The Loogaroo Express runs both directions: It can take passengers out of Forgotten Hollow just as easily as it can bring them in, and Sylvia’s mother had thirteen reasons why Sylvia was not going to be riding it out of the valley this weekend.
Sylvia concentrated on calculus while her mother rattled off her concerns. “Firstly, the curtains fit the windows poorly! Light always seeps in through the cracks. And knowing you, you’d have your nose in a book and wouldn’t even notice when your arm was bathed in sunlight!”
Her mother’s voice, especially when she was trying to make a point, or rather, thirteen successive points, became musical. Sylvia reflected on something she’d read in a lesson by Knill in her advanced math textbook: “Calculus plays a role in music because every music piece just is a function.”
If that’s the case, then her mother’s voice could be described by calculus.
“Reason number seven,” said her mother, “The vending machines on the train don’t sell plasma packs! Not on the train, not in the station, not in San Myshuno. What if you get hungry?”
Her mother’s voice rose in pitch towards the end of that statement, but it was quite melodious. The last syllable, “ree,” actually sounded like it was pitched at 440 A.
Assume g(x) is a 2π periodic function, we can generate a sound of 440 Hertz when playing the function f(x) = g(440 · 2πx). If the function does not have a smaller period, then we hear the A tone with 440 Hertz. (Knill)
“And finally, reason number thirteen,” said her mother, “I like having you here. I just don’t want you out gallivanting through the strange world with all those strangers.”
“All right, Ma,” said Sylvia. Her own voice was lower, softer.
We can not hear the actual function because the function changes too fast… [to allow us to] notice individual vibrations. But we can hear the hull function… We can generate a beautiful hull by playing two frequencies which are close. You hear interference. (Knill)
Sylvia began to hum softly, barely audibly, while her mother canted on about all the reasons to stay off the Loogaroo.
The frequencies of their voices combined into a beautiful resonance, and the glasses on the shelves began to vibrate slightly.
“It’s settled then,” said Sylvia’s mother, and she walked upstairs to tuck in Zap.
The moon shone on the patio. Sylvia wondered about the logarithm for moonlight.
Mathematics made her world feel less small, somehow. It defined it, surely, but it also connected it. If she knew the logarithm for the observed moonlight here on the patio of her mother’s ancestral home in Forgotten Hollow, then she could imagine the logarithm of the moonlight shining over San Myshuno Bay, and so it was through the moonlight that she might explore all of the world onto which it shone.
The old lore held fascination, too, connecting her to time, rather than space. The practices she read about in her mother’s old leather-bound encyclopedia were every bit as precise as mathematical formulae.
“Have you practiced any of these things?” she asked her mother when she joined her with her own volume of the esoterica.
“Yes, dear,” said Miranda. “My father ensured that my own upbringing was very traditional, and very thorough. I wasn’t allowed to be some young wild thing, like you are!”
“Oh, Ma!” Sylvia protested. “I’m not so wild.”
“But you are,” said her mother, wrapping her in a hug. “Your father and I spoil you so. We wouldn’t have it any other way. But still. Wouldn’t you like to go shopping, dear? Get something a little less nomdish and a little more rune?”
“Oh, Ma! You know I’m happiest in my old sweats.”
Sylvia escaped to the garret, where her thoughts could wander in solitude. She found an old easel, a stack of canvases, and a cabinet with fresh acrylics. Her grandfather was an artist, in addition to all his other myriad accomplishments.
The next day, after school, Sylvia found her mother deep in study down in the cellar library. Her father was napping in the cryptorium.
Back upstairs, her math text didn’t bring the comfort she sought. The daily problem for their homework was the old train travel puzzle:
A train leaves the station at 8 p.m., travelling north at 90 miles per hour. Another train starting from the same point at 10 p.m. travels east at 100 miles per hour. Find, to the nearest mile per hour, how fast the two trains are separating at midnight.
Oh, to be on a train at midnight, instead of stuck here, in this prison of a house!
If she caught the southbound train at 8 p.m., how many kilometers away from this stockade would she be by 10 p.m.? They didn’t call it the San Myshuno Rapid Transit system for nothing: the Loogaroo pulled into the city an hour after leaving the Forgotten Hollow station.
She’d do it. She had permission, of course, to roam the valley all she wanted during nighttime, and as long as she was home before sunrise, neither parent cared. They were, after all, creatures of the night.
“Ma!” she called down to the library. “I’m heading out! I’ll be back before breakfast!”
“That’s nice, dear! Have fun!”
She raced through the square down to the station, hopped onto the last car just as the doors were closing, and in a little over an hour, she got off at the station in the art district, near a bright building that housed the modern art museum.
A woman dressed in hippie clothes, with her hair wrapped in scarves, approached her.
“Are you here for the organ?” she asked.
Sylvia felt self-conscious. Was it that obvious what she was?
“No, I’m not hungry,” she said. “No need for organs.”
The woman laughed. “Pipe organ! The new pipe organ! The one that Bach selected himself!”
Sylvia had read about this on the web. The St. Catherine organ, which sixteen-year-old Bach traveled 50 kilometers, mostly on foot, to play and listen to had been donated to the museum, which boasted of ideal acoustics for such an instrument. Sylvia confessed that she hadn’t come for the organ, but to see the city.
“Well, the organ is amazing,” said the woman. “I’m terrible on it–it’s nothing like a piano. But still, the sound!”
Sylvia began talking about the calculus of music, and how a person’s voice can travel the same frequencies as any instrument, and how we hear not just with our ears, but with all the empty spaces within our bodies, and how we listen not just with our minds, but with our stored cellular memories, too. And then, they remembered that they hadn’t yet introduced themselves.
After the introductions, Sylvia and Cathy Tea fell silent for a moment. They shifted in that awkward space that’s shared when two strangers have dove into the deep end and find themselves friends before they, really, know anything about each other except that they are kindred spirits.
“I’m going to meet some friends,” Cathy said at last. “Would you like to come? We’re meeting in Willow Springs, but it’s a short ride on the express. And the organ will be here later.”
Sylvia felt swept along, and besides, saving the organ for another night would give her an excuse to return.
The two new friends never stopped talking on the quick ride to the next town. They talked of everything that mattered: Bach’s music and the mathematical patterns found within it and the replication of those patterns in the songs of wrens and the influence of bird songs on abstract concepts in art and the ways that painting shapes the arc of a story or a line of poetry and what does it really mean to be an artist, anyway? Is it something one does or something one is? They both agreed: It is who one is.
Sylvia felt just as comfortable with Cathy’s friends.
They were so wholesome: fit, and tan, and cheerful. She envied Cathy, being a part of this group, and the longing she felt seemed to clarify something for her. It was like looking into her grandfather’s old stereoscope. Suddenly, she saw her future in 3-D. This was what she wanted be.
“Where you say you be coming from?” asked Davion.
“I didn’t,” replied Sylvia. “But it’s not too far of a ride on the train.”
Davion didn’t say where he was from, either.
“At present snotch of time,” he said, “I live on the isle of the Windenburg quenya.”
“I’ve always wanted to listen to the sea,” said Sylvia. “I’ve never seen the sea.”
“Never seen it?” asked Davion, in amazement.
“No,” said Sylvia. “But I’ve read about it! I can imagine it!”
Cathy sat alone at the next table, and Sylvia excused herself to go join her.
“I like your friends,” she said.
“They’re your friends now, too!” said Cathy.
“If only it were that easy,” replied Sylvia.
Soon, her new friends began marking the late hour. “I’ve got an early morning,” said Cathy. “I have little ones, you know, and they never sleep past the thrush’s first song!”
An hour before midnight, and they left Sylvia there alone. She pulled out her i-phone and looked for videos of the ocean.
It’s the same pattern, she thought, watching the waves rolling in and out on the shore, only trochoidal, not sinusoidal.
The Trochoidal shape [of a wave] can be approximated to the shape of the Hyperbolic Tan Function graph, tanh(x). (Passy)
She wasn’t sure which she liked best, the smooth wave of the sine or the peaked wave of the trochoid, but the pulsing of each reminded her of what drew her most strongly to these new friends: their own internal bright red oceans, salty as the sea, moving through the veins in their own continents of flesh.
The heart beats in fractal patterns as it flows through the fractal pathways of veins. The fractal dimension of the human voice produces resonance. The fractal dimension of the cerebellum receives the fractal patterns of Bach’s fugues. “Everything is sound and light.”
“Hey, there!” said a bright voice.
Syvlia turned to see a young woman seated at the table.
“I was supposed to meet my mom and my neighbor here,” the young woman said. “Have you seen them?”
When she discovered that her new friend Cathy was the neighbor, and her friend was the mom, Sylvia joined the woman. The woman talked about her garden until after midnight, and then, complaining of the late hour, she, too, headed home.
Sylvia was alone in the night. Soon, the last train would stop to pick up passengers heading back to the shadowed valley of her home. She would be on it, returning well before the first sun. But part of her dreams would remain here, to walk out beside the willow in the morning light, listening to the thrush’s song.
Through the fractal dimension of her cerebellum, this imagined self might be felt to be real, just as real as a Bach fugue played on the St. Catherine’s organ in the Museum of Modern Art in San Myshuno.
And it was that imagined self, she thought, that would get her through the stretches of long, sunless days that awaited her back home.
Knill, Oliver. “Lecture 33: Calculus and Music.” Math 1A: Introduction to Functions and Calculus. 2012. Web. 2 Feb. 2017.
Passy. “Mathematics of Ocean Waves and Surfing.” Passy’s World of Mathematics. 4 Dec. 2013. Web. 3 Feb. 2017.
Bieberich, Erhard. “Structure in human consciousness: A fractal approach to the topology of the self perceiving an outer world in an inner space.” Fractal.org. Web. 3 Feb. 2017.