An endless cascade of snowflakes falls onto her head. The crunching sound of snow underfoot gives way to the squelching sound of slush. Inga gazes at the gray mass, listening to the muffled sobs of people descending into the subway. Soon, the rustling fades into the roar of the trains.
Put on gloves, hold the door, tap the card, avoid stepping on the tile lines (just in case), and stick to the right side of the escalator. These mechanical actions lull her, and even the roar of the trains cannot shake Inga from her morning stupor. Snow melts on the fur trim of her hood, trickles running down her shoulders. Moving down the gear teeth of the escalator, Inga drowsily listens to the drawn-out echoes of a violin.
A vast white arch, a transition from one station to another. The whistle of the wind and the flow of passengers. «We are wandering and rambling, ” Inga thinks. „Like a mass of sleepy snowdrops disturbed by someone under a snowdrift. How deep are we in this drift? What if we can’t crawl out when spring comes?“ „You dropped a mitten, ” a voice rang from the edge of an imagined hole, and Inga, pulling her head out of the abyss with great effort, looked toward the sound. The stranger held out a dark blue mitten to her. Inga found it strange that the snowflakes in the girl’s curly hair hadn’t melted during the journeys across the escalators and corridors. It was as if she had just arrived at the edge of the platform. But her black-and-white scarf looked even more unusual. It was woven from whimsical curls and intertwined spirals, and the longer Inga examined the interplay of abstract patterns, the more details emerged before her eyes. The stranger lightly waved the mitten in front of the mesmerized Inga. The girl took it, mumbled a thank you shyly, and, unable to contain her curiosity, asked the girl where she got the scarf. It was woven from whimsical curls and intertwined spirals, and the longer Inga examined the interplay of abstract patterns, the more details emerged before her eyes. The stranger lightly waved the mitten in front of the mesmerized Inga. The girl quickly tucked it back into her pocket, murmured her thanks sheepishly, and, unable to contain her curiosity, asked the girl where she got the scarf. The stranger smiled: „Do you like it?“ „Very much.“ „I’m afraid it’s an exclusive piece.“ „I see… It suits you very well.“ „Thank you!“ The train still hadn’t arrived. „How do I interact with people? How do I interact with people? Should I say something else?“ In a panic, Inga couldn’t think of anything better than the classic approach: „… What is your name?“ „Julia, ” the girl turned her head toward Inga and looked deeply into her eyes. The irises of her eyes were speckled with dots, and it seemed they were slowly spinning, like dust or snowflakes in the sun’s rays. „Julia?“ „Exactly! And you?“ „Inga.“ „Nice to meet you, Inga.“ „Likewise.“ Finally, the roar of the approaching electric train was heard. „Well, have a nice day!“ Julia said loudly, waving her hand. „Thank you, you too. And thank you again for the mitten!“ „You’re welcome! Take care of it.“ „Attention, doors closing…“
Inga quickly blended into the crowd of passengers and lost sight of her new acquaintance. She settled in the space between the carriages, leaning against the accordion-style wall. Under the rumble of the wheels, she managed to drift off, and in her dream, she saw spiraling threads of a wondrous scarf. They danced in circles within the underground tunnel, and as the train passed through, the twists leaned against the walls, transforming into shadows.
The path ran through the park. The dark trunks of the trees were etched through the blanket of snow. The walkway was slippery, so Inga moved in small steps, spreading her arms out to the sides. The day would be the same—slow and monotonous, cautiously shifting from one second to the next, hiding under the cloak of night as soon as possible. January, what could you do? Inga imagined herself scurrying like an clumsy squirrel along a branch. The branch forked—a turn to the left, another bifurcation—a turn to the right. «If only the branch didn’t end but kept dividing into an ever-increasing number of twigs. How many possible routes would exist then? Where would they lead me? Perhaps they would continue infinitely, and I would run like a squirrel on a wheel, down an endless path… Though no, sooner or later the twig would become too thin and snap, and I would tumble down.»
Crunch. Inga mechanically looked toward the source of the sound. Forgetting the icy carpet beneath her feet, she slipped and fell onto her side, clutching at the air and the trees, which waved dismissively at her. Fortunately, a noble drift of snow lined the edge of the path, absorbing the impact.
Inga liked that fleeting blend of vertigo and a spark of fear, which was quickly replaced by relief. «It’s as if I am falling into the arms of a faithful friend without even looking. A kind of trust test.» The girl didn’t hurry to get up, waving her arms around to create a snow angel.
The sky above Inga was an impenetrable white, and the girl felt like a parrot in a paper-covered cage. And yet, she was drawn to this side of winter: the world seemed to begin on a fresh page. But instead of creating a new version of herself, it froze, crystallized, quieted. Only the birds reminded one of the life existing in the winter forest. Inga nodded at a crow, whose caw matched her thoughts, that landed on a branch above her head. As if roused by an alarm clock, the girl rose from the drift and glanced at the imprint of her coat. Inga liked looking at herself with wings, even if that imprint wasn’t entirely real. There is a certain charm in simplification.
Next to her snow angel, Inga noticed boot prints leading away from the main path. She had never seen such a detailed sole pattern. Even the imprint looked impressive—triangles of spiral galaxies, each swirl encompassing even more swirls… Inga mentally congratulated herself on the day of intricate footprints.
Shaking off her backpack and adjusting her scarf, the girl followed the trail. She stepped carefully beside the footprints, trying not to trample them. They meandered quite a bit, stubbornly refusing to follow a straight line. Passing the bird feeders, where large titmice and siskins usually ate, Inga paused to give the birds seeds. The feathered friends knew her well, and sometimes they even ate directly from her hand. This time, Inga decided to play with the titmice. She tossed the food into the air so the birds could catch it mid-flight. Even when the seeds fell into the snow, they didn’t stay there long. One of the birds quickly found them.
The lapwing traced the trajectory of one of the seeds. After flying several centimeters from the branch to the ground, it disappeared into a hollow in the drift. Moving closer, Inga noticed that the contours of the drift were sharp, as if cut from paper. The longer she looked at it, the more irregular it seemed. Its entire mass looked as if it were composed of triangles. New vertices sprouted from the sides of each peak, and from those, even more small vertices. The bird, meanwhile, vanished into the ground, leaving not a single feather behind. The strange tracks that had led Inga here ended near this small mound.
The girl tried to regain her composure. «It’s just a drift; a snowdrop monster isn’t hiding in it. I probably just didn’t notice how the lapwing flew off to its own business.» Nevertheless, she decided to check and touched the surface of the drift. But her hand passed right through the vertices of the triangles, which felt slightly prickly.
Inga was certain she was about to reach a solid surface, when suddenly her entire body plummeted downward, into the false canopy of angular snow. For a moment, it seemed the titmice had frozen mid-flight, neatly bisecting a white veil of clouds. All reality transformed into a color photograph, whose reverse side was a white sheet riddled with infinitely broken lines. There was no comfort in the fall’s landing. Drops of fear melted into one raging sea of terror.
The piercing brightness blinded her, and after confirming that besides the negligible patch of azure, the space was absolutely empty, Inga closed her eyes and continued falling. The fear subsided because there was nothing to fuel it. A multitude of possible explanations for this flagrantly absurd situation surfaced: she was dreaming an extremely long and monotonous dream; she had died and this was the afterlife; the hollow Earth theory was correct; she had fallen down the rabbit hole and would soon find herself in Wonderland. She had been kidnapped by aliens who, instead of flying saucers, inexplicably used snow-trap pits. The girl tried to imagine what such extraterrestrials would look like and what purpose she might serve them. She pictured spider-like bears waiting for prey in their webbed dens. Having frightened herself enough, Inga opened her eyes and surveyed the expanse of light to confirm her relative safety from menacing aliens.
Then she noticed a streak of turquoise. At first, the girl hoped it was the long-awaited piece of sky and that it would soon eject her to the surface. But, flying closer, she recognized it as a piece of fabric and desperately clung to it. Suddenly, a barely audible whisper, no louder than the crunch of footsteps in the snow, broke out: «Don’t touch it, please. All the seahorses in the valley will freeze without a scarf.» Inga stared in astonishment at the snowman, whose «neck» was wrapped in the turquoise material. The surface of the two snowballs was very uneven, as if children had kept adding smaller and smaller snowballs and twigs to its sides. Bitter experience had taught the girl to touch such objects cautiously. The mere thought of the endless fall made Inga’s stomach ache. She held onto the scarf more tightly. «Happy day of inexplicable geometric nightmares, ” she grimly thought.
«Completely explainable, and hardly nightmarish! Of course, there are difficulties with navigation and spatial orientation, but any mode of existence has side effects, ” the wounded voice echoed through the absent walls. It sounded vaguely familiar.
Suddenly, Inga felt the ground beneath her again—or rather, a floor paved with flawlessly white tiles. Each tile fit perfectly against the next, even though their boundaries were twisted to an impossible degree. The girl decided not to step on those lines under any circumstances (perfectly justified caution). The joy of standing on a solid surface briefly distracted her, but she soon looked around, searching for the source of the voice. A front of curly clouds approached her. One of them almost brushed Inga’s nose and hummed:
„I see you’ve already met my relative. I call him Snsnsnegegeg, though among humans he is better known as the Mandelbrot Set. A rather boring name, just like mine.“
„I think I offended him by hanging onto his scarf. What’s around his neck? He said something about ice skates… I hope I didn’t hurt them.“
„Oh, you mean the Seahorse Valley!? What a charming name; I adore the associations people come up with. There’s a story here, similar to the etymology of the word ‘hippocampus.’ The patterns on his neck have nothing to do with these sea creatures, but their shape is a little similar. So, poor Snsnsnegegeg convinced himself that he was responsible for an endless army of seahorses. He started complaining that, you see, the climate was too harsh for his wards, so I had to procure a scarf for his precious pets.“
„Is that so…“
Inga stubbornly couldn’t grasp what was happening, but she intended to fix it. The spiral clouds continued to hover over her, drawing near and then receding.
— So, I’m not in the afterlife? — Nonsense, it’s not an afterlife. But if you want, you can call it an in-between place. Inga couldn’t help but laugh at those words—a little hysterically, but genuinely. —That was a good joke, right? Terribly funny, I know. But you’re right; I really should introduce myself properly.
My name is Julia, or rather, a multitude of Julias. I am a multitude of points. My DNA is a mathematical formula.
Одно из облаков Джулии спустилось на землю и начертило уравнение на плитке:
f (z (n+1))=z (n)²+c
Механизм моего создания очень прост, основные ингредиенты:
- the complex plane (something like a canvas upon which one of my countless portraits will be painted)
- a fixed parameter $s$ (the canvas number, which defines my form)
- any point $z$ on the plane
To begin, we toss an unsuspecting $z$ into the pot—specifically into this equation, as Julia pointed to the symbols she had previously drawn—and observe its behavior. After multiplying it by itself and adding a certain $c$, we toss $z$ into the pot again. If it begins to grow exponentially, mindlessly approaching infinity and swelling with pride, it is left out. This point is not part of me. But if it settles on a single value or fluctuates among a limited number of options, then this point is one of my cells; it belongs to my set. Yet, the real marvel lies in how difficult it is to predict the fate of a point on my boundary. One could say that my skin is incredibly sensitive to the slightest changes. A point might be me or it might not be, and the closer you look, the more irregularities you find. But these fractal patterns charm people, even if they don’t fully grasp how this beauty is formed.
— So you gave me back the mittens? — Of course! And I hope now you see why the scarf is only on me. This is part of my shell, of which I actually have infinity—a different garment for every value. Beings from fractional dimensions find it hard to adjust to a world with three. The place we are now exists on the threshold of thresholds. The laws of nature, mathematics, and other human inventions operate here whimsically. That is why I can take on any form here, while you can calmly exist here in your flesh-and-bone suit. «Not very calmly, ” Inga thought, shuddering as she recalled falling through the infinite void. For this very reason, I managed to encase Snssnsnegegega in crystals of water. You see, to me, it is like a map with biometric data. On it, the position of 's' is determined, and consequently, my appearance. I managed to fix its magnificent nose in one single position where my fractal boundaries take on a human silhouette. But soon it will begin to melt, and its nose will slide downward. Weather is one of those annoying parameters that has a significant influence on the snow-covered world.
— But why do you so want to be human? — I don’t understand so much about your world. I have never seen life as a limited stretch of time, where existence is confined to one blue planet, one fragile body, and thousands of ideas by which humanity organizes itself. Likewise, I cannot explain to you what it is like to be a fractal, even if I cover you from head to toe in formulas and metaphors. But understanding is not as important to me as the experience itself. Learning languages and shaping my thoughts to fit their grammatical structures. Growing hair and looking at my hands. Watching snow fall and taking the subway. Finding lost items and knowing where the right exit is. To you, this kind of existence might seem foolish, detached from reality, meaningless, and weightless. For me, however, it is an exciting adventure. Humans are drawn to infinity, boundlessness, and complexity, yet they fail to appreciate that their grounded and limited experience allows them to feel the essence of existence more deeply.
Inga sensed Julia’s fondness for monologues, so when she fell silent for a few seconds, the girl said nothing. The floating harness continued: — There are also fractal structures in physical reality: broccoli, bronchi, nerves, clouds, coastlines, mountain ranges, corals, snowflakes. But their complexity has a limit; you can’t study them with a magnifying glass forever. They know when to stop. I lack their worldly wisdom. With the arrival of spring, this dimension will collapse, and we will return to our familiar existence in the world of abstractions. But I would really like to go back. I already miss trees that don’t branch out to infinity. In order for me to find the right position on Snsnsnseghehege, I need to leave a mark in the material world. For this, I need your help, Inga.
With these words, the clouds thickened, swirling and mixing. The metamorphoses formed a figure familiar to Inge—a woman with curly hair and a flowing scarf. The lines around her body were still fractal, moving like waves. Julia carefully stepped onto the floor and gazed into the girl’s eyes. «I’m sorry I’m only telling you this now, but on the inside of your mitten, there is a Koch snowflake embroidered. It is a shape constructed from branching triangles. (By the way, the snowdrift you bounced into here was also glued together from its pieces.) This pattern encodes one coordinate of a point where I look as I do now—teeth, arms, scarf, and other organs. The threads are made from my hair, so they contain my DNA, the formula of my set. If you allow me to embroider a second snowflake, a second coordinate, on the other mitten, you will have the key to this dimension. When winter comes again, you will simply need to turn your mittens inside out and let the falling snowflakes touch the embroidery. Then, natural and mathematical fractals can exist together, the threshold of the threshold will open its door to me, and we can see each other again.» «But where will you be until next winter?» «Honestly, my self-awareness emerged along with the perception of the physical world, so I don’t remember what it was like to exist without it. I don’t even know if I’ll remember anything from everything I’ve learned this winter. But, as they say, a try is no harm.» Tiny Koch snowflakes formed in Julia’s eyes. They circled her head and settled into her hair.
«I really hope this isn’t torture for you, ” Inga said sympathetically, adjusting her hood, „I’ve always wanted to be a bird—to cut through the air with my wings, to soar in the clouds. My favorite birds are swallows. They fly so high, but they build their nests under rooftops. I love gazing into the summer blue sky until I spot even a single swallow. It reminds me that the sky isn’t just a dome above our heads, but a vast expanse. That thought makes it easier to breathe. Still, I like being me, I like how my thoughts and emotions feel. But if I could, I would take the form of a swallow.“ Julia listened with absolute concentration, her intricate outline almost ceasing to move. Inga continued: „I’ll help you. It sounds silly, but I feel like your determination to be human is what makes you human. Even if you don’t remember me, I want you to be able to live in our limited fractal. Maybe we can build a snowman together. Or feed the birds.“ „It’s a shame there are no swallows in the winter, ” Julia tried in vain to stop the stream of snowflakes falling from her eyes. „Blue tits are nice too.“ „Oh, I just remembered that one chickadee snuck through the empty doorway! Don’t forget to pick it up when you go out.“ „Okay, but try not to leave entrances everywhere. Otherwise, a spider-bear will definitely fly to you.“ „Are there things like that?“ „Of course! All sorts of creatures wander in the forest.“ Inga laughed, and Julia joined in, making a hoarse, resonant sound.
It seemed like she was having trouble maintaining her form, so it broke apart again into several separate fractals— «Hahahaha, this shape is called the Duadi Bunny,» Julia laughed. «For some reason, people notice animals in fractals with surprising frequency.» «These ovals look a bit like bunny ears, so I don’t blame the mathematicians for their naming choices.» «Would you entrust your precious mitten to one of my bunny ambassadors?» «I would consider it an honor!» Inga carefully passed over the mitten. Julia let her hair down and began to embroider. Bunnies danced around her, helping to weave the thread. The pattern was finished before Inga could take in Julia at work. The snowflake on the back was almost invisible; it gave Inga a fleeting impression of dizzying infinite descent and ceaseless complexity. But the cold of the fractal white stung the girl’s fingers, so she was glad to put on the mitten. «Thank you so much!!!» Julia boomed. «You’re welcome!» «Well, do you have to return to the surface?» «I’m afraid so.» «I can give you a lift (with the bird, of course). Before you know it, you’ll be outside. Snsnsnegegeg will miss you terribly. Right, Snsnsnegegeg?» «Seahorses need shrimp.» «There are no shrimp on a branch.» «Slowly, through snow and cold, The sea will catch them with a scarf.» The snowman’s whisper was so faint that the words were difficult to make out. «Oh, well, if he’s reading you his farewell poems, then you are in a very select circle of trusted people!» «I am incredibly flattered.» They stood nearby, hesitant, fiddling with their scarves. Finally, Julia said: «I will miss the birds, the hoods, and the words starting with the letter I.» «I will miss you too. The perfect fractals named after living organisms.» «See you next winter!»
