Title: Disquisitiones Arithmeticae: Projection
Author: Eve11
Date: Fri, 12 Feb 1999
Category: S
Rating: G
Spoilers: mild fifth season
Keywords: Minor character story
Archive: Yes to Gossamer. Anyone else may do so by informing me
Disclaimer: The X-Files and characters therein are property of Chris Carter and 1013 productions. Use is unauthorized but hopefully not unnoticed.

Summary: In projective space, even parallel lines meet at a single point. A family trip has unforseen consequences.

Author's note: For those of you who've noted the Erdos quote in my signature, it's pretty easy to tell that I'm a math major. And the abstract world of mathematics abounds with strange phenomena. 'Disquisitiones Arithmeticae' is a going to be a series of stand-alone stories with nothing more in common than the fact that they are mathematically inspired. But that's as far as the math goes, I promise. You don't need to be a math major to understand the story. I have a few more ideas, but 'Projection' may be all alone for awhile...

This story has roots in geometry. The projective plane is just like the regular 2D, x-y plane that we are all familiar with, except for one little addition: a line at infinity. Because of this, every pair of lines intersects at exactly one point, even parallel lines, which have a common point at infinity. Parallel lines, parallel lives... it's just a one letter difference. The exact situation in the story mirrors a novella I read years ago: "Anna to the Infinite Power" by Mildred Ames.

Thanks to Thyme for beta-reading! I will repeat my keywords from above: this is a minor character story, i.e. Moose & Squirrel don't really show up. If you still want to read further...

I open the door and the air tells me I am far from home. Florida air is heavy. Thick. Not like Colorado at all. It's so different; I want to leap out of the car and drink it all in. My excitement grows with each breath. We flew in a stale airplane, rode in a stale car. But this tells me we're here. Somewhere new. I grab Jamie's hand and tug on his arm.

"C'mon Jamie, we're here! Do you see the castle? Can you see it from here?"

I let his hand go so he can undo his seatbelt. He grumbles. "I saw it on the way in, Emmy. You were asleep. Too bad, you missed your chance." He looks at me solemnly. "They're tearing it down tonight. Moving it to Euro-Disney."

I clutch my backpack tighter with one hand, balling the other into a fist. "No, they're not!" I hit him on the shoulder. I can do that because I'm smaller than him and a girl. He can't hit me back, and boy, does he hate that. "You're just teasing me," I say. I mean, they wouldn't tear it down, would they? I get out of the car and scan the horizon beyond the parking lot, shading my eyes with my hand.

I don't see it.

"I'm not teasing," Jamie says loudly, scooting over to follow me out the passenger side door. "They need it in France -- they're losing money over there big time. They share it, you know." He swings long, gangly legs onto the pavement and stands up. Mom says he's grown four inches in the past year -- whatever it is, he's taller than me. He can see over the cars.

"Can you see it?" I ask him again.

"No, squirt." He pokes the top of my head. "I told you, you missed it."

I push him away. "Nuh-uh! You're lying! I'm gonna tell Mom you're teasing me." The front doors open, almost at the same time, and Mom and Dad step out. Mom bats the air in front of her with a map. Dad sets his ever-present briefcase on the roof of the car. "Okay, kids, we're here!" he says. "Our first Disney World vacation. Look at those palm trees!" I try to follow his hand, but I can't see over the cars. I start to run forward, but Mom stops me.

"Stay close to the car," she says.

Jamie looks at me and rolls his eyes. He wants everybody to think he's too old to be excited. He thinks twelve is all grown up. I stick my tongue out at him.

"Mom, Jamie was teasing me," I tattle.

"Jamie, don't tease your sister," Mom says tiredly. "You could try to get along for a change. Be friends."

"Mom," he whines, "I'm twice her age!"

I know he's two times older than me. But that's still not grown up. Two times older than him is just twenty-four, and Mom says that life doesn't even begin until you're forty. He's still complaining, but my mind sticks on the numbers. Two times older now. Six years apart -- constant. From now on, six years apart will be less than the number of years we've been together. I smile, watching the limit in my head, and grab his attention.

"From now on, Jamie, we'll just get closer together. You're twice as old as me now, but that will just get smaller. One and a half, one and a quarter..."


I was always good at math. Dad says it's kind of scary, but it's pretty clear to me. A fraction with both parts getting bigger. That little six won't even matter if the numbers get really big.

"If we lived forever," I explain, "we'd be the same age at the end."

I smirk as he tries to figure that one out. But Dad breaks his concentration, opening the trunk.

"Jamie, help me with the bags while your mom and Emmy go check in."

Jamie groans, and Mom takes my hand, heading for the hotel. That's one good thing about being small. All I have to carry is my Sleeping Beauty backpack.

The hotel room is big. It has a bedroom and a living room, and even a little kitchen. Mom makes us some sandwiches and juice. She tries to make me and Jamie sit at the table, but I wander over to the big curtain on the far wall.

A balcony! While Mom and Dad unpack and argue about plans, I get Jamie to help me open the sliding glass door. I walk outside, happy to feel the hot, heavy Florida air again, instead of boring old air conditioning. Jamie follows me out. I run to the railing. I've got to be able to see the Sleeping Beauty castle from here. I'd heard about it so much. I watched 'Sleeping Beauty' twice a day for the entire week before we got here.

I hear Jamie snort. "Are you still looking for the stupid castle, Emmy?"

"It's not stupid, Jamie!" I still can't see it. All I see are palm trees. It's got to be here. I know Jamie was only teasing, but I can't help myself. "They aren't really going to move it, are they?"

He laughs. "I swear, Em, how you can be so smart and so gullible at the same time, I'll never know." He comes up behind me and puts a hand on my shoulder. "Nah, they're not gonna move it, silly. They've already got one in France." He points off toward the left. "It's over there. You just can't see it from here."

I knew he was just teasing. I feel really dumb for asking. I'm not dumb. I don't want him to think I'm dumb. "*Poisson* means 'fish' in French," I say, trying to save face. I remembered that from 'The Little Mermaid'. I could sing the whole song if I wanted. I'm good at remembering things. "How do you say 'Sleeping Beauty' in French?"

<La belle dormante>

He shrugs his skinny shoulders. "I dunno. 'Ley Sleepeeng Byootay'?" he says in a bad French accent. But I don't listen to his answer. I'm thinking.

Something's missing. Something important. Something. . . I gasp. "Jamie, I forgot about my princess costume!"

He rolls his eyes. "Emmy, you're not going to put that on now , are you?"

"I have to! We're going to the castle!" I turn back toward the door, intent on my mission. "Help me get this open. I can't believe I forgot about it!" I stop at the door and turn around. Jamie is biting his lip. He looks like he's... laughing.

I put my hands on my hips. "What are you laughing for? This is important!" But that just makes him laugh harder.

"Nothin', nothin'," he says through giggles. He grabs the door and slides it open. I do my best to ignore him and head for my backpack, which I left on the table.

"Watch the modem cable, honey." Dad's voice scares me and I almost trip over the black cord. He's got his laptop set up on the table. "What'cha need, pumpkin?" he asks. I start to say that I need my backpack because it's got my princess costume in it, but his phone rings. He puts a finger in the air, turns around and heads for the bedroom.

I hear Mom from inside. "Jack, you're supposed to leave work at home. That is the definition of vacation."

"Not now, Beverly." A small click. "Clark here... mmm-hmmm. I see..." Mom comes into the living room, slowly shutting the door behind her. She looks annoyed, but when she sees me, her mood lightens.

"Hey Emmy, you want your backpack?"

I nod.

"Daddy moved it onto the couch," she answers, picking it up. "Here you go. Are you going to put on your princess costume for the castle?"

I nod again, unzip my bag and carefully remove the costume. It's so beautiful. It's satiny pink with lace on the outside, and it has a beautiful pointed hat with wisps of pink lace coming out of the top. It's not quite the Sleeping Beauty outfit from the movie, but I love it anyway. And I need to get dressed for the castle.

"Bev, could you come here a minute?" Dad calls from the bedroom. Mom leaves, and Jamie flops onto the couch. I take off my t-shirt and try to pull the dress over my head, but my arms get stuck half-way.

"Jamie?" I say with a mouth full of pink satin.

"You stuck, Emmy?"


"You could just go like that..."


"Okay, okay, here." I feel his hands tug on the dress, and it comes on the rest of the way. I put the hat on by myself.

"You could go as Prince Philip," I offer.

He only laughs. "No way you're getting me in tights, Emmy. I don't care how much we're supposed to get along."

"Hmph," I say. I twirl around, watching the dress fan out. The bedroom door opens, and Mom and Dad both come out. Dad heads straight for his laptop. Mom hangs some raincoats up on the bar next to the room entrance. I run toward her.

"Mommy, I'm ready now! Look! Look Dad! Can we go?"

"Very pretty," Dad says. His eyes never leave the computer screen.

Mom straightens the coats on the rack. "Not today honey," she says simply. "We need to unpack and get settled, and Daddy's got some work to finish. We'll go tomorrow, I promise."

Tomorrow!? I can't believe what I'm hearing. Tears spring to my eyes. "But I want to go today!" I cry. "Now! You said..." "Emily..." Dad gives me a warning. I shut up. I don't like to hear him yell.

"We're not going to do anything today?" Jamie asks, discouraged. I knew he was more excited than he seemed. "I thought we were going..."

"That's enough!" Dad's voice is stern, but not angry. "No arguing. We'll see Disney World tomorrow. Let Mommy and Daddy wind down for today, kids."

"There's a pool downstairs. You can take Emmy swimming," Mom says helpfully. But I've already decided that I am not getting out of the princess costume until I see the castle.

I'm crying. I try to use my sleeve to wipe away the tears in my eyes, but the lace is scratchy. All I do is smear tears over my cheek. "I don't want to go swimming," I say, hugging my arms around me to protect the dress.

Mom sighs. "Jack..."

Dad reaches into his pocket and hands Jamie twenty dollars and a plastic key. "I saw a game room when we came in," he says. "And I know they've got some kids' programs here. Why don't you two get some snacks or something and explore the hotel, see what's around?"

Jamie's shoulders droop, and he grabs my hand. "It'll be better than hanging out here, I guess."

"Don't forget... we're in room 512," Mom reminds us, ushering us out the door. "Be back by six." We find ourselves in the hallway, staring at pale blue walls and a dark blue carpet. I hear Mom's voice from behind the door. "I don't believe it, Jack. I just..." It fades out.

"I bet they're having a fight," Jamie pouts. "Time for us to disappear, huh princess?" I wish I could disappear. I turn around, looking up at the door.

512. Five-hundred-twelve. Two times two times two times two times twotimestwotimestwotimestwotimestwo.

"Five-twelve is a lot of twos," I mumble.

"C'mon." Jamie yanks my hand and we head for the elevator.

There's a playground next to the pool, with swings and a see-saw and one of those metal merry-go-rounds. I want to play on the swings. Jamie makes me take off my princess hat. He says it'll get caught up in the chain and broken if I'm not careful. When I give it to him, he tries it on.

"You look silly," I say, giggling.

"Not as silly as you," he answers back, ruffling my hair.

I stick my tongue out at him and head for the swings. I pump as high as I can go, and right at the top...

"Jamie I can see it!"

"The Sleeping Beauty castle?"

"Yeah! Just the top tower!" I let the swing slow down and leap off as gracefully as a princess should. "It's over there," I point. I smooth the wrinkles on my dress and give Jamie a smile. "Let's go!" I start running in that direction, but Jamie catches up with me quickly.

"Emmy! It's like a mile away! Disney World's a big place." He kneels down so he can look me in the eye. "We're supposed to stay here."

I know he doesn't want to stay at a dumb, boring old hotel. "'Till six o'clock, Jamie? C'mon, they won't even know we're gone. I just want to see it. That's all."

He scoffs. "How're we supposed to get there, genius?"

I look at him as innocently as possible, twisting my sneaker in the dirt. I should be wearing princess shoes for the castle, but that's not going to stop me now. My eyes follow a large metal track that leads to the hotel exit.


"Please stand clear of the doors."

"Por favor mantenganse alejado de las puertas."

After three times, I can mimic the monorail voice pretty well. Jamie just stands next to me, muttering. "...I can't believe I'm doing this..." The car speeds up.

I see the castle out of the window, tall and huge and wonderful. I'm so excited I want to fly.

"The next one's our stop, Jamie." I tug at his sleeve. "C'mon, there it is."

"...I can't believe I'm doing this..."

The monorail slows down gradually, and stops right in front of the castle. The doors open. I ignore the electronic warnings to stay clear of the doors and drag Jamie out onto the platform, toward the castle. He stops me for a moment, pats some dust from my dress, and straightens my princess hat.

"Five minutes," he says, "and then we're going back."

"Okay!" I run forward, leaving him to catch up. Up the steps to the huge castle doors. The turrets are so tall that when I look up, I get dizzy. My eyes find the topmost tower. "That's where she gets put when the evil spell puts her to sleep!" I say, pointing. "There's a secret passage up to it from the fireplace!"

Jamie isn't interested. "Four minutes," he says calmly.

Ignoring his wet blanket attitude, I yank open the door and rush inside. It looks like a big ballroom. There's even some grown-ups dancing. I recognize the music -- there's no words, but I remember the lines from the movie.

<"I know you, I walked with you once upon a dream...">

My heart is leaping in my chest. This is where a princess belongs. Not in some dumb hotel. There's mirrors all through the hall, and so many people! They can all be my subjects. <"I know you, the gleam in your eyes is so familiar a gleam...">

I head toward the mirrored wall, to a spot that's not too crowded. I need to make sure I'm presentable to my kingdom.

<"And I know it's true that visions are seldom all they seem...">

Hat -- straight. Dress -- beautiful. I smooth the lace in front.

Shoes -- I'll just have to cover them up.

Hair -- I release it from its pony tail and billow it out around my shoulders. My hair is a lot lighter than Jamie's. He's got dark hair and dark eyes. My eyes are blue. He says it's because I cry too much.

I give myself one last look in the mirror. And stop.

There's someone else there.

Another little girl, standing next to me, with light hair and light eyes. She's dressed up -- she's got a purple princess dress and a tiara. And a magic wand. I really like her dress, but my hat is definitely better. But her face... her face looks...

The music repeats. <"I know you...">

She looks like me. Exactly like me. I stare at the mirror, trying to find a difference. Eyes, nose, mouth, hair, shape... this can't be real.

I turn and look at her. She looks as surprised as me. Our eyes meet, and my heart starts racing. Is she real? Slowly, I reach out and touch her hand.


She brings a hand to my cheek. I don't know what to do... I don't know what to think... ohgod, I think I need to disappear.

She closes her eyes and starts murmuring words.

I close my eyes and start seeing numbers. Everywhere. Cascades of numbers. I wrap myself in sums. It's the easiest way to hide.

I gather my thoughts, steady my mind in the numbers.

When I open my eyes again, she is looking at me. I take a deep breath and meet her stare. Her eyes are wide. I think we both want to run away.

"What's your name?" she asks. It's almost a whisper.

"Emmy... Emily. Emily Clark."

"Emily," she repeats. "I'm Emily Rogers."

Our voices are the same.

Suddenly, I need something to be different. I need to have something that is just me. I think she feels the same way... I know she does. I can tell. She speaks first.

"But... my name is weird. I've got two middle names," she says triumphantly.

I can almost hear Mom's voice in my head when she's angry. <"Emily Dana Katherine Clark youcleanupthismess..."> Her face falls. She heard me -- I know I didn't say the names out loud. She can hear my thoughts.

I can hear hers. I already have.

And I know how she's different. I remember. I remember the balcony.

<La belle dormante>

Languages. "You don't know numbers like I do," I say. "You know languages. Words. You know French, I heard you. I heard you earlier today. La belle dormante."

She nods. "Sleeping Beauty," she translates. "I think I heard you too. Five-twelve is a lot of twos."

I nod. "Nine of them," I whisper. "You're going home today, aren't you?" I see a picture of green hills, and I hear a name.


"We stayed an extra day. I wanted to see the castle again."

That's why... the phone call... the fight.... that's why we couldn't go. I hear Jamie's frantic voice, calling for me. Nononono I can't do this... I can't leave. The room starts spinning, and I need the numbers again. We sit down, and everything is questions.

<why... who... what are we?>

We both need the answers. We both have no way of knowing them. Mom and Dad know, they know about this. Her thoughts echo mine, and suddenly I am afraid, terrified of my parents. They know, and they don't want us to know... I used to have nightmares that they were aliens, just pretending...

just pretending... and then I'd wake up. We can't tell them about this. Our minds are one stream. For a moment I can't tell where I end and she begins.


Then I hear her voice, frantic in my mind.

<we need to forget this... this wasn't supposed to happen... >

Later. Bury it now. Eyes wide, we stand and back away from each other, carefully, slowly. I turn around, close my eyes.

pleasepleaseplease let me forget this...

I build a wall of numbers, and shove Emily Rogers behind it.

"Emmy... Emmy please..."

For a minute, I can only see the numbers. I think I'm saying them out loud. Then I feel hands on my shoulders. I hear a voice.

"Emmy, snap out of it!"

I open my eyes and see Jamie. I think I'm ready to cry, and I have no idea why. I'm shaking. Jamie sees recognition in my eyes and practically yanks me to my feet.

"I want to go home," I say quietly.

"Good thing, Em," he growls. "'Cause we're leaving. Now."

I follow him out of the building, confused. Down the steps, to the platform. The monorail snakes toward us and stops. The entrance opens with a rush of air.

"Please stand clear of the doors."

"Por favor mantenganse alejado de las puertas."

In the back of my mind, I hear the whisper of a phrase that isn't English, isn't Spanish. Through the window, I look back at the castle, watch it retreat from view.

I remember going in. I remember the mirrors and...

<"And I know it's true that visions are seldom all they seem...">

Numbers. Five-twelve is a lot of twos. Suddenly, I'm afraid to look any further than that.

Jamie is silent for the ride, and I feel cranky. I think the lace on my dress is making my ankles itch. But princesses don't scratch their ankles in public. I let it go for as long as I can before reaching down, very unladylike, and bunching the dress up so my ankles won't feel it anymore.

At the hotel, Jamie draws me aside before we get on the elevator.

"Mom and Dad are not finding out about this. I'm forgetting it happened." "Mm-hmm," I say. "I won't tell, I swear."

For a second, just a second, I feel like he is the only one I can trust. I push it away.

"You better not tell, squirt," he says, pulling on my princess hat. He snaps it back onto my head.

"Ow! Jamie, quit teasing!" I kick him. The elevator opens silently in front of us.

"C'mon! Let's go see the pirates! C'mon, Daddy! Let's go, Aunt Em-lee!"

I feel Nicky's small hand tugging at my pants. Four years old, and he's already strong and strong-willed. Jamie says it runs in the family. He usually cites me as an example.

Funny, but I don't think the Clark family genes have much to do with it in my case.

"Emily?" I hear Jamie's voice behind me. "You coming?"

I stand in front of the castle. It looked so much bigger when I was six. If I listen closely, I can hear Tchaikovsky's Sleeping Beauty waltz filtering out through a tinny intercom.

For the longest time, that waltz only ever made me think of geometric progressions. It was only two years ago that I let myself see past the numbers. Now I wonder what she heard in it for all those years. Latin conjugations? Chinese characters?


His voice draws me out of my thoughts. "Sorry," I say. "Go. Take your son to see the pirates." I know he catches a tinge of bitterness in that statement. "I need to stay here for a minute. I'll meet up with you guys outside Splash Mountain in an hour."

"Okay..." he sounds unsure, but he lets the subject drop. "I think you can probably get a good princess costume at the shop down the street," he teases.

I laugh. "You're terrible, you know that?" As I head for the castle doors, I say over my shoulder, "I should have told Nicky they moved the Pirates of the Caribbean to France." That gets a questioning glance from Cheryl, but he shrugs it off. I see him smile and take Nicky's hand, and the three of them melt into the crowd.

I open the doors and step inside. I don't know why I chose here. Symmetry, maybe, but the paranoid person inside me says it is a nice public place on neutral ground. Nobody will think twice about a trip to Disney World. I'm humming to the tune, now.

<"I know you, I walked with you once upon a dream...">

I scan the crowd. In my mind, I'm looking for a little girl in a purple princess costume. I bet she's looking for a little girl in pink with a triangular princess hat. I remember everything now: the fear, the helplessness, the confusion. I felt it all again two years ago, but at least then I was old enough to deal with it.

Suddenly, my mind catches a hint of words I don't know. But it is enough. I know it's her. For the first time in twenty years, I feel her presence. I look up and see a perfect clone of myself walking toward me.

She's not in purple; instead it's a modest blue sundress. Well, I'm not in pink, either.

"Emily," I say.

"Hi," she answers, taking in my appearance. "Looks like you found out pink just isn't our color."

It's enough to break the ice, for now. We move to a bench and sit down, each feeling the other's nervousness.

"I'm glad you got in touch with me," she says quietly.

I smile. "I've been doing some research for the past two years or so," I start, "and I've come up with... a name." I falter over the words. It's not what I was going to say, but I can't bring myself to say that part out loud. Not yet. She hears me rolling the word around in my mind. For a moment, I hear her thoughts in tandem with my own.


I hand her a manila folder, and she opens it without comment.

We both need answers.

We've both been waiting for a long time. She reads the name aloud, letting it hang in the air between us. A bridge.

A connection.

A beginning.

"Dana Katherine Scully."

End notes: The limit Emily's trying to describe in the beginning is this:

lim (n -> infinity) of (n+6)/n = 1. (i.e., when n gets big, n+6 and n are pretty much the same)

That, and 512 = 2^9.

That wasn't too bad, was it? I imagined the two Emilys leading parallel lives; they were never supposed to meet. But, projection being projection, they do meet at one point. Also, this is probably set in the future, so if the Sleeping Beauty Castle doesn't look like that on the inside (actually, I'm not even sure you can go in ), I'm just going to pretend that renovations were made in the year between now and when this story takes place.


"A mathematician is a device for turning coffee into theorems." -- Paul Erdos

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