This is the hardest of the three because it is about the beauty of abstraction. It’s difficult to pin down that allure with words. What is intrinsically appealing is that it doesn’t need words, that it eludes simple description, even though it is so simple.

Math was always fun. It was not secret, like reading and spelling obviously were; all of the rules were easy to understand. There were no exceptions. No silent e’s or g’s. No doubled consonants, long or short vowels. Of course, what they called “math” was really arithmetic. Math came later and was a lot more than just fun.

Arithmetic was the simplest rules imaginable. In a day you could understand nearly all of it. Once you learned to add numbers of three or four places you could add the largest numbers you could imagine. The understanding was entirely extensible across the other operations. I remember in sixth grade racing with two friends to see whether we could finish the extra credit long division homework before the class ended, so we could hand it in without even taking it home.

That’s great, and the teachers spend a few years making sure that you are comfortable traversing the landscape of large numbers, fractions, decimals and negative integers. Nothing breathtaking, but it was necessary to develop a certain familiarity with the questions and their answers.

At first, word problems are similarly simple. Integers being added or multiplied. But to me they were so banal, just pieces fitted together to make the mute answer with the units tacked on: Sally has six dollars left. Rudyard walked eleven miles to school. Anne will be on fire for forty-five minutes.

Then I was introduced to x. First, x made it possible to look at more complex word problems. That was great. Sally had twice as much money as she did when she started, and it is the same number of quarters as she used to have dimes. Well, that was complicated enough to be interesting. Now we had a conversational place holder, a way to easily discuss something which we did not yet have a value for.

Symbols which could represent any number were great. Locking them into relationships with one another until they represented a definite number was even better. It had the pleasant feel of a polished piece of wood sliding into a Chinese puzzle. But we put aside these interesting pieces and took out some graph paper. It was nice to draw in math class. It became a calm spot in the chaos of my seventh grade days, a time to sit down and take out my fine point markers, my compass, and my clear, gridded ruler.

The first day the teacher wrote a simple equation on the board and drew one line on the permanent graph etched up there: x = 2. There were other things that day, that week, but that was the best of them. That contained so much of what was great about symbology, abstraction and the visual representation of a small piece of truth. The line was an idea, not a line. The line as drawn had a width, but the truth is that the line was conceptual. Everywhere on the line, x was equal to 2. It stretched off the page with arrows at each end and it went on forever. It was infinite, and the area to either side, where x did not equal two, was also infinite. It was a ball bearing which rolled along in a groove in the wood floor, pleasingly steadily supported and guided, it hummed with certainty.

I read Stephen Hawking’s book “A Brief History of Time.” In the preface he mentions having to include a few formulae, and that a colleague had warned him that every time he included a formula he would lose half his audience. That seemed so sad to me, because what could be more interesting and beautiful than an equation which explains some part of the random, natural universe? The first equations I saw weren’t tied to anything concrete, they were just mathematical doodles and were still beautiful:

x = 2y

That was a steep line. Too steep a roof to walk up. But so fascinating. And we started to draw more than one equation at a time so we could look at the intersections:

x = 2y

y = 5

It wasn’t difficult to figure that x was 10, but you could draw the two lines and see the answer as the pulsating point where the two lines met. Maybe the pulsating was only happening in my head, but it was there. And a pair of equations which had two possible solutions would graph as two lines (at least one of them a curve) which met at two different points. The truth, drawn on a cartesian plane.

It was great, and it got better and better as the equations became more complex. Trigonometry offered the curves of sine and cosine waves. The next course of geometry delivered parabolas and the quadratic equation. And then calculus, a world of beauty all its own, where complex questions can be reduced to simple answers. Oddly recursive questions like, “A snake is eating his tail, when will he feel full?” worked out to very satisfying answers like, “In five minutes.” (In truth, the most fascinating question I remember answering in calculus was about high rise buildings. If it cost two thousand dollars per floor to build, plus another thousand dollars every floor higher, and given the rent per floor collected, what was the ideal height to build? Calculus delivered the answer so easily while trial and error would have taken so long and, really, would never be certain the way the calculus was.)

Eventually, though, it was clear that math was static. That is, the answers were there, and you just needed to find them. You peered down the narrow tube of

4(ab) + c = xy

and the answer was there, a bright spark where the tube was pierced by

x = y^2 + (ac)/b

There was some drama in the revelation, but as I began to reach the limits of what I was able to grasp (somewhere beyond trigonometric identities and the simple parts of topology) my continuing curiosity was not enough to animate the subject matter for me. It’s difficult to bring to life that which you do not fully understand.

(As an aside, I did sit through a thesis presentation for a Bachelors in Mathematics. The discussion was about the Hairy Ball Theorem, which posits that if you have a sphere covered with hairs evenly spaced over its surface it is impossible to comb them all in a single direction without leaving one standing straight up. Puzzle over that for a moment or two.)

Fortunately, computers came along at the right time for me. A generation earlier math would have slipped away from me, unused and atrophied over several years, the beauty of abstraction would be lost to me. But starting in seventh grade there were computers in every school I attended, available with different degrees of effort, but they were there. I still remember typing

if x > 80 then vx = -vx

in 1978. That meant that if the little blip I was steering reached the right edge of the screen it should bounce off of it, changing direction but not losing any of its velocity. That was written in BASIC on a TRS-80, a very simple little computer from Radio Shack. Later it was referred to as a Trash-80. They had very limited memory and the only way to store the programs you wrote was to save them to a cassette tape. I rarely did that, I just typed them in again. Simple games where you tried to steer your block over to touch a blinking block. Or you ran from the blinking block. One cold winter afternoon I stood in the Radio Shack at twenty-third and seventh avenue and typed in a hundred or so lines so that my brother and sister could play my latest little game.

The computer was a compatriot. Someone I could explain the pure abstraction to and who would agree, expound, and draw what I meant. It was fantastic. with a few lines

x = 7

for y = 1 to 20 ; gset x,y ; next y

the computer drew the same plot I had on my page. It was not infinite, but in some ways the areas where it insisted on being discrete made it even more satisfying. It was an exploration of the abstraction, a peering into a virtual world with a concrete display of what was in there.

For the next decade I would struggle, torn between the easy construction of truth in the virtual world and the bloody, sweaty work of construction in the real world. Ultimately, I found a way for one to help with the other, and in the times that the frustration of physical building was highest I was able to fall back to construction things out of assertions, little lines of code and abstraction which illuminates truth with almost as much satisfaction as a wall reaching for the sky.

I know there are a lot of pilots who “always wanted to fly.” One of my favorite aviation quotations (there are a pile of them out there) is from Gary Powers, the pilot of the U2 spy plane shot down over Russia. Describing his first flight to his father he said, “Dad, I left my heart up there.”

I wasn’t like that. I flew in planes, even a few little planes, growing up. One flight all the way from New York City to Dublin when I was seven should have been momentous, but what I remember was the small paper tray which the meal arrived in. As soon as the meal was discarded I was able to start drawing on the inside of this open-top cardboard box, turning it from an Aer Lingus child’s meal into a hovercraft for some action figures I had with me. Controls, gauges, piping, wires, screens… my hovercraft probably had more instrumentation than the Boeing I was riding over the Atlantic.

So I liked the machines, but I had no real love of the sky or of flight.

I was always around water. My first seven years were spent on the Jersey shore, a two minute ride from the beach. Summer days were spent on the sand with the Atlantic either lapping at our feet or pushing over the walls of our sand castles. As breaks from the beautiful, endless beaches that stretched all the way up to Sandy Hook, we were taken up to Cape Cod to my father’s family place. As a way to get away from all of that, we were taken further north to a tiny lake in Ontario where we had a more solitary fifty feet of sand. Fresh water lapped my feet there.

At the beach club, three years after the first moon landing and those flickering images of men leaping about in reduced gravity, my cousin and I spent as much time as possible in the other world of the swimming pool. The “adult pool” was off limits some of the day, but the kiddie pool was too shallow to get our heads underwater. So we were covert and stayed in the shady end of the big pool, hyperventilating and then sinking to the bottom. I would look over and Trey’s hair would float from his head like Medusa’s locks, and he would mime relaxing in a chair, or reading the paper, or driving a car. Surfacing, we would gasp for breath through laughs and then plunge down again and I would pretend I was asleep on a bed, or eating a sandwich at the table, or scooping ice cream onto a cone.

Eventually we were exhausted. We had been fueled by lunch and the magic of a Dusty Miller, the sundae the beach club served with vanilla ice cream, chocolate sauce and malt. Mouthfuls were dry from the powder, but powerfully sweet and sluiced down by the bland creamy vanilla. On one Dusty Miller you could swim all the way out to where the lifeguards would blow their whistle at you. Or you could spend close to an hour suspended in the teal blue of the swimming pool, taunting gravity with one-handed hand stands or triple flips. When we were exhausted we would collapse into the adirondack chairs, dark green or bright white, surrounding the pool. Shivering shoulders wrapped with the thin beach club towels, we waited for the sun to bake us back to mammalian temperatures.

As the chlorine drained out of my ears I would reconnect with the world. Conversations would turn from the muted warbles we heard in the pool to words, phrases and eventually sentences I could understand. The weight of understanding would press me a little deeper into the slope of the high-backed chair. As a woman within earshot complained to her neighbor about her husband, or his job, or the tennis club’s wardrobe rules, I felt the brittleness of similarity. The odd congruence of so many lives being lived in such parallel fashion drove me back to the pool, to a world these people seemed unaware of, a world where communication was only mimed and gravity was a mere shadow of itself.

As I grew older and my lungs grew stronger, I was able to stay longer in that world. Long enough to really see the mirrored quilt of the surface, one of the features of the underwater experience that fascinated me most. And as I learned to snorkel and skin dive up at the lake, and on vacation in warmer waters, I found this other world infinitely more varied and intriguing than the beach club pool suggested.

The most engrossing part of the experience was the calming absence of gravity. Other than the slight tug toward the surface (if my lungs were full), or gradual sink (if I had exhaled enough), I was neutrally buoyant. Eventually I was propelling myself with serious swim fins, peering at the other-world through a large mask, a far cry from the swim goggles my cousin and I used. The sunlight, honey golden shafts tracing across the mossy rocks of the bottom, was dispersed by the silt in the water and didn’t provide a definite up or down.

I spent months at the lake. There was not much to do other than read or play in the water, and whenever I could I was under the water. Swimming the same route from the dock to the beach or the dock around to the back bay I should have been passing over the same set of rocks, but they always seemed different. Visits to the other world were periodic, punctuated by the need to surface, clear the snorkel, get a fresh lungful of air and flip back down to my cruising depth of eight or ten feet. At that depth, with a flick of one flipper, I could start a roll and as I torpedoed over the vast, curving granite of the submerged Canadian shield I would swap the up and down. The glinting, mirrored surface would straighten out and stretch ahead like a rippled ground plane. The ballooning, mossy monsters of the bottom became dark clouds of a strange sky. I navigated between the two with absolute authority, belonging to neither and analyzing both.

Careful exhalations led to a neutral buoyancy where there really was no up or down. I could move in any direction, explore any path. Again, after dozens of trips around out abbreviated coast, this should have become familiar (submerged) ground, but it didn’t. I believe I was so focused on the sensation of being lost, unanchored, that I did not try to orient myself. I knew the moment I popped my head above the surface I would know where I was, and that wasn’t the fun part.

The fun part was the flying.

When I was younger, just a boy, I would draw all the time. Constantly. It was how I made sense of the world. What made a cow so like a cow? I drew one. What made cars look like they were moving? I drew one. Airplanes flew, guns shot bullets, waves crashed, buildings had doors, stairs, windows, and people waved, shouted, and floated gently to the earth beneath their parachutes of silk. I couldn’t really understand the world, but I understood a line making its way across the page. I understood the object enclosed by a line, the balance of the figure and the ground, the edge of the page and the movement of a story through the still image.

All I wanted was another set of markers. Mine ran out. Dried out, when I forgot to put the caps on tightly enough. They were mashed flat to shade the large areas, so they were useless for the detail lines. A new marker was filled with ideas. It had all of the hundreds of drawings it was going to make all coiled tightly in its bright plastic cylinder, ready to be pulled slowly out through the squeaking fiber plug at the end, as it was dragged over a clean white page.

My mother was indulgent. I am sure I had new markers all the time. For a few years the markers came in a box with a simple line drawing of a smiling cartoon face on the front. The smile was huge, and a cutout, so that the markers in the box were the brightly colored teeth of his grin. As I lost or exhausted the colors, the man’s grin would wobble, marred by gaps and sloping canines. When he had only a few left rattling in his head I would start lobbying for a new box. One would often show up in the car after a doctor’s visit or when I was home sick from school. Oh, the big grin tight with all the rainbow teeth again.

When I drew, the world receded. I knew my father was still dead (a large event in my short life; since he died when I was four it loomed over the rest of my single digit years), and I knew the kids at school were still teasing me because I was short and fatherless, but as the ninety degree arc of yellow filled in the color of the page, the sun lit up a new place, a place no one else controlled or knew about, a place where I made all of the decisions. Before I was ten I drew boats which lumbered out of lakes on wheels. I drew the Irish in an epic aerial battle with the Nazi war aces. In those worlds the worst thing that happened was when the lighter colors smudged the darker ones, causing a rare disconnect between what I saw in my head and what would slowly fill the page. Nothing so tragic that I couldn’t fix it on the next drawing.

As I got older, the drawings told fewer stories about people and more stories about machines. There were boats with cars hidden in their bellies, and helicopters ready on deck to hoist them out. There were houses with large wheels on the corners and elaborate cistern systems and filters to allow their occupants to wander afield from the rigid connections of municipal water and sewer. I loved mechanical things and once sat for hours watching the repairmen in the pit of the small passenger elevator in our building in New York. I asked them what needed fixing and the foreman said, “There’s a piece on the axle that helps with the voltage to the motor.” I asked if he meant it needed a new commutator, the spilt ring which shifts the direct current from one set of magnets to the alternate set. He didn’t answer, but he said to his workers, “I should have sent this kid down here yesterday and we would have known sooner what was wrong.” I looked at all of the litter which had collected in the bottom of the elevator shaft and thought about how it was a giant, collaborative collage we had all been making. I wished there were more of the silver gum wrappers and more of the brightly colored Starburst wrappers.

The more I understood, the more the drawings had to tell. For weeks I worked on drawings of a vehicle for arctic exploration. Page after page for the plan of the living space it would need. The bathroom was modeled on the bathrooms I had seen up at the Boat Show with my stepfather. The metal treads were wider than the skis in the front for steering. The heater was tucked into a space under the tank for drinking water, which would keep it from freezing. The space for the driver was connected to the living space by a small opening with a sliding hatch so that the two explorers shared the view out the front while the machine glided and grunted over the packed snow.

As I entered puberty (after a two year delay, more short jokes) the machines were sometimes replaced with odd, gangly spiders with Don Martin eyes and Richard Scarry costumes for their occupations. To properly date it, there was a spider speed skater who would have given Eric Heiden a run on the rink, if it had ever managed to escape from my math book. A few cute, alien creatures tumbled down the margins of my composition books, but in high school I started imaging things which were beyond my abilities to draw. And then I took a drafting course.

We had to, at my high school. It was originally a high school for Math and Industrial Arts, back when Industrial Arts rated capitalization. One of those arts was drafting. There were two drafting teachers for the freshman at Stuyvesant: Flash Gordon and Mel Gordon. Flash Gordon was also the photography teacher, and he didn’t really need his drafting students to do more than learn how to produce a decent bar graph by the end of the semester. Most were able to do that after the first week and Flash mostly strolled between the drafting tables while the kids socialized. His thick grey mustache would dance on his upper lip as he told them funny stories and if they came to class a little early they’d find him brushing his careful, grey pompadour.

Mel Gordon was the devil. Or, at the very least, he looked like the devil. He had just a few wisps of hair left near the back of his head, slicked back with something which made it almost as shiny as his pate. He had thin, high-arching eyebrows. Maybe they were plucked. They would curl down over his cinder-black eyes as he explained the twenty different ways we were going to spoil a drawing, before we had even taped the blank sheet to our drafting boards. (The first way was that we were going to tape it down incorrectly.)

So I had Mel. Not at first. At first I had Flash, but my older brother was at the same school two years ahead of me. He was in tight with the head of the Industrial Arts Department and said I was probably going to be good at drafting. Mr. Wright, the vice principal he was chummy with, said, “If you want him to be any good, I would suggest switching him into Mel’s class.” So a week after classes had started I carried my partially completed bar graph upstairs to Mel’s drafting studio, where he tossed it in the trash and told me I would be producing something finer, something without compromise. And that it certainly wouldn’t take me a semester to make a couple graphs on a page: that was expected by Friday. Morning.

Mel Gordon was amazing. He was an old-school instructor from the days of vocational programs, where how well you learned the trade the school was teaching you directly affected how much money you were going to bring home to your family. He still carried that responsibility in how seriously he took our efforts. As we toiled over our ninety square inches of bright white paper, he would stalk the aisle between the tables, promising us that he could “see a sixteenth of an inch error, from across the room, on an angle.” My favorite moment of the entire year (because I stayed the next semester for the advanced, elective course) was when he kicked a student off of her stool and demonstrated, for less than five minutes, how he expected us to be able to draw.

“The page is taped down, there are no wrinkles, I am just picking up the triangle. There’s a dusting of eraser bits on the page, so the triangle glides above the paper, it doesn’t smudge any of the work I have already put down. I am not moving my hands quickly, but I don’t stop moving, either. There’s no time to sit back and admired the lines that I’ve drawn, because I’m always getting ready to draw the next line. I use the ruler as a measuring tool, not as a straight edge because it hasn’t got one. I do a light enough tick to mark this dimension that I don’t need to erase it later, it will become a memory on the page. And when I make these guidelines for the title text, I don’t press down on the pencil, I just pull it across the page, the mass of the pencil is enough to give the line the weight I need.” He was the sort of teacher who made a distinction between mass and weight, even though he was drawing not teaching physics. He had nearly finished the entire assignment, due in a week, in less than five minutes. He peeled the tape neatly from the corners, pinned it up next to the blackboard, and said, “I didn’t do anything that any one of you couldn’t do. So get to it.”

It was in Mel Gordon’s class that the magic happened. He had some artifact. Some piece of a machine. A chunk of metal with a purpose. It had a tunnel through it, a flange, a couple holes for bolts. It was the size of Mel’s fist and he held it up against the board. He talked about needing to make it, that somewhere someone had an idea for this piece and they needed to tell someone in the metal shop how to make it. He set the piece on his desk. He pulled down the drafting machine attached to the blackboard. This was a contraption which provided a straight edge and an adjustable triangle so that he could demonstrate drafting techniques on the board with his chalk in a chalk holder.

“The would need to know what it looked like from the front…” the chalk hissed along the board. “They need to know where to drill the large hole…” hiss, bang, bang, hiss as the chalk danced a centerline for the tunnel. “They need to know about that arc…” bang, squeak as the compass described the arc in the front view. “You would have to dimension this… label that…” He stepped back. There were three views of the piece, a top, front and side, draw in what I learned was called orthographic projection. With a few dozen lines he had captured reality on the two dimensional black slate.

I gathered my books slowly at the end of class. We had started our own drawings, each working on a chunk from the metal shop. Mine had a chamfer, a fillet, and a few odd channels routed into one surface. I had blocked out the views and was starting to fill it in. It was very calming, pleasing work. The real world was being mapped to my other, virtual world on the page. The best part of Mel’s demonstration, though, was an implication: the drawing came first. We were working backward. This was how to describe things which you wanted to make real.

I know other things happened in high school. I still have the text book from my Physics course because it was so interesting I believed I would want to reference it throughout my life. I know that I went to one dance because for the rest of my time in high school I was “the weird kid who showed up on roller skates.” I had a friend. So I know things happened, but what mattered was the page. Now it could do so much more.

There were all these objects to be described. I was involved in the science fiction magazine so I drew the cover, a carefully drafted spacecraft was in orbit around a distant planet, a small shuttle craft angling toward the docking bay. There were more vehicles, some rockets and planes, but once you understand scale there’s really only one thing you want to drawing: buildings.

Buildings have everything.