Topo vs. Graph

As a kid, I got my introduction to hiking in a particularly spectacular setting, the Wind River mountains of Wyoming. As soon as my sister and I were old enough, my parents would prepare us with local hikes, and drive us west in a marathon road trip – sometimes straight through, sometimes with a night in a motel in Grand Island, Nebraska – and we’d disappear into the wilderness.

In most peoples’ imagination, trails are an integral part of hiking. But one striking thing about the western mountains is that above a magical elevation, trees thin out and trails became optional. We could pick a particularly remote and interesting-looking spot and just go directly there, navigating through the landscape visually, with greater or looser precision as dictated by the terrain and our haste or leisure. The glacier-scoured contours of those mountains seem custom-configured to encourage this sort of loose-jointed bushwhacking or aimful wandering, with open glades, bold ridges, and frequent grassy benches to offer a path down even quite steep slopes.

I didn’t fully appreciate the freedom of travel and freedom of thought that this sort of terrain encourages until much later, when I started hiking and backpacking in the White Mountains as an escape from city life in college. Hiking in the east is all about trails. Off-trail travel in the eastern mountains is difficult to impossible given the heavy growth of trees and underbrush – and unethical above treeline, given the fragile soil and heavy traffic. Views are limited, and navigation is a matter of keeping track of which trail you are on and not missing the junctions. When the rigidity of trails is combined with infrequent vistas, humidity, crowds, and restricted camping locations, the overall comparison isn’t favorable.

I think these two modes of navigation are actually quite general. The first, western version I’ll call Topo, after the special color-printed topographical maps that we would order in preparation for those Rocky Mountain trips. A topo map offers comprehensive, uniform fine-grained knowledge of the entire landscape, down to the resolution of the contour interval and scale. A Topo landscape facilitates broad freedom of travel (albeit with natural obstacles and salients), and trails are of secondary importance or non-existent. The eastern version might uncharitably be called ‘Maze’ but more neutrally I’ll call it Graph, since in a mathematical sense that’s basically what the AMC’s hiking maps are, with thick lines representing trails and segment mileage printed on, the scale relatively broad and the contour lines recessive. The idea of Graph is that getting from place to place is less about the details of the landscape, and more about the paths through it (which are discrete and relatively few), the quality of those paths, and the logic of linkages among them.

These two ways of thinking about a landscape don’t depend on holding a map of one type or other. The broad meaning of Topo is to have an actual overall layout in your head at some fidelity, and a sense of your physical position on it to some accuracy. I came to appreciate it in the wilderness, but it’s just as useful in cities: ‘I’ll just head south until it gets kind of downtown-ish, then go east until I hit the river.’ Graph means discrete knowledge that you’re on trail X between point A and point B, combined with a finite list of instructions for getting certain places: ‘go out the door, you’re on Prospect St. Turn left, then walk until you hit Mass Ave. Take another left and walk to #77.’

Some landscapes (open, legible) lend themselves to Topo, while others (cluttered, opaque) lend themselves to Graph. A train on a railroad is the ultimate Graph – there are only so many tracks, and you are either on a track or you are in very bad shape. The ultimate Topo is the sky for a bird, or perhaps the ocean for a fish, though given a chart and GPS the ocean is Topo for a boat. With their orderly grids, some cities encourage Topo-type facility (e.g. Salt Lake), while in others (Boston) perhaps only particularly-talented natives eventually achieve Topo, and everyone else operates in Graph. When I first moved to Portland, the cluster of roads around PWM was an impenetrable tangle in my mind, and I had the barest Graph-based proficiency in how to get to a few places. As time has passed I am gradually approaching an overall Topo sense of the layout, and a facility with back-streets and connections.

I believe that the ideas of Graph and Topo extend usefully to the contours of landscapes that are not geographical. Topo is broad confidence, legibility, and freedom to ramble through the terrain of the solution space. Graph is narrow efficiency, stick-to-known-paths, and so tends fragile. Speaking of fragile, in my amateur attempts to learn and play fiddle tunes, I am almost purely Graph – the line of the melody might as well be a railroad track. If I am lucky, my mind and fingers can stay on the rails and keep the tempo. If I fall off the track I must back up, retrace my steps, and attempt to navigate the junctions until I make it correctly twice around the A part. Then the B part traverses what is dimly recognizable as the same landscape, albeit at higher or lower altitude – the exact relationship between the two is obscured by dense thickets of wrong notes. But it’s obvious that talented musicians navigate the same musical landscape in Topo. They see not just the single set track of the melody but rather the entire terrain of chords and harmony, and while they follow the cycles by instinct, they can just as easily jump the rails and ramble freely over the entire glorious landscape – and even take flight, to improvise and soar ecstatically like an acrobatic bird, then land back on the rails without missing a beat.

The same modes of thought apply in engineering realms as well. When I was a kid I had one of those Radio Shack ‘200-in-one’ electronics project kits with the little springs and jumper wires for making connections, and to me the projects were literally Graph – use wires to make connections following a sequence of numbers, then throw the switch and see if it works. While the manual also had narrative descriptions of how each circuit functioned, maddeningly they were written at a level suitable for a competent analog electrical engineer, not an elementary school kid. So my knowledge remained Graph, and in analog electronics I never got much beyond that, while the best engineers I know operate in pure and glorious Topo, flipping effortlessly between digital and analog, frequency and time domain, with a gut feel for every nonlinearity of a magnetic core or an insulated gate semiconductor. If only those old kits shipped with a copy of The Art of Electronics, I might not be a mechanical engineer.

Being invisible and mostly insensible, electricity is by nature hard to grasp, but chemistry is probably worse. My knowledge of aluminum alloys is strictly Graph: ‘Use 6061 for structural, 6063 for cosmetic. Use 7075 for airplanes.’ But I am sure that a skilled metallurgist has Topo knowledge of aluminum alloys – understands intuitively what the copper, zinc, silicon, etc. are doing in there, and I bet in a pinch they could whip up something pretty good in a crucible the way a skilled cook improvises in the kitchen. Chemistry might be where Topo knowledge is most impressive – while mortals scan an impenetrable text for clues and pray that nothing explodes, like Harry Potter on the first day of Potions class, the master seems to have an intuitive feel for what molecules want.

Cooking is domestic chemistry, and in the kitchen some people are strictly Graph, and wouldn’t dare make something without a recipe, while others are casually Topo, adding a dash of this and a sprinkle of that. I fall somewhere in between – when making soup or yeast bread I operate in Topo, while in quickbreads I’m more cautious and hew close to a recipe (Graph). To me the distinction feels inherent in the landscape/solution space of those foods, but perhaps there are others who see them oppositely?

While I’m firmly Graph in music, analog electronics, chemistry, and countless other areas (literature, small engine repair), and transitional in cooking, I enjoy solid Topo facility when working with wood, metal, and manufacturing technologies. I’ve got pretty good familiarity with rough and finish carpentry and working knowledge of many adjacent trades, and can play with the techniques to solve unique problems. I know what most ordinary kinds of wood are good for, and I can fell a tree, get out a bolt, slice it up with a bandsaw, dry it, join, plane, saw, assemble, and finish, and make something nice out of it. In metal I can MIG, TIG, stick, solder, and braze, and could mill and turn with decent precision when I was in practice. In manufacturing I know how most things are made and why they’re made that way, and can mix and match to come up with something new. Alex Slocum who taught the famous 2.70/2.007 design class used the metaphor of Legos – each material, technique, structure, or mechanism you learn is like having another Lego piece in your toy box. The more pieces you collect, the more freedom you have to solve problems efficiently, and once you collect enough, you transition from ‘I know A way to solve this problem’ (Graph) to ‘Which of the Many ways to solve this problem do I feel like using in this instance?’ (Topo)

Skill in prediction seems central to having a Topo understanding of a landscape: “There will likely be an annoying boulder field under that cliff face.” “If the screws are too short they are likely to pull out of the end grain.” Professor Slocum also talked usefully of having a ‘Mental FEA Module’ – a good engineer should work to build, exercise, and test their intuition for predicting how their designs will behave, so they can mentally iterate through potential solutions, rather than ‘guessing and checking’ experimentally or relying heavily on closed-form math or computer simulation. A big box of mental design Legos is like a good topo map, providing fine-grained coverage of the landscape, and a well-tuned Mental FEA module provides advance judgment about what parts of that landscape will make for easy walking. Design then starts as a playful mental exercise of trying to sense what physics will work well, what reality is struggling to communicate to us – in the words of the Indigo Girls, ‘trying to feel what’s coming next’.

Beyond any one person’s knowledge, I have the sense that the human scientific project started off groping in darkness with the barest fragments of Graph knowledge and has rapidly built out an ever-more-detailed Topo map of big chunks of reality. The alchemists could see very little of the landscape of Chemistry, and fumbled around hoping to find One Weird Trick for turning dross into gold; now the physicist Sean Carroll has provocatively claimed that the laws of physics underlying everyday phenomena are completely explained. That’s not to say we understand emergent phenomena, or the Big Bang or dark matter, but rather that essentially there are no blank spaces on the Topo map of forces and particles that make up our everyday lives. Fortunately or unfortunately, that still leaves plenty of work to do – e.g. Prevent Pandemics. Cure Cancer. Build a Sustainable Global Economy.