simple radial chart, looking like a radar scope, showing the Fibonacci Sequence as a curve, a little like an airplane wing, the Prime Sequence a curve like a snail shell, and the Ugly Sequence also showing a curve like a snail shell.
He put his head down and began to draw again. In a moment he looked up. “Now, here is the same illustration but subtracting prime and ugly.”
It looked like a picture of a splotch, a little bit like Antarctica. A zig-zagging random shape.
He breathed out heavily. He hesitated, as if to resist revealing something, and then said “Sequence you see is called Reficul sequence. I don’t know origin of name exactly. Maybe name of scholar. . .”
“I have never heard of it,” I cut him off.
“Yes, this is not surprising. I learned about this sequence at university. The Communists taught many things you were not taught here in West . . .” he trailed off.
“Not only,” he started again, “does subtraction of orderly progression of ugly number and prime number create chaotic sequence, but numbers appear in natural phenomena, or, perhaps . . . failure of nature. The sequence is opposite of Fibonacci. Sequence is chaos, ugliness . . .” he lowered his voice, “and death.”
I shifted, and raised my head slightly.
“Pattern appears many places. People choose not to see it because they see disorder as absence of order. But sequence suggests order. Yes, chaos has order, unique order, hidden order all its own.”
“There was study done in Bucharest at the university there,” he went on, “about accident rate at busy intersection. Numbers of sequence appeared over and over at intersections with high accident rates. Crime, too. Crime pattern follows sequence. Ugly things, hideous things, have proportions of sequence. Medieval gargoyles, pictures of demons, disorienting colors and sounds have sequence. It is not clear if this is coming from intention or because such ugly things have such proportions. The numbers, they are, what is word? Pr-r-r-ofane,” he said.
He paused for a moment. I shifted uncomfortably. It was silent for a second or two. Then he continued.
“These numbers show up in infant mortality. They are found in patterns of starvation in famine and mortality from disease! These numbers are found in conflict and natural disasters. When they appear in crystalline structures, the structures fail. The extinction of animals, the failure of agriculture,” he said, his voice climbing the octaves. “In all these things, terrible things, we find these numbers.”
“These numbers,” he said, nearly shouting and turning back to the Steinway jabbing a finger at the numbers above the pins, “this sequence . . . it is written in your piano.”
I paid the man and he vanished, but I never forgot what he had said. The piano never played well again, and I never saw this strange man again, either. The piano is long gone, sold and probably sold on and on, to trouble other households. But the fear remained. The fear of bad numbers, and the order of chaos, lingers.