Fourth-Dimensional Tessellations of the American College Graduate

Marie Vibbert

Alana’s ex-boyfriend, Steve, met her at the driveway to Windermere Farms, his three-acre orchard wedged between the Rapid tracks and East 15 Street. “Thank you for coming so fast!” He backtracked like an excited border collie, leading the way up the weed-choked brick driveway.

“Is it taxes?” Alana’s friends were always asking her to do math for them, like it took a degree in mathematics to calculate a percentage, and since Steve’s text had said “Urgent!! Need U ASAP!” she assumed he was in legal trouble.

“No. This is a real math mystery, I promise! With bees!”

Steve’s latest scheme involved buying different strains of honeybee from around the world. He hoped to solve global bee collapse and make enough money off honey to pay his rent. Neglect and white flight had left this farmhouse preserved and cheap. It was surrounded by chain-link fences topped with broken segments of barbed wire weighted down with wild grape vines. Vacant-eyed apartment buildings stared over it all.

“The closed ends of honeycomb are trihedral sections of rhombic dodecahedra,” Alana said. She loved the sound of the words, the hard consonants repeating like the repeating angles bees built into structures.

Steve’s beehives, like a miniature apartment complex, stood in three ranks on the concrete slab where the home’s garage had once been. “Huh? Yeah, you are so the only one who can solve this mystery.” Steve threw a bundle of mesh over her. Alana struggled with the netting while Steve lifted the wooden cover from a beehive. Then he knelt to unsnap clasps like old suitcase closures on the front. “Look!”

Alana felt a welling of feeling—of magic, sexual excitement, of fear. Like the feeling when you crested the first hill of a roller coaster and looked down the track.

“Well?” asked Steve. “What is it?”

Alana knelt. The wax was warm and sensual with glistening honey. Thin lines formed, not hexagons, but elaborate shapes, something broken, with polyhedral sections branching off and crossing through each other and seeming to undulate.

“It’s not even tessellating,” Alana said. “Honeycomb is supposed to tessellate.”

“I know!” Steve bounced on the balls of his feet. “Is it completely random? Do I have chaos bees?”

“I’m not . . . .”

“I took a section for you!” Steve ran to the house.

Alana stared. The shapes were impossible to track, but there was some order; it lacked the grace of pure chaos.

Steve pushed a Lucite cube at her nose. “This was a photo frame of my dad’s. Photo-cubes. He had them all over the house. Not the point. Look at the section. Is it fractal?”

Alana turned the cube. In motion, the seemingly random shapes tessellated—they repeated.

“Alana?”

“Hecatonicosachoron,” Alana said. Syllables locked into each other, building meaning like links build a chain.

She felt Steve move the wooden hive-door in the grass and kneel beside her. “You know what it is?”

“I think . . . .”

She reached out and poked a hole in the beehive with her finger. The hole closed. When a fourth-dimensional object rotates, it will seem to heal because not all faces are perceived in the third dimension. “I think it’s a 120-cell. A hyperdodecahedron. Maybe. Tiled with other shapes. I need a camera.”

“I already took some pic—“

She shoved Steve. “Quickly!”

“Hey, there’s no rush. It’s been like that for . . . .”

“Go!” She slapped at him until he ran for the house. She didn’t want to take the time to explain her fears. It wasn’t hot enough to melt the wax, but wax deformed with weight, with gravity. What if the shapes went away, like unobserved quanta?

By the time she got all the angle shots she wanted, she was sticky with honey from pressing close and pulling sections of wax apart to capture the structures within.

She looked down at the smeared camera. She tried to lift her veil and it got tangled in her honey-slimed fingers. Steve was standing near at hand with the smoker pot. “Wait, where are the bees?” she asked. The next rank of hives were covered in a soft, moving matt of bee bodies all along their fronts, but there were none on the open hive and only a few in the air nearby.

“I don’t know.” He laughed. “I haven’t seen a bee on this hive in months. I thought the queen was dead. I found the weirdness when I started taking it apart. The honey is just . . . appearing. Isn’t it wild? I was so scared there wouldn’t be enough honey to sustain the hives, or enough pollination for my grapes.”

Alana walked away from the hive. She had to think and she couldn’t look and think at the same time. The complex geometries and the inner glow of wax and honey were too mesmerizing. She sat down on an overturned tub. A police siren wailed in the distance.

A glass of iced tea, sweating, presented in front of her, attached to Steve’s hand. She took it. “So,” he said, “You still haven’t explained it—I mean, you gave it some names, but what is it? What could have made my bees do that? At first I thought it was disease or poison . . . the messed up area gets bigger each day. It’s jumped to the two closest hives. But the honey is still good!”

“It’s a fourth-dimensional shape,” Alana said.

The hollow tub echoed with the force of Steve’s butt dropping onto it next to her. “Whoa. Fourth-dimensional bees? Are my bees invisible? I could be stung by invisible bees!”

“No, it doesn’t mean that. We perceive three dimensions. Fourth-dimensional shapes appear to us as three-dimensional. It’s . . . if you poked your fingers into water, a being who perceives only the water surface would see five circles, not the one shape of your hand. We see what intersects our plane of existence.”

Steve shook his head. “Wait, wait—the fourth dimension is time, isn’t it? Fourth-dimensional bees sting you in the past! I’ve already been stung! My bees are dead! It’s a paradox.”

“The fourth dimension isn’t time.” Alana had to breathe out slowly and calm herself down. This wasn’t the fiftieth time she and Steve had had this conversation. Usually some movie or TV show was involved. “Look—this is—yes. Thank you. Fourth-dimensional geometry is my specialty. I never thought anyone would ever call me about my specialty.”

Steve smiled goofily. “What are you going to do?”

“Analyze the photos. Write a paper.”

“Okay, but I’m going to start advertising honey from beyond our dimension, like, yesterday!”

Alana’s current boyfriend, Huy, smelled of turpentine, as did his attic apartment on Hessler Street. Splatters of paint—a past resident’s work in a mathematically improbable evenness—decorated the slanted walls. A bower window looked out on the top of an oak tree and let in the smell of someone grilling on a porch below.

Huy was as excited as Alana. He propped the painting he had been working on against the sofa and put a fresh canvas on the easel. “Fourth-dimensional bees.” His fingers moved quickly, just a tremble, a quick wiggle, and golden lines formed on the canvas. “Bees change gender chemically, you know.”

“Only at birth, and you wouldn’t want to be a drone,” Alana said, lying on the bed, which was musty from not being changed, perfectly boy-like. “As soon as food was scarce, the hive would kick your gigolo ass out.”

Huy laughed with his tongue against his top teeth, which were white and strong and always set Alana’s heart racing. “You know, Steve is cute,” he said, teasing her back. A mutual attraction to Steve had been their first connection. Art was their second. Once Huy painted water on her with his favorite sable brush. It was a sweltering day, and what started as a cooling technique became breathless foreplay.

The same brush strayed through sunshine yellow and goldenrod and ochre. “What are they building, these mathematician bees? What do they think they are making?”

“Bees don’t understand geometry,” she said, “But they make perfect hexagons. These bees . . . they can’t know fourth-dimensional physics or what they are creating.”

Huy crawled onto the bed. He teased her with the possibility of brushing paint on her belly where her shirt lifted. “Blind, passionate instinct,” he said.

“So where have the bees gone?” Alana asked. “They aren’t dying. They are just . . . gone.”

He shrugged. “That is the weird question.” He bent backward, unfolding in a boneless way that made her want to touch him. Huy had been a dance major, first, before painting. It left him unfairly graceful. “What are you going to do?” He went back to his easel.

“Find them,” Alana said. She twirled a finger in her hair, locking the curls into a column. “No one else has published on this. I checked.”

“Better get cracking, then.”

Alana twirled another lock, then another. She remembered twisting her younger sister’s locks, making tessellations of little puffs of afro. She glanced at the cracked mirror above the dresser. Could she make a hex pattern in her hair? She was procrastinating. She picked the twists apart as she looked for her laptop. She only had an hour before her shift at the coffee shop, and after that she had a MATH 102 class to teach.

The oddest question was: why hadn’t the bees made a 24-cell? That could tessellate on its own in four dimensions. Mathematicians called the pattern “tesseractic honeycomb.”

Alana was in her favorite coffee shop—not the one she worked at—a reclaimed Victorian druggist’s, the teas sorted in tiny little drawers with brass label-holders. The corners were braced, making flat hexagons, and she unfocused her eyes, turning her head to appreciate the honeycomb of wood.

She stopped. She looked down at the photographs and notes splayed out in front of her. She had printed out perspective drawings of three-dimensional projections of hyperdodechedra and other hyper-objects. She had been laying these, on tissue paper, over the photos and marking similarities and differences. And there were differences.

She got up without paying for her second cup of mocha.

Steve didn’t answer her text, but he showed up shortly after she was arms-deep in one of the hives they hadn’t opened yet.

“What are you doing?” Steve asked. “Careful! Don’t waste the honey.”

“I’m . . . rotating it.” Alana huffed with effort. She did damage the honeycomb, and spill honey, and there was a terrible ripping sound of two layers coming apart, but then she had the comb turned. She sat back and licked a glob on her arm.

The tessellation was off. There were interruptions in the pattern. Of course there were—120-cells didn’t tessellate. They didn’t stack like hexagons; they needed structures between them. Joinings. Ligatures.

Wiping her hands as quickly as she could on the grass, Alana got the camera out and took pictures. “There’s . . . something in there. In the math. It’s not simple repetition, there’s chaos in it, randomness.”

Steve said, “Well, yeah, it looks random to me.”

“But it’s not perfectly random. Do you know what that means?”

Steve shook his head.

“It suggests intelligence. It suggests direction. It suggests . . . .” Alana almost couldn’t breathe. “It’s a message.”

“Cool! From who? The bees?”

“I don’t know!” Alana’s sticky fingers hung in the air in front of her face. Evidence of non-human intelligence dripped and sagged in front of her and she didn’t know how to prove her theory.

Steve guided her onto the back porch and sat her down. The screen door opened and shut with a yelp of springs and a slap of wood. He brought out a bowl of warm water and washed her hands and arms and face. The rag smelled of beeswax and lemons. “There we go. Easy. This was supposed to be fun.”

Alana sniffled. She felt so comforted. Why had she ever broken up with Steve? “I don’t know how to interpret the message. I have data . . . but is it language? How do I decode it?”

“Hey, you know who is a whiz at that stuff? Laurel. The programmer? Why don’t we call her and have her take a look?”

“Are you guys still dating?”

Steve rolled his eyes. “You’re joking. Laurel was before Christian.”

“Oh. Right. Is he around?”

“Alana. Seriously? Christian was two girlfriends and a boyfriend ago.”

Laughs bubbled up through Alana. She bent over. Misunderstanding, Steve set his rag aside and hugged her. “Hey, hey. It’s okay. You’re okay.”

Alana got control of her breathing with a snort. “You have a repeating pattern,” she said. That was why she’d broken up with him. He didn’t get the joke.

Huy peered into the hives with all the absorption of an artist. “It’s more meaningful in three dimensions,” he said. “I should sculpt. Or just . . . heavy paint. Slather the canvas in dripping amber.”

He backed up into Laurel, who had been looking over his shoulder. She was taking a break to smoke a cigarette while her computer ran a dozen pattern recognition programs on the series of ones and zeroes Alana had interpreted from the presence and absence of ligatures in the honeycomb. Laurel was very well groomed. Her eyebrows were like black tildes marking her blue eyes as operators.

Alana didn’t like the way Laurel’s eyes travelled down Huy. Worse—he blushed and looked away.

Steve came out of the house with a tray of glasses. He puffed his chest up. “Who knew in undergrad we’d be unlocking the secrets of the universe at my farm?”

“The odds were good it would involve your exes,” Laurel said. She took a glass. The lemonade caught the sun. “Was there anyone you didn’t date? We could use a biochemist.”

“What for?”

“Alana thinks the honey might be alien.”

Alana felt everyone turn their attention to her. She hoped she hadn’t been scowling. She cleared her throat and sat up straighter on the hay bale. “It might be helpful to see if the honey was formed from different pollen than Steve’s other, normal hives.”

Laurel’s computer beeped. “Oo! That’s the compiler.” Laurel handed her lemonade to Alana and bent over her laptop, which sat on top of another hive. They were all ignoring the usual safety precautions around the strange hives. No one had been stung by an invisible bee yet.

The morning had been spent in figuring out the order of the message—how do you start and end in four dimensions? Top-bottom, left-right? Horizontal or vertical? There were 384 combinations in a fourth-dimensional grid. Steve presented the solution: time. “The fourth dimension!” he crowed, though really it was a first-dimensional solution. He talked endlessly of travel-stain and brood-comb—beekeeper jargon that helped track the age of each section of wax. Steve traced the ligatures into an order, oldest to newest. Alana had assigned each ligature shape a number, creating a string of numbers for Laurel to put in her computer.

Codes. Line up a string of numbers. Guess that the three represents an E because there are more threes than other numbers. You start to see two numbers always before 3, they are T and H. You’ve accomplished the code-breaking of middle school passed notes. Alana had thought—and gotten excited thinking—that she would have to write new algorithms to detect patterns, but the business of code-breaking was well-trod ground for computer scientists and Laurel hadn’t even had to download any new software.

Laurel stepped back from her computer. “Huh. Um. Crap.”

“Did it not work?” Steve asked.

Laurel frowned and typed furiously, ash scattering from her cigarette still held between two fingers.

“What is it?” Huy asked.

Alana tried to see over Laurel’s shoulder. Lines of text scrolled up the screen.

“I don’t like that look. That’s a bad look,” Steve said. “She looked like that when we broke up.”

“Shut up,” Laurel said. She picked up the laptop and carried it to the porch.

Alana, Steve and Huy followed. Alana drank the lemonade. It was too tart and cold.

Laurel had the laptop balanced on her knees. Her cigarette burned down, forgotten between two fingers. Alana was amazed at how she carried it like a ring, tossing smoke curls as she gestured. “There’s a pattern, all right. I’m sorry, guys. I don’t know how else to interpret this.”

“Is it a message?” Alana asked.

“Yes, and the message is: Danger. Quarantine.”

“Are you sure?” Steve asked.

Laurel gave him an insulted look.

“But who is the message for?” Huy asked, “Is it for Steve? Humanity in general?”

Laurel flicked her brassy hair away from her tildes. “We’re the ones who are killing them off with pesticides. It’s a warning to other bees.”

Alana got out her phone. “Steve dated a biochemist. I’ve got him on twitter.”

“Maybe we won’t need him,” Huy said. He opened Steve’s refrigerator. “I was researching honeycombs and color for my project. There was this wild story. It was in Salt Lake City. Suddenly, all these bee hives were producing red honey, in three different counties.” He stuck his head deep in Steve’s refrigerator, pushing and sorting jars and take-out boxes. “Dude. Clean up now and then. Anyway . . . yes.” He straightened, brandishing a fat jar half-full of red liquid. “Turns out, it was a maraschino cherry factory dumping its wastes. The bees would fly past miles of flowers just to eat corn syrup and red number five. Or . . . whatever is in this.”

Steve and Laurel had gone back onto the porch, talking over each other. “That’s idiotic,” Steve said. “Alana! Do you know why she’s interpreting this message that way? Semaphore.”

“It’s a beautiful visual system.” Laurel leaned in the doorway. “What would you use to communicate across languages? Shakespeare?”

Huy pushed between the two of them. “It’s certainly human to jump to the conclusion that everything is a warning. Everything is out to get us, so spray the bullets and the chemicals!”

“Easy, art school,” Steve said. “This is about data, not emotion.”

Alana helped Huy lay a cookie tray in the middle of the hives. The cherry juice shone like stained glass in the sun, like it had depth beyond the scratched tin sheet.

“The pattern repeats in a perfect checkerboard,” Laurel said. “It’s exactly the semaphore signal for danger or quarantine, minus the color.”

“Maybe it’s not a message at all,” Huy said. “Maybe it’s art.”

“How long until we can tell if the invisi-bees are eating the cherry juice?” Steve asked, argument forgotten.

“Invisi-bees?” Huy said. “Really?”

“I haven’t patented it or anything, dude.”

Alana stopped at the corner store after work and loaded two jars of maraschino cherries in her backpack alongside her laptop. She had wanted three, but there wasn’t quite room. As it was, she worried about the clinking sound when she went over potholes on her bike.

Huy and Laurel were already at Steve’s, playing cards with him on the screened-in porch. A jar of de-juiced cherries sat in the middle of the table, like the eggs of some alien insect.

“Hey,” Steve jumped up. “Do you think it’s been long enough to check the hives for red honey?”

“No,” Huy and Laurel both said, glaring at each other. They glared like the rest of the world had vanished. Alana felt uncomfortable. Steve ran into the kitchen. Alana followed.

Steve lifted an old-fashioned microscope onto his counter. “They’ve been arguing all day. I think they like it.”

Alana bit her lip. She set her backpack, laptop, and jars of cherries on the counter that separated the old butler’s pantry from the kitchen.

Laurel swung the screen door open with a bang of her hip. She held two glass slides and a lit cigarette aloft. “Anyone who looks at the math can see the pattern.”

Alana opened the file of data.

“Hey, you can’t smoke in here. We’re doing science.”

“Oo, so commanding.” Laurel bumped Steve with her hip, too. She played at holding the slides away from him.

“Quit it. You’re getting ash on the science.”

Huy shouted from the door, “Is anyone going to help me clean up?”

Steve waved Laurel away like she was a fly and bent to the microscope, adjusting the knobs. “Yes! There’s red in there. So is this one the weird hive or the normal hive?”

“If I tell you before you check both samples it will bias you. Here’s slide B.”

Alana had been trying to concentrate and listen at the same time. Something jumped out at her in the numbers and she stopped listening. She checked her figures. “It’s tessellated.”

“What?” Laurel said. “Shut up,” she said, when Steve tried to tell her about the slides.

Alana said, “The message—the pattern you found. It’s not one pattern, it’s a repeating pattern of a repeating pattern.”

Laurel nudged Alana out of the way and nearly burned her with her cigarette. Laurel said, “Loss-prevention. Redundancy. That’s why it looked like a checkerboard. They really cared about the message coming through.”

“Makes sense,” Steve said, “If you’re sending a message through beeswax. I mean: one fire and kaplop.”

Alana nudged Laurel out of the way as she saw her about to mess up a perfectly good formula. “I’ve been doing work on tessellations and fourth-dimensional geometry. Wait . . . stop.”

Laurel tried to push her aside again, then stepped back. “Okay, but don’t forget the second-order changes.”

Alana felt even more anxious than the first time, higher on a steeper roller coaster. A clean line of numbers formed. “There it is. Our message.”

“But . . . .” Laurel’s tilde eyebrows were tighter, almost square-root signs. “But what does it mean? What language do we decode it to?”

“How about English?” Steve asked. Alana realized she’d forgotten about the boys. Steve shrugged. “I mean, these are American bees, right? What other language would they have seen?”

Huy shouldered his way into the kitchen with the honeycomb separator and eyedropper. He pushed Alana away with his elbow when she tried to touch him. “I’m sticky,” he said. He bent over the kitchen sink.

“I love a man doing housework,” Laurel said, luxuriating on every vowel she spoke like it was the dirtiest joke.

Huy gave her a heated look. “Thanks for not helping, assholes.”

Alana left without looking at them.

Alana stayed in her own apartment that night, a joyless single-bedroom box all in cream and taupe. She wanted to move in with Huy, but for now she was glad she hadn’t. Math was a better boyfriend. Numbers didn’t care if you understood them, didn’t get angry or jealous. They just were.

She had dozens of texts. She didn’t read any but the one from Steve saying that both populations of bees had fed on the red cherry juice and to please stop buying cherries. She’d read it because she saw it was cc’ed to everyone.

There was a knock at the door. She ignored it. Then there was the sound of a key in the lock. She put her pillow over her head.

Rustle of paper bags. “I brought Chinese,” Huy said.

Clink of plates being set out on her table. She was not going to give in. But she was hungry and now she could smell ginger and soy sauce. She burrowed deeper under the pillows to block the delicious smells with her own hair-oil and sweat.

Huy touched her elbow. “Come on. I got dumplings. You love dumplings.”

Alana made a muttering sound she hoped could be interpreted as both ‘go away’ and ‘thanks’.

Huy sat next to her. The mattress dipped toward him, creating a gravity well she could easily fall into. She wanted to wrap around his comforting solidity. She held firm and scooted toward the wall.

Huy plucked the pillow from her face. “I flirted with Laurel. Laurel, I think, flirted back. It was childish and stupid. I’m sorry.”

Alana wanted to stay mad so she scowled and stayed put. He backed up, pulling her off the bed. “I’m not interested in her. Be more worried about Steve. Or dumplings. There’s your smile.” She fell against him naturally. His hand on the small of her back, he rocked her in a four-step. He started singing, “Dumplings . . . dumpling dance come dumpling dance with me.”

Alana froze. Huy frowned. “What?” he asked.

“Dance. Bees communicate through dance.”

Huy laughed as she clambered over the bed to get to her laptop. “I thought you were mad at me!”

“I am,” Alana said. “Bee dances are all about repetition and angle.”

Angles. She had been treating the ligatures as binary—presence and absence. Then she had tried shape—thin, fat, hourglass. What if it was the angle of each that mattered? Relative to what? The center of the polyhedron? She scampered to her dresser and found her initial photos and drawings still laid out.

“I’ll put the food in the fridge,” Huy said. “And call Steve.”

Laurel was at work, so they met in her research lab. “All right, I admit, I was totally wrong about the danger flag. It’s English.” She frowned at Steve’s smug grin and said, “Hush. Anyway, the language predictor said 98% chance of English, so I did the translation assuming that, and there are, like, sentences.”

She laid a tablet on the table. “This is real arm-hair-raising shit.” She backed away.

We have gone where you cannot kill us. You see us but we also see you. We could take our food. We could starve you.

“I . . . need to check your math,” Alana said.

“Please,” Laurel said. “But once I saw there were 26 distinct angles, I just assigned a letter to each and brute-forced it. I cold have done this on a napkin with a pencil.”

“We chased them into the fourth dimension,” Steve said. He slumped against a slate counter. “Guys? Humanity just got dumped.”

“There’s more to the message,” Alana said.

Laurel said, “I wanted to be sure. There’s about fifteen percent more, if I got the punctuation marks right.”

“It’s not going to end with ‘lol kidding’,” Steve said.

“Have a little faith.” Huy put his hand on Steve’s shoulder. “Let’s get coffee while the girls get their math on.”

An early winter storm lashed the storefront as they gathered four months later. Laurel chewed her nails, staring hard at the “no smoking” sign while Steve fetched their combined coffee orders.

“To us!” Steve declared, holding a glass aloft over the magazine spread on the table. It was as glossy as a freshly painted nail. Real print. Huy’s painting graced the facing page at the beginning of Alana’s article. They were all co-authors.

“I like Margot,” Huy whispered, glancing at Steve’s new girlfriend. She was a rubenesque beauty with a passion for organic foods. “But I’m not going to grow attached. Seeing him with a new partner is like seeing a kid with a new goldfish.”

Alana poked him in the ribs and he kissed her bicep.

Laurel said, “What gets me is the large size of fourth-dimensional space. It’s huge, right? Way more room for way more bees. Enough to build a hive mind larger than ever before. Forget writing us messages—these bees could be creating singularities! We could have a whole other article on measuring hive-intelligence.”

“What gets me,” Steve said, marveling at the article, “is how the weird hives stopped. I mean, for a while there . . . .”

“We should have taken measurements,” Alana agreed.

“More weird hive by ratio each day,” Steve said. “And now . . . it’s steady. Maybe it’ll shrink.”

“Maybe they know their message was received,” Huy said.

All four friends grew quiet, reading the displayed page. Above the technical title, “Fourth-Dimensional Aperiodic Tessellations in Geometry of Honeycomb,” was the text of the bees’ final message, as near as they could figure, in English.

The end of the message made Alana’s heart clench. Made her squeeze Huy’s hand under the table.

“We love you,” it read. “We forgive you.”