Interpolation

The Projection

Chapter 15: Interpolation

The first contour line was the 2,140-foot line, the lowest contour on the map, running across the mouth of the tributary where it entered the Koyukuk. Silas drew it with the Rapidograph 1 — the heaviest pen, 0.50 millimeters — because the index contour, every fifth contour line, was drawn heavier than the intermediate contours, a visual hierarchy that allowed the reader to count contours quickly, the heavy lines marking the hundreds and the light lines marking the twenties, the system as old as topographic mapping itself, a convention so deeply embedded in the practice that violating it would be like writing a sentence without capital letters, technically legible but wrong.

He started at the left edge of the map, where the 2,140-foot contour crossed the western ridge. He had a spot elevation on the ridge at 2,387 feet and another at 2,093 feet, and the 2,140-foot contour passed between them at a position he estimated from the slope angle, the contour crossing the ridge at a point approximately thirty percent of the way from the lower elevation to the higher. He placed the pen on the vellum and began to draw.

The line moved to the right, following the ridge slope, curving slightly south as the slope curved, then turning sharply north into the tributary valley, crossing the creek — which he had not yet drawn, but whose position he knew from the station data — and then curving south again, following the opposite ridge slope, climbing out of the valley, the line closing on itself in a broad U-shape, the U pointing upstream, the classic contour pattern of a valley, the lines bending upstream where they crossed the drainage because the drainage was lower than the ridges on either side and the contour had to accommodate the shape of the ground, had to detour into the valley and back out again, the way water had to detour through a valley before it could reach the next valley.

The line was not straight. No contour line was straight, because no terrain was flat, and the curvature of the line was the map's way of expressing the curvature of the ground, the three-dimensional shape of the surface projected onto the two-dimensional page. Where the line curved gently, the slope was smooth. Where it curved sharply, the slope was broken, interrupted by a gully or a ridge or a change in angle. The curvature was information. Every inflection point, every bend, every change in the radius of the curve told the reader something about the terrain, and the reader — if the reader knew how to read contour lines, which was a skill that took years to develop and that most people never developed — could reconstruct the three-dimensional shape of the ground from the two-dimensional pattern of the lines, could see the valley and the ridges and the cliff and the saddle, could understand the terrain without standing in it.

Silas drew the 2,140-foot contour in a single motion, the pen moving continuously from one edge of the map to the other, the line unbroken, the ink flowing steadily, the hand guided by the data and the judgment, the data saying where the contour crossed the measured points and the judgment saying what it did between the points.

Between the points. That was where the art was. That was where the interpolation happened. Interpolation was the process of estimating values between measured points, of filling in the gaps, of imagining what the terrain did in the spaces where no instrument had been set and no measurement had been taken. The word came from the Latin — inter, between, and polire, to smooth — and it meant, literally, to smooth between the points, to create a continuous surface from a set of discrete observations.

The smoothing was a lie. The terrain was not smooth. The terrain was rough, irregular, fractured, a surface shaped by geology and erosion and permafrost and ten thousand years of freezing and thawing and the slow, relentless work of water on rock. The smooth contour line was an idealization, a mathematical fiction, a representation that traded the roughness of the real surface for the legibility of the drawn line. But the lie was a useful lie, because the smooth line conveyed the general shape of the terrain even as it glossed over the specific roughness, and the general shape was what the map reader needed, the overview, the gestalt, the big picture that let you understand the valley without having to stand in it.

Silas drew the second contour line: 2,160 feet. Then 2,180. Then the index contour at 2,200. The lines accumulated on the vellum, the valley taking shape between them, the terrain emerging from the data the way a face emerged from a portrait — line by line, feature by feature, the representation becoming recognizable, becoming real, the map beginning to look like the thing it represented.

He worked slowly. Each contour line took fifteen to twenty minutes — longer for the complex sections, shorter for the simple ones. The work required sustained concentration, the kind of concentration that excluded everything else, the hand and the eye and the mind focused on the single task of placing the line in the right position, the right position being the position that honored the data at the measured points and represented the terrain faithfully between them.

The faithfulness was the hardest part. The data said: at station seven, the elevation is 2,312 feet. The data said: at station eight, the elevation is 2,389 feet. Between these two stations, the 2,340-foot contour line had to cross, and the question was: where. And the answer was: it depends on the terrain. If the terrain between stations seven and eight was a uniform slope, the contour would cross at a point fifty-two percent of the way from station seven to station eight, because 2,340 was fifty-two percent of the way from 2,312 to 2,389, and a uniform slope distributed the elevation linearly. But if the terrain was not uniform — if there was a bench, or a steepening, or a cliff — the contour would cross at a different point, and determining that point required knowing the terrain, which required either additional measurements or judgment.

Silas used judgment. He had walked the terrain. He had seen the slopes. He had noted the benches and the steepening and the cliff. He carried the visual memory of the valley in his mind, a three-dimensional model built from fifteen days of walking and looking and measuring, and the model was not exact but it was detailed, and it was the source of the judgments he made as the pen moved across the vellum, the pen tracing the shape that the model provided, the model shaped by the experience that the data could not contain.

This was what he had tried to explain to Jin. This was the difference between the hand-drawn contour and the computed contour. The computed contour drew on the data alone. The hand-drawn contour drew on the data and the experience. The experience was the surplus, the additional information, the thing that the cartographer brought to the map that the algorithm could not, and the thing was not mystical or romantic or sentimental. It was practical. It was the knowledge of the terrain that came from standing in it, walking through it, looking at it with eyes that had learned to read the landscape the way a musician's ears had learned to read the sound.

He drew through the morning. He drew through lunch, eating a sandwich at the drafting table with his left hand while his right hand held the pen. He drew through the afternoon. By four o'clock he had twenty-three contour lines on the map — from 2,140 feet to 2,580 feet — covering the lower reach of the tributary, the broad, gentle section where the valley was wide and the gradient was moderate and the contour lines swept in long, graceful curves across the vellum, the curves parallel and evenly spaced, the visual pattern of a smooth slope, a readable terrain, a landscape that the map could represent without strain.

He stopped at the steepening. At 2,580 feet the terrain changed. The valley narrowed. The gradient increased. The contour lines would crowd together from here on, the spacing decreasing, the curves tightening, the pattern shifting from the open, relaxed geometry of the lower reach to the dense, compressed geometry of the steep sections, and the drawing would require more care, more precision, more judgment, because the closer the contour lines were to each other, the more critical the placement became — a millimeter of error on the map was still a millimeter of error, but when the lines were two millimeters apart, a millimeter of error was a fifty percent displacement, and the map would look wrong, the terrain would read wrong, and the reader would misunderstand the ground.

He cleaned the pens. He capped the ink. He stood up from the drafting table and stretched his back and his hands and walked to the window and looked at the yard. The fireweed was a foot tall. The birch trees were in full leaf. The evening light was golden on the grass that had grown up around Margot's garden beds, the grass taller than the remains of the perennials she had planted — the lupine, the wild geranium, the Jacob's ladder — the cultivated plants losing ground to the uncultivated, the garden reverting to meadow, the way a cleared field reverted to forest, the way a human presence reverted to absence when the human departed.

He thought about the contour lines he had drawn and the contour lines he had not yet drawn. The lower reach was done. The easy part. The part where the terrain cooperated and the data was dense and the judgment was straightforward. Ahead was the steep section, the cliff, the cascades, the saddle. The hard part. The part where the terrain resisted representation, where the contour lines piled up and overlapped and became unreadable, where the cartographer had to choose between accuracy and legibility, between showing what the data said and showing what the reader could understand.

This was the projection problem. Not the mathematical projection — the UTM, the Lambert, the Mercator — but the representational projection, the broader problem of mapping, which was the problem of converting a complex reality into a simplified representation. The reality was always more than the representation could hold. The terrain was always more than the map could show. The life was always more than the memoir could tell. The marriage was always more than the two people in it could describe.

He had interpolated between the measured points of the terrain. He had drawn smooth lines through rough data. He had filled in the gaps with judgment, with experience, with the accumulated knowledge of a career spent looking at landscapes and turning them into maps.

Could he interpolate between the measured points of his life with Margot. Could he fill in the gaps. Could he draw the contour lines of their marriage — the lines of constant emotion, if such things existed, the lines that connected the moments of equal happiness or equal difficulty or equal understanding, the way the contour lines connected the points of equal elevation. Could he smooth the rough surface of thirty-three years into a readable representation, a map of the marriage that showed its shape and its slope and its steepening and its cliff.

The metaphor broke down. It always broke down, eventually, because the terrain and the marriage were not the same kind of thing, and the instruments that measured one could not measure the other, and the contour lines that represented one could not represent the other. The terrain was objective. The marriage was not. The terrain could be measured from the outside, by instruments, by anyone with the training and the equipment. The marriage could only be measured from the inside, by the two people in it, and the two measurements — his and Margot's — disagreed, the way the GPS and the theodolite disagreed at station fourteen, by an amount that was small on the map and large on the ground.

But the metaphor did not break down entirely. The metaphor persisted because the underlying process — the process of representation, of projection, of turning a complex reality into a simplified map — was the same in both cases. The cartographer chose what to preserve and what to sacrifice. The husband chose what to preserve and what to sacrifice. The choices were different in kind but identical in structure, and the consequences were the same: the map was incomplete, the marriage was incomplete, the projection preserved some things and lost others, and the lost things could not be recovered by re-surveying, by re-measuring, by going back to the field and standing at the station and reading the angle again, because the station was gone, the instrument was packed, the field season was over, the wife was dead.

He went to the kitchen. He made coffee. He drank it standing at the window.

He thought about the steep section of the map, the section he would draw tomorrow. The cliff at station eight. The quartzite face, twelve meters high, vertical, a step in the valley wall. The contour lines at the cliff would converge to a thick brown band, the lines so close together that individual lines could not be distinguished, and he would replace them with hachure marks — short, heavy lines perpendicular to the contours, pointing downhill, the conventional symbol for a cliff or a very steep slope.

The hachures were a confession. They said: the contour lines cannot represent this terrain. The terrain is too steep for the chosen contour interval. The map's language breaks down here, the way language breaks down in the face of extremity, and the hachures are a substitute, a different symbol, a different word for a thing that the standard vocabulary cannot express.

Margot's painting of the quartzite cliff in the Wrangells — the painting in the hallway — did not use hachures. It used brushstrokes. Thick, textured, loaded brushstrokes that mimicked the rock face, that captured the verticality not through symbols but through the physical quality of the paint on the surface, the paint standing up from the canvas the way the rock face stood up from the valley floor. The painting did not confess its limitation. The painting transcended its limitation. The painting used the two-dimensional surface of the canvas to create the illusion of the three-dimensional surface of the rock, and the illusion was convincing not because it was accurate but because it was felt, because Margot had felt the rock — its weight, its hardness, its indifference — and had translated the feeling into paint.

He could not translate feeling into ink. Ink did not have the vocabulary. Ink had lines and symbols and conventions, and the conventions were powerful and useful and sufficient for the map's purpose, which was to represent the shape of the terrain, not the feeling of the terrain. The map's purpose was limited. The map's purpose was defined. The map's purpose was the projection — the chosen preservation and the accepted sacrifice.

But the legend.

The legend could use words. The legend could say things that the lines and symbols could not say. The legend could acknowledge what the map sacrificed. The legend could confess.

He put the coffee mug in the sink. He walked to the study. He looked at the vellum on the drafting table, the grid and the stations and the twenty-three contour lines of the lower reach. The map was emerging. The valley was taking shape. The terrain was becoming visible in the representation, the three-dimensional ground becoming the two-dimensional page, the projection performing its ancient, necessary, incomplete work.

He turned off the light.

He walked down the hall. He stopped at the closed door. He stood there for a long time, longer than he had stood there before — a minute, two minutes, five minutes — his hand at his side, not reaching for the knob, just standing, standing in front of the white door with the brass knob and the closed room behind it, the room where Margot's paintings waited and her journals waited and her unfinished canvas waited on the easel, the canvas that was its own kind of map, its own kind of projection, its own record of a landscape seen through particular eyes at a particular time, the landscape of the Brooks Range, the same landscape that Silas was mapping on the vellum in the study, the same terrain, the same ridges, the same sky, seen from a different point, measured by a different instrument, projected onto a different surface.

Two maps of the same territory. One on the vellum. One on the canvas. One complete. One unfinished.

He stood at the door. He did not open it.

He went to bed.

Tomorrow the steep section. Tomorrow the cliff. Tomorrow the hachures that confessed what the contour lines could not say.

And after that, the legend.