The Mathematics of Absence

After Ellen disappeared, I set about erasing doorways—not violently with a palette knife and black paint, nor with the spectacle of demolition, just the silent precision of someone cauterizing memories. Under steady strokes of my brush, entrances transformed into walls like sea foam receding into sand. I worked methodically through the predawn hours, building by building, transmuting possibilities into dead ends. I found a terrible, unexpected relief within the muted violence of my erasures—how one feels when finally naming a pain that has lived too long without language.

The phone vibrated against the polyethylene table, severing my focus. It was Hiro, calling from Tokyo at what must have been his 3 AM. The soft blue glow of his name on my screen felt like an accusation. He was constantly worrying about me. Not that I ever gave him a reason to stop.

"The paintings,” he said, his accented voice deliberate, each syllable carefully placed. “They're ready, yes? The courier arrives in two weeks. The gallery is waiting.” No pleasantries, just clipped words with polished edges. "Tell me you haven't—" he hesitated, searching for the right word in English,"—gone unplanned again."

Against the faux brick paneling of my Lower East Side studio, six canvases stood in a line, their surfaces still wet under the cold, clinical glow of the overhead fluorescent lights. The Mathematics of Absence—my signature series—depicted New York tenements that betrayed the laws of nature. Buildings connected by Escher-like staircases that led nowhere and everywhere, fire escapes that doubled back upon themselves, and windows that promised escape but offered none. One by one, I had sealed every doorway, leaving no exit, no way forward. Six months of painstaking work, undone in a night and transformed into something unrecognizable. Something that refused to comply.

How could I explain to Hiro that the paintings he'd been promoting to collectors across four continents no longer existed? That they were entombed beneath layers of paint so fresh they reflected my recessed eyes each time I passed?

In truth, I'd begun altering the paintings long before Ellen shook her head no, before she unthreaded our entanglement. I'd caught her studying those same paintings one night, catching details I was only beginning to understand.

"These buildings," she'd said softly, her pinky tracing the canvas frame that dwarfed her, "what are they turning away from?"

"They're just shy," I'd answered, dry humor masking unconfronted truth.

She'd tilted her head, unconvinced. "Jalen, all the buildings feel distant," she'd replied, more observation than objection. My name in her mouth always sounded like something rescued from drowning, each syllable pulled from depths and given air. Even in criticism, there was intimacy. "Like they were never meant to be entered at all."

She was right, of course. She was always right. Every structure in every reimagining of New York was hinging away, facades proposing retreat. I'd been preparing for her departure long before the day came. My buildings started to turn away because I'd always sensed impermanence in connections. I painted retreat before experiencing it—a premonition in brushstrokes that I tried to ignore each time Ellen entered my studio.

We'd first noticed each other eight weeks earlier at The Apthorp on the Upper West Side. At first light, its stone façade seemed to turn away from the city, as if keeping the garden enclosed secret. I was sketching the grand arches that framed the entrance when I caught my first glimpse of her taking photos, occasionally pausing to scrawl quick notes in a small black Moleskine notebook. For three mornings, we completed our rituals in silent awareness—her documenting the building's details, me sketching increasingly distorted doors and walls. No words exchanged, just occasional glances when one thought the other wasn't looking.

Our actual meeting came outside the west gate on the fourth day. I was hunched over my sketchbook, shoulders curved from years of leaning into my work, when her voice cut through the morning air.

"Beautiful ironwork, isn't it?" she said, not to me but to someone on her phone.

She stood tall with perfect posture, one hand motioning while the other held her phone. Dark hair pulled back, wearing a white hat and black leather coat that seemed explicitly tailored for her frame. I slouched further over my sketch, feeling suddenly aware of my perpetually bent spine, the antithesis to her architectural straightness.

When she caught me looking, I didn't look away.

"The ironwork catches your eye first," I said, surprising myself. "But I keep thinking about how the light changes the stone's color. In the morning it kind of glows."

She ended her call and approached, curiosity flickering across her face as she studied my sketch. "You noticed." A pause as she extended her hand. "Ellen."

"Jalen," I sheepishly replied, half-heartedly attempting to straighten my spine as we shook hands. Her grip was firm, deliberate. "Occupational hazard," I added, gesturing to my back. She laughed softly, displacing the tension between us.

That evening, we met at Washington Square Park because all inevitable things start with hesitation. Each conversation pushed us deeper into the park until we found ourselves beneath the iconic arch long past when the lamplights flickered on, past when the last stragglers had gone home. The space between the trees felt suspended outside of time. And yet, when I returned alone weeks later, it was just a park again, indifferent and unchanged.

From then on, the city conspired to keep us moving together, like it had been waiting for us.

"Rousseau wasn't painting a jungle," she'd stubbornly insisted one afternoon at a café outside Grand Street, leaning forward over her cooling coffee. "He was painting the space between waking and sleeping."

"Isn't that what the painting's about?" I'd countered. "A woman dreaming of structure in a place without walls?"

She'd smiled then, “You sound like an architect.”

"And you sound like a painter."

That café closed permanently soon after, as if it had existed purely for our conversation.

During a meandering evening stroll several weeks into knowing each other, she opened her Moleskine for the first time. "Your paintings remind me—," she said, pausing to look at her doodles, "—they remind me of things I draw when I'm not supposed to be drawing."

The pages were filled with meticulously detailed structures that defied expectations—buildings that seemed to breathe, changing shape based on the viewing angle. Practical architecture mutating into something closer to prayer.

"These are stunning," I'd said, softly touching the edge of a page where the building seemed to peel open like the spine of a well-worn book. "Why aren't you building these?"

“I would if I could," she replied, kicking at a loose pebble. "I suppose not everyone gets to paint their ideal world. And don't get me started on clients." She turned to another sketch, this one of a steel, terracotta structure caught in mid-breath. "Sometimes I think architecture should feel more like poetry and less like a contract."

"Then what would be left?" I said, gesturing to the skyscrapers surrounding us.

She tapped her pencil against the paper. "Maybe just possibility. And isn't that better than an ending?"

I don't know if I ever told her that her sketches made me want to return every painting to canvas. To restart with a new understanding of what walls were. If she could think outside the rules, why was I still painting within the lines? I guess that's why I was obscuring the doorways now—a deferred response to possibilities I hadn't been courageous enough to acknowledge.

Sometimes she would appear at my studio late at night, settling into my one decent chair, sketching structures that folded into themselves like origami. Sometimes, I would listen to her take client meetings and hear her translate impossible dreams into possible structures. We were learning each other's languages, though with accents neither could shed.

The week before she was set to fly back to Milan, we'd grown into a habit of silence. Not the comfortable kind that settles between people who know each other well, but something deliberate. We'd both recognized that words would only cement what we were trying to avoid.

One of those nights, she fell asleep in my studio chair, her notebook open to a drawing of a building that looked like it was trying to dig its own foundation—its steel and concrete beams drawn like scrawny arms. I worked until dawn, stealing glances at how her breath changed the air around her. The painting I made that night was the first one I destroyed after she left.

"You keep painting things trying to escape," she'd remarked another late night, resting her neck over my shoulder, studying a cityscape in progress.

"You keep designing things that want to stay," I'd replied, though I wasn't sure what I meant.

The truth was messier than either of us could confess. Her precise architectural renderings had started sprouting unexpected growth—ivy-like vines that defied perspective, polygonal windows that suggested chasms I doubted her clients had commissioned. For all its apparent motion, my work had begun mutating in ways I hadn't intended. We were rubbing off on each other in reverse.

The morning she arrived at her decision, she showed up at my studio before sunrise. I had never left from the night prior, not from painting but from staring at the canvases as if they might offer some solution I hadn't considered.

She stood at the edge of the room, by the door.

“I need to tell you something about Milan,” she said.

I nodded, my chair refusing to turn around. A barely-started building wavered between approaching and retreating on the canvas before me. "You're going."

"I am. It’s a two-year project. The kind I grew up dreaming about," she said as she came to stand beside me, her shoulder not quite touching mine. "It’s a lovely place, you should visit."

"Congratulations," I said, meaning it and not meaning it.

She stepped closer, looking at my canvas rather than me. "You already knew."

I cleaned my brush. "I've been painting it for weeks."

"You once told me possibilities matter more than endings,” I reminded her.

"I know." Her reflection appeared in the window, distorted by morning dew. 

She moved to stand beside me, close enough that I could smell the mineral scent of her skin. "This has been important to me. You have been important to me—you know that, right?"

I let the words settle, heavy with the shape of what they implied. "Don't turn this into something neat," I said finally. "Not when it wasn't."

"Jalen."

"If you're going to leave, just leave," I added white to my palette, too much.

"That's not fair." Her voice remained level, unclouded. "I'm not reducing what happened here. I'm acknowledging it."

"While walking away from it."

"Yes." No hesitation. "While walking away."

I exhaled, steadying my grip on the brush. I couldn't look at her nor did I want her to see the condensation forming in my eyes. Instead, I studied the painting before me, pretending to be caught in some technical dilemma, something, anything would be easier to solve than this.

"Some people build," she said after a long silence. "Some people move. I want to stay, but I've always been the second kind." 

I let the words settle, then exhaled softly. "And I've always been trying to freeze moments before they disappear."

She touched my shoulder, barely making contact. "No. You've been a way of seeing things I didn’t know I could envision. I'll always carry that with me."

I desperately wanted to tell her I didn't want to be carried. I wanted to be present tense, in the moment, and not a memory she'd occasionally reference when looking at odd buildings. But I couldn’t bring myself to say anything. She left before sunrise.

The next day was when I started painting over doorways, turning exits into walls—not to erase or to trap anything in, I realized later, but to test if absence could hold its own sort of presence. 

Hiro called again: "The courier will be here in a week. They're expecting the series exactly as discussed. As commissioned." Each word carried the weight of contracts signed, promises made.

I hesitated, watching the paint start to crust into permanence. "The work's changed. I need more time." 

A pause. Then: "Changed how exactly? How much more time?" Panic tinged his barely restrained voice as if it was bracing for impact. 

I examined my newest piece, tracing my finger over the texture of everything I had erased. "Structurally," I finally answered. 

Outside, raindrops unexpectedly materialized, disbanding the edges of the city, transforming it into something liquid. As I watched the storm render buildings as impressions, I couldn’t help but be reminded of Ellen's way of seeing—how she studied spaces not just as structures but as something malleable, shifting with time and light.

After convincing Hiro to postpone the gallery opening to the Spring, my eyes uncovered faded pencil marks in the recesses of my notebook. Upon closer inspection, it was undoubtedly one of Ellen’s measurements. Not her typical, precise notation, but something hastier: “49.9m from corner to absence.” I obsessed over it for hours, creating and recreating possibilities in my head, chasing meaning where I could uncover none. In the end, I conceded it didn't matter—she was measuring something I couldn't see.

As cold gave way to March warmth, I hesitantly walked into a pristine, neat gallery, adrift in the hum of polite chatter and white wine. The invitation had been delivered earlier that day—addressed to my studio, not to me, as if the physical space had been summoned to witness what came next.

I walked past rows of detailed models complete with perfect plastic trees and human figurines until a scale replica in the corner held me still. A public garden shaped like a double diamond and accessible only through a single wrought iron gate that functioned as barrier and invitation. No name attached, but I didn’t need one to recognize the artist behind it. 

I peeked at the dimensions. The distance from the gate to the central fountain? 49.9 meters.

"The architect was adamant on those explicit dimensions," the curator explained to a small group that had formed. "Something about the intersection of mathematics and love."

I stood back, unable to form thoughts as I watched others crowd forward picking apart something they’d never understand. Ellen had once explained in detail how public spaces are defined far less by what’s present, but by what isn’t—how absences create places to gather as much as presence does. With my eyes, I traced the invisible pathways she’d designed, imagining how people would move through the space, how the sun would dapple the stones in the morning. She never showed up. She was in Milan and maybe that was okay.

In my studio that evening, I started a new canvas meant to hang directly opposite the six I'd altered. I mixed colors I rarely used—phthalo blue, fluorescent orange, a transparent beige that seemed to disappear under the light. Where those six paintings had been about sealing exits, this one created entrances—pathways of light cutting through solid forms, thresholds that invited rather than rejected. I worked through till morning, the paint still moist when Hiro called.

"What have you done?" His voice carried across continents, tight with disbelief but measured, always measured. "I just saw the photos. You told me you needed more time to polish. These are…unrecognizable. Why is there a seventh piece? I can’t move the opening date again."

I looked from the phone to the six canvases lined against the wall, their doorways sealed with paint—then to the new seventh canvas taking shape. "They were incomplete," I said simply.

"Incomplete?” A rare crack in his composure. “You never told me you painted over—"

"Hiro,” I interrupted gently. “They're part of something larger now." I couldn't explain that the sealed exits were only half the conversation. That something had started to make sense in the space between what was closed and what could still open.

He sighed, a sound I'd heard many times before. "Send them all. Seven instead of six."

The city, in its perpetual reconstruction, carried on as I worked. When the paintings arrived in Tokyo, Hiro arranged them as instructed—six on one wall, the seventh opposite, creating a corridor of tension between what was sealed and what remained open. Critics wrote about "architectural dialogue" and "spatial counterpoint." The gallery, which had originally commissioned scenes of New York, now spoke of what they called a "transformative installation." But they missed what mattered most—that the real work existed in the empty space between the walls, in the invisible corridor where viewers stood, looking back and forth between absence and presence.

"What changed in you, Jalen?" he asked.

What could I say? That I'd stopped seeing gates as things that should be locked and started seeing them as thresholds of movement? That I'd learned absence could be foundation rather than void?

My newest canvas waited by the window, still taking shape. Unlike my previous work where buildings stood isolated, here they reached toward each other across deliberately empty space. I'm not sure if Ellen would ever see this painting, but if she did, she would understand immediately what others might miss—that she existed in it not as subject but as negative space transformed into structural element. The composition didn't depict her but was fundamentally shaped by her absence, the way cathedral arches are defined by precisely measured emptiness. The way she'd taught me to see separation not as ending but as a necessary counterpart to connection.

I try not to look her up, though once in a blue moon, I stumble upon one of her digital remnants. That's how I found myself reading an article on an architectural blog with her name on the headline. A profile on her approach to public spaces, describing her work as revolutionary for its emphasis on "interstitial experience"—the moments between defined spaces where possibility lives. No mention of New York, or a slouching painter who once sealed doorways out of grief. No need. It was there in the methodology, in the attention to how absence shapes presence.

In my dream that night, I stood before the wrought iron gate of her garden courtyard. Sunlight streamed through the bars, creating dappled shadows that danced with time. I understood what the measurement meant—49.9 meters, not quite reaching 50. Deliberately incomplete. Half full. A space left open for what comes next.

If Ellen were here she’d say that every architect knows that the most crucial part of a gate isn't the barrier but the opening. I’d respond that every painter knows that negative space is never truly empty but alive with potential. And now I know that absence isn't what remains when something is taken away, but what continues to build everything that follows.

When visitors ask about the notebook page bandaged to my studio wall—Ellen's last gift—I tell them it's neither beginning nor end, but the exact measurement of distance necessary for two separate things to exist in perfect relation to one another, should they choose to measure the distance between them again.