Xn+1 = RXn(1-Xn)
Riley Matthews got to school earlier than most. Her dad taught History at Abigail Adams, and she preferred to ride with him than to walk or take the subway (she never knew what to do with herself on the subway, never knew where to sit or what to do with her hands). But arriving early allowed her to sit on the bench near her locker and text her uncle, who'd either be heading to his classes or to work as her Uncle Eric's PA (Uncle Eric was the Mayor of New York City, and Josh was only three years older than her). It also gave her time to put up posters for Debate Club (in spite of this being her first year in Debate, and in high school at all, the head of debate had taken a liking to her, seemed to be training her up to be his successor).
Riley loved Debate Club, and not only because she was good at it. She loved hearing other people's opinions on all nature of matters, loved to learn new perspectives. She also loved arguing her point, so really, Debate Club had been the natural booth for her to gravitate towards at the Student Organisations Fair.
As she stapled a poster advertising this week's topic (Should churches (defined as churches, temples, mosques, synagogues, etc.) remain tax-exempt?) to the notice board in the stairwell, she watched the student begin to stream in, picking out certain faces amongst them. Noah Argota, the head of the Debate Club, waved at her and shot her a wink before disappearing into the wave of students. Noah was great as the Debate Club president, never chose topics like abortion or LGBTQ+ rights, because he knew they weren't up for debate, and he always shot down those who tried to propose them as valid debate ideas.
She smiled at Chai, who nodded back. She and Chai had kissed, once, a couple of months ago, but nothing had ever come of it. Chai wasn't ready to be out yet, which Riley respected wholeheartedly, and Riley, well…
Riley had a huge crush on the girl who sat behind her in History.
Her name was Maya Hart, and she was so pretty. She has curly blonde hair that she brushed less than she should and wore tight leggings and leather jackets and heeled boots every day, come hell or high water. She painted her lips bright red and her eyes dark, and when she deigned (was forced) to join class discussions, she was clearly highly intelligent, well-read, and had fascinating opinions. She was angry at the world and didn't talk much, but she was nice, beneath it all, and it was all Riley could do not to kiss her every time they made eye contact. Especially when she saw Maya smiling softly at people from her vantage point against the pillar, longing to be part of a group but scared of making the first move.
And the reason she hadn't asked her out yet was walking into class now. Lucas Friar.
He'd moved here from Texas in middle school, but hadn't ever seemed to make friends- mostly he just jumped around from girlfriend to girlfriend (Missy Bradford, Sarah Carpenter, even Isadora Smackle from Einstein Academy, for a brief period). But he always came back to Maya Hart.
Riley could never understand their relationship- they fought all the time, and she didn't think that they had ever actually put a label on what they were, but the two of them sat side-by-side in most classes, flirting and fighting and touching and ignoring each other. There was nothing that could tell the rest of the class what mood they'd be in on any given day- they weren't friends with anyone except each other. Most of the time, Riley's dad said he just hoped for a day when they were happy because when they were ignoring each other they somehow dragged the whole class into it. He'd tried to work their dynamic into his lessons before -Thomas Jefferson and Alexander Hamilton, who'd worked together when it counted- but none of it worked. Maya Hart and Lucas Friar were simply untouchable, unfortunately for Riley.
On the other hand, she thought, as she caught glimpse of a figure rushing into school after the bell rang, as always. She and Farkle seemed to have accidentally become a different sort of untouchable. Riley wasn't sure when it had happened. Farkle Minkus had been a family friend, once upon a time, but now he was just her best friend.
Her mom and his dad had been intellectual rivals in high school, and she sometimes got the feeling that his dad had sort of expected them to get married one day. Her dad and made fun of/been friends with Mr Minkus through heir entire school career, and seemed to be terrified of Mrs Minkus (although, to be completely honest, Riley hadn't seen Mrs Minkus in nearly two years) so their families had pushed Riley and Farkle together as kids. Back in middle school, when Farkle had retreated into himself, she'd been the one to deal with the kid who was pressing him against the lockers to whisper cruel things in his ear. Later, he'd been the one to hold her when she cried about the texts she was receiving (just stop breathing/no one likes you/do you even have friends/bet your sleeping with him slut/gonna cry to daddy bitch/your pathetic/everyone will be happier if you just dont wake up), had been the one who took her to the mall to get her number changed, had immediately blocked the contact in her phone (but not before tracing it back to the bully and threatening to ruin her - not that Riley knew that).
They weren't even done with their first year of high school, but somehow, everyone knew her and Farkle. She was on Debate and had started SAGA, and he was on the mathletes and in Physics Club and Chess, and both of them had joined the yearbook committee and the school magazine (Riley as a photographer and Farkle as a junior editor). They were both on the Honour Roll, and both took part in sport (she did girls' soccer and Farkle had been strong-armed into joining cross-country).
Riley wasn't quite sure how it happened. She and Farkle had gone into high school intending on joining the clubs they'd been part of in middle school but had then befriended seniors and had been recruited by teachers. And their willingness to participate seemed to have earned them a weird place of honour in the school. She hadn't even realised until a couple of days ago.
She'd been approached by Trev Rivera, two years her senior, and he'd asked her on a date to Olive Garden. She'd politely declined (there was no way she was going on her first date with a junior she didn't even know), but he'd started harassing her, popping up after every class and walking her to her next class, trying to persuade her to say yes. Farkle had tried to step in and it hadn't helped at all. Eventually, she'd been leaving the cafeteria with Farkle while the juniors and seniors had been walking in, and he'd asked her again, and she'd told him to fuck off.
That, of course, had led to him grabbing her wrist and telling her that she should "be more polite, he was trying to be nice". Farkle had immediately told him to let go, and she'd been gathering her strength to rip her arm from Trev's grip when Antione Grant and Thor Benward had appeared behind her, telling Trev to let go (he'd done so without another word, slinking away to lick his wounds). They'd nodded at her and continued to their regular table, and Farkle had pulled her away to Art (where she spent the lesson watching Maya engaged in her own, beautiful world, and wishing she was part of it).
She smiled at Farkle and grabbed his hand, pulling him into History.
Her father had written NORMANDY LANDINGS on the board. He stood up from his desk as the students all made themselves comfortable, and began his lecture.
"The Normandy Landings, codenamed Operation Neptune and commonly referred to as 'D-Day', was the largest seaborne invasion in history. "D-Day" was an Army designation used to indicate the start date for specific field operations. In this case, the "D" in D-Day doesn't actually stand for anything—it's merely an alliterative placeholder used to designate a particular day on the calendar. But because this invasion was so big, the word "D-Day" took on a new meaning. We now use it in everyday life to refer to an important day, the day something big happens."
He moved to the window, his preferred vantage for teaching, mostly because he said he felt that the "kids in the back" couldn't escape him from there, "We learn about history because it's important to learn from our mistakes. But we forget that most events in History affected our lives today in some way."
Riley could feel Maya shooting spitballs at her, could imagine that Lucas was laughing at her from where he sat behind Farkle, but said nothing. She didn't want to be a distraction. Rather, she leaned over and took Farkle's hand as Yogi began to question her dad on their assignment (write a 400-word essay on a word from history that took on a new meaning because of what happened). She heard Lucas fake-wretch, and couldn't resist turning around to glare at him. He just smirked in return, raising his voice to call her dad's attention.
"Hey teacher-man, how come Maya and I aren't allowed public displays of affection, but your daughter and her boyfriend can basically sit on each other's laps without taking any shit? I'm pretty sure that's nepotism or something."
Riley could feel herself flushing scarlet, and Farkle dropped her hand just as fast. They'd considered it, once, and had decided that it might be quite nice if they weren't both severely crushing on other people. She hadn't known that there were rumours like that about them, and couldn't resist glancing back at Maya to see what she thought. She was just staring silently at the back of Riley's head.
"Well, Mr Friar, I don't personally feel that friendships should be discouraged in this class, but there is a difference between holding hands and you actually trying to keep Ms Hart on your lap during class. And I would appreciate it if you'd refrain from swearing in my class."
"Sure thing, teacher-man," Lucas replied, not sounding at all as if he intended to keep his word, "but only if you give me my hat back." Lucas wore a black cowboy hat to school every day, and her dad confiscated it at the beginning of each class and gave it back at the end, otherwise, he wouldn't take it off.
Somehow, that was what got Maya's attention. She barked a laugh and fired her shot, "Hey Friar, you're not as tough as everyone thinks you are."
It didn't seem to get to Lucas though. Riley wondered, again, at the dynamic between the two of them, "Hey Hart- neither are you."
The bell rang before the class could devolve any further, and everyone shoved their things into their bags, eager to be out as soon as possible while Lucas and Maya were glaring at each other like that. Riley still couldn't remove the blush from her cheeks.
Xn+1 = RXn(1-Xn)
Later, Riley was at Topanga's, waiting for her shift to be over. They had several waitresses, but her mom usually gave them off on Friday evenings, when business was slow, and let Riley earn her pocket money by working on weekends. There was currently no one in the cafe, so Riley was bent over the counter, working on her essay, when she walked in.
This was a secret the two of them shared.
Riley knew that Maya was heading out to a night shift at the same diner her mom worked at. She knew that Maya was coming from a shift at Village Cleaners Laundry, where she'd been since school let out. That was why Riley did this- she knew that Maya couldn't. Maya hadn't asked, but when her dad had told her to find out why Maya just wasn't turning in assignments, and Riley had found out, she'd offered.
She'd tried to consistently keep the work at the level Maya had been handing in before she'd taken a second job, but sometimes she slipped up, and Maya got upset (Riley hated to upset Maya).
That was why Riley silently passed Maya the typed-out essay that she'd done at home and smiled at her, even as Maya slipped into the bathroom and put on her uniform for the diner. She also pushed a danish at Maya on the way out (maybe she couldn't date Maya the way she wanted to for fear of Lucas Friar, but she could damn well take care of her).
But today was different.
Because today, Lucas Friar stormed in not five minutes after Maya left.
"Where is she?" he was practically snarling. She tried to act nonchalant.
"I don't know." The part half of their deal relied on no one finding out, especially not Lucas Friar.
"You're lying. I know you are. Listen, Maya's my-" he paused and she leaned in, hopeful, wondering if she'd finally figure out what they were to each other, "she's mine. And I want to help her. Please."
Riley had never seen Lucas so blatantly emotional about something. It was a complete 180° on his usual attitude, and she didn't know what to do, because maybe he could help Maya better than she could, but then Maya wouldn't talk to her again, and Riley couldn't let that happen.
She settled on a shrug.
"Sorry, can't help you with Maya. But can I help you with anything else? Coffee? A danish?" she busted out Customer Service Riley.
Lucas Friar shot her a dark glare.
"Listen, Matthews," he began, his voice low and rough, "let me tell you a story."
"In this world, there are sheep, and there are wolves. Sheep stick together, and wolves stick together. But they don't cross over. There has never been a sheep and a wolf that stuck together against the rest of the pack, or the rest of the flock. And it's not hard to tell which is which.
I'm a wolf. Maya's a wolf. We stick together, we're a pack. You and Minkus, well, you're sheep. You're part of an easily misguided flock. It's not your fault. That's just the way it is."
He braced his hand on the table and leaned forward, to the point where their foreheads were nearly touching, "But you're a sheep, and Maya's a wolf. And she'd drop you in a moment, so stop trying to stick with her. You're part of a flock, not a pack." Riley wasn't quite sure what to do with the insinuation (more like the blatant statement) that she was a sheep, so she just asked again- "I'm sorry, sir, but can I help you with something regarding our service?" She plastered on the smile of all waitresses but remained tense, ready to defend herself.
"I regret talking," he said, sounding as if he thought she couldn't understand what he was saying. She kept the smile on an waited.
Eventually, he relented. "Coffee, black. And the WiFi password, if you have one."
She rattled off the password and turned to make coffee. He retreated to a corner and pulled out his laptop, clearly settling in.
"We close at ten!" she called to him.
He didn't acknowledge her.
Xn+1 = RXn(1-Xn)
It went like this. A butterfly flapped its wings in Central Park. A four-year-old girl pointed at it, pulling her mother to a standstill. This meant that they didn't see the cyclist, who had to serve to avoid hitting the mother and her child. He fell over three yards later, right at the feet of young, stressed attorney, Topanga Lawrence-Matthews. She spilt her coffee all over her blouse, which meant that she had to duck into the nearest bathroom to try to clean herself up before walking home.
This detour to the bathroom meant that she took an extra fifteen minutes to walk home. In that extra fifteen minutes, her husband was worried sick, but couldn't reach her because her phone had died after being doused with hot coffee.
Because she arrived home fifteen minutes later than usual, her husband, Cory Matthews, was stressed beyond belief when she finally arrived, shirt stained and tired. Because he was stressed, Cory Matthews welcomed her home with a "where were you, Topanga?" instead of an "I missed you, babe".
Because she was greeted in slightly harsher terms than usual when she arrived home, and because she was still covered in coffee, Topanga Lawrence-Matthews didn't immediately go up to say hi to her six-year-old daughter. Rather, she started crying in her husband's arms, the stress of working two jobs and keeping their family afloat (even with the help of his parents) overwhelming her.
Because his wife was crying, Cory stayed to comfort her, instead of checking on his daughter, who'd been playing in her room for half-an-hour, where he'd sent her after he'd begun to worry about his wife.
Because the two of them didn't go up to their daughter's room, Riley Matthews never opened her windows on the first day of spring.
Because Riley Matthews didn't open her windows, Maya Hart didn't hear her singing while she was crawling around the fire escape of the building next to hers.
Because Maya Hart didn't hear the singing, she never went to investigate, which meant that she never crawled in through a bay window and never met Riley Matthews.
Because a butterfly flapped its wings in Central Park, Riley and Maya didn't meet until they were far older.
Because a butterfly flapped its wings in Central Park, Topanga Lawrence-Matthews came home and cried into her husband's shoulder about the stress she was under.
Because Topanga Lawrence-Matthews came home and cried into her husband's shoulder about the stress she was under, she and her husband had a long talk about ways they could ensure that she'd be less stressed.
This led to them deciding not to have any more surprises like Riley (they could always adopt if they desperately wanted more kids).
A couple of months and several long talks after a butterfly flapped its wings in Central Park, Topanga Lawrence-Matthews made the appointment to have her tubes tied.
Because a butterfly flapped its wings in Central Park, August Matthews was never born.
Xn+1 = RXn(1-Xn)
Riley was at a Debate meeting with Einstein Academy, and Farkle was there to support her. She was up against Isadora Smackle, well-known as the toughest of her competitors. Their topic was Churches and Taxes- Smackle arguing for churches being taxed, Riley arguing against it.
The debate ended in a stalemate, and Riley took it in her stride.
She went up to shake Isadora's hand after the debate, and Farkle joined her. She laughed quietly- Farkle's crush on Isadora Smackle was the least subtle thing in the world, and it was deadly obvious that Isadora returned his affections. So, once the generic platitudes were through, Riley stepped aside and let Farkle engage her back into Churches and Taxes. She smirked, and when she caught Farkle's eye, she mouthed ask her out. He went a light pink, but nodded slightly, and turned back to Isadora.
Riley smiled softly and began her walk home.
She was near Topanga's when she bumped into Maya. Literally.
But it seemed that Maya wasn't going to say anything, and so Riley wouldn't either. Except- "Thanks for not telling Lucas where I was."
Riley turned around to face Maya, "You didn't want me to."
Maya shook her hair out of her face and shrugged, "Yeah, but still. He said he gave you the whole speech about wolves-and-sheep, and you still didn't crack. So good on you." It looked as though that was all Maya had to say, but Riley wasn't ready to be done with the conversation "Maybe I'm not a sheep."
Maya smirked at her, all pretty cheekbones and sparkling eyes, "Oh yeah? What are you, then?"
"Maybe I'm a hummingbird."
Maya snorted, and Riley thought it was the most beautiful thing she'd ever heard, "A hummingbird?"
"Yeah," Riley nodded once, decisively, "Fast, busy, with a nose in everybody's business." That got a proper smile out of her.
"Well then, Hummingbird. Don't let me stop you."
Xn+1 = RXn(1-Xn)
"Well done on your B minus, Ms Hart," her dad was saying, holding Maya's test paper up for the whole class, "You've got some serious potential. I appreciate your take on the term Bourgeoisie, especially when you pointed out that most people fall into a class that would be considered the bourgeoisie according to Marx. Well done."
Lucas Friar jammed his elbow into Maya's side, snickering to her about actually having intellectual opinions. Maya shot a confused look at Riley in return (and no wonder- Riley had overheard her talking about that very subject a couple of weeks ago- it had been how she'd decided what to write).
History was their last class, and when the bell rang, Maya and Lucas damn near sprinted towards the exit. Riley stayed behind. Her dad took her home, too, and he only ever left after four. Riley had soccer practice today, but only from 3:30. So she sat in her dad's class quietly, and wondered about Maya.
When she left the class at 3, already changed, sports back over her shoulder, she nearly bumped into Maya.
"B minus, Hummingbird?"
"Well, yeah." Riley smiled softly at Maya, leading her over to an alcove, "Maya, I only used stuff I'd heard you say before. I'm just the one who wrote it down. You're a B minus, Maya, if not better."
Maya Hart refused to meet her eyes, choosing instead to look at Riley's jean jacket. It was old, had belonged to her dad in the 80s and 90s, but neither he nor her mom used it anymore, and it had somehow made its way into Riley's closet.
"I know you're there, cowboy," Maya suddenly said, and Lucas Friar stepped out of the shadows. Riley wondered, again, at how they did that. How they were so in sync.
"You're doing her homework?" He asked, confusion in his tone. Riley half-shrugged, nodding.
"She doesn't have time." Lucas held her gaze, both of them refusing to break away. After thirty seconds of tense silence, he nodded.
"Guess you're part of the pack after all."
But Riley shook her head at him, smiling at Maya. "Nah. I'm a hummingbird."
Xn+1 = RXn(1-Xn)
Chaos is the science of surprises, of the nonlinear and the unpredictable. These phenomena are often described by fractal mathematics, which captures the infinite complexity of nature. A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our familiar dimensions. Recognizing the chaotic, fractal nature of our world can give us new insight, power, and wisdom.
In another time, Farkle and Isadora stay in a relationship all through high school, but part ways on friendly terms when he chooses to go to the Massachusetts Institute of Technology and she to the University of California, San Francisco Medical Center. They wanted different things from life.
In another time, Lucas Friar doesn't become a veterinarian. He moves back to his family's ranch and takes over the running of the place when his grandfather dies. He's good at it, has a head for business. He eventually marries Vanessa Kinney, the cheerleader who dated his middle school best friend.
In another time, Isaiah "Zay" Babineaux never quite develops his passion for dance, and he too stays on the family ranch and handles things. He and Lucas never quite reconnect, Lucas angry about missing a year of his life and Zay pining over Vanessa.
In another time, Maya Hart loves art but never tries to make a career out of it. She gets into a teaching college and becomes a History teacher at Abigail Adams after Cory Matthews was promoted to vice principal. She never tells him that Riley wrote the essay on the bourgeois.
In another time, Farkle doesn't take to quantum mechanics the way he did when he was teaching Riley about the universe, so he takes over his father's company. He's good at what he does and increases Minkus International's charity donations by over 35%.
In another time, Riley Matthews moves to London in 2016, partly glad to be out of the shitstorm that is the USA elections. But without her friends, she never gets into astronomy, and she majors in fashion design (she's good, excellent even. She's eventually hired to work for Victoria's Secret, and people love her designs). She's at the Fashion Show when she bumps into her childhood friend, Farkle Minkus, and things evolve from there.
In another time, their baby girl, Ada, into the world 12 months after they're reunited.
In another time, Riley Matthews and Farkle Minkus married in 2029, three years after Ada Matthews-Minkus is born.
In another time, Riley Matthews-Minkus and Farkle Minkus had five children; Ada, Rosalind, Hugo and Henrietta, and Alexander.
This is not that time.
But that doesn't mean that, in this time, things were worse.
No, they're just… different.
All because a butterfly flapped its wings in Central Park.
Xn+1 = RXn(1-Xn)
