IV. Adjacent Territory Essay 23
Zoom
What changes when you change scale
You look at a coastline from a plane. It is a smooth curve, two or three long sweeping arcs, and it takes perhaps ten seconds to draw. You look at the same coastline from a boat. Now it has headlands and bays and a rock with a bird on it. You look at it from the shore, at your feet, and the edge is a chaos of pebbles and weed and a small hole where a crab is. At no point did the coastline change. At every point it did not look smooth.
The famous measurement puzzle follows from this. The length of that coastline depends on the length of the ruler you use to measure it, and a longer ruler gives a longer answer, and the answer keeps getting longer as the ruler gets shorter, and there is no limit to it. The coastline is infinitely long in the limit, which is a strange thing for a shape made of sand to be.
This is not a paradox about coastlines. It is a fact about what a length is, and the fact is that length is a property of a pair: the thing, and the scale at which you measure it. Changing the scale does not change the thing. It changes what the thing can be asked about, and it changes which questions are even well formed.
Take a workshop that makes one thing. A chair, say. A plain wooden chair, four legs, a seat, a back, made in batches of twenty.
Looked at from across the room, it is a chair. That is the whole of what the room needs, and the word is efficient, and the chair is a chair at that scale. Nobody in the room is asking what the chair is made of, and nobody would be helped by the answer.
Now move closer. The chair is still one chair, and there is no new object. But now the join between the leg and the seat is visible, and it is not a good join, and it is the joint that will fail. At this scale the chair is a joint. The question has changed from what is this to what holds this together, and the answer is a fact about the object that was there all along and was invisible from across the room.
Scale and level are not the same thing
The ladder is about what you are thinking about. The tangible level is one particular thing. The abstract level is what several such things share. Moving up the ladder changes the subject: you stop looking at this chair and start looking at chairs, and you have gained reach and lost resolution.
Zooming is about how close you are. You stay on this chair and you get closer, and the subject does not change at all, and what changes is the amount of detail you can hold at once.
They interact, and the interaction is where the confusion comes from. When you zoom in far enough on one chair, you find yourself looking at a board, and a board is a thing you can generalise about, and you have moved up the ladder without deciding to. When you zoom out far enough on a set of chairs, you find yourself looking at furniture, and furniture is a category, and you have moved up the ladder by moving back.
But they are separable, and the separation is worth holding, because the two have different failure modes. Climbing the ladder without zooming gives you a category with nothing under it. Zooming without climbing gives you an infinite supply of detail and no way to use it. The skill is knowing which one you are doing and whether you meant to.
What is real at each scale
There is a temptation in all of this towards a story in which the small scale is the real one and the large scale is the illusion. The coastline argument is usually told that way, and it is worth resisting, because the claim does not hold up and because it leads to a bad conclusion about attention.
Consider the three scales of the chair. At the room scale, the chair is a thing to sit on, and that is true and it is what the room is for. At the joint scale, the chair is a joint, and that is true, and a workshop that only knew about the room scale would make chairs that fail. At the grain scale, the chair is wood, and that is true, and a workshop that only knew about the grain scale would take a week to make a chair and would not know why.
Nothing is more real than anything else. Each scale is the scale at which certain questions are answerable, and the questions are different, and the answers are all true. A thing is not "really" a chair or "really" a joint. It is a thing that can be described at several resolutions, and each description is correct about the things it can see.
What follows from that is a practical rule: the question comes first and the scale follows from it. If the question is whether it is worth repairing, the joint scale is right. If the question is whether it is worth buying, the room scale is right. If the question is why it cracked, the grain scale is right. Choosing the scale without a question is how people end up measuring the wrong thing beautifully.
What if you could only ever work at one scale? Say you were a person who could only see the grain, and never the whole chair. You would know everything about the wood and nothing about chairs, and you would make boards and never chairs. Now the other way: a person who could only see the chair and never the grain would make chairs that fail and never know why.
"Zooming in always gives more truth." Sometimes it gives more information and less meaning. A painter who zooms in far enough on a face sees only skin, and skin is not a face, and the painting stops being a portrait. The detail is real and the painting is worse. This is a real and well known failure of a particular approach to drawing, and it is also the failure of a person who has become very good at noticing particulars and has stopped being able to see the shape.
The other misconception: "the large scale is the abstraction and the small scale is the concrete." It is the other way round, and this matters. The room scale view of a chair is an abstraction: it is a compressed answer that works for most purposes. The grain scale view is the concrete one, because it is the actual stuff. And the ladder, which starts at the particular and moves up towards the general, is doing something different from zooming, which starts at the general and moves down towards the actual. Two of the three essays in this series, and several of the figures, are about the ladder, and this one is about the other axis, and the two axes cross at the bottom where the particular and the actual are the same thing.
The ladder as a zoom
It is tempting to see the five levels of this site as five zoom settings, and the temptation is reasonable, because the levels do move from particular to general. But the resemblance breaks in an instructive place.
A zoom changes what you can see about the same object. The ladder changes what object you are looking at. When you move from one chair to the category chairs, you have not zoomed out of the chair. You have changed the subject, and the chair has not moved, and you are no longer looking at it at all.
Which is the more useful of the two moves? Zooming, most of the time, because it is the one that keeps you in contact with the real thing. Moving up the ladder is powerful and it is also an abandonment, and the abandonment is easy to forget because the new subject is so interesting. The whole argument of this site is that you have to come back down, and the reason you have to come back down is that the ladder moves you away from the thing rather than bringing it closer.
Zooming never takes you away. It takes you in. Which is why zoom is the safer of the two moves and also the more dangerous, because there is no moment when you notice that you have left.
Limits
Zooming is limited by the observer, not by the object. There is always a scale at which you cannot resolve the detail, either because the instrument cannot or because the eye cannot or because the mind runs out of capacity. At that scale the detail becomes noise, and the coastline argument turns into its own answer: the length of a coastline is finite once you specify the finest scale you will actually measure at, and that specification is a decision about your instrument and your attention, not about the coast.
Second limit: the choice of scale is not always free. A surgeon cannot choose to look at the patient from across the room. A bridge engineer cannot zoom in on one rivet and call it a system. Where the scale is fixed by the job, the interesting question becomes what the fixed scale makes invisible, and that is a question with a specific answer that can be looked up in the other place.
Stuck
You are stuck at one scale when you can see your situation perfectly at that scale and cannot see the situation at all at any other. The classic case is a person with a microscope for a problem that needs a wide view. They have made a beautifully detailed map of one cell, and the cell is fine, and the organism is dying.
The way out is to deliberately change the scale and to look for something that should be there and is not. At the room scale, a chair is a chair. Go wide and ask what the room is for. Go close and ask what holds the thing together. Each time, look for a thing you would expect to see and cannot, because that is where the mismatch between scales lives, and the mismatch is where the problem usually is.
Sources
- The coastline measurement problem is the standard example of a fractal boundary, and has been discussed in the mathematical literature since the late nineteenth century. No specific author, date, or measurement is cited here.
- The description of the underground rail diagram in the previous essay and the claim about drawability in this one are general observations about drawing practice, not citations.
- All examples are ordinary and hypothetical.