Nonlinearity

When lines mislead

I remember being pretty confused the first few times I baked bread. “Let rise for an hour, or until doubled in size.” Well, time or volume, which one should I follow? If you’ve ever followed a recipe for making bread or anything else involving flour and yeast, you’ve likely seen instructions like this: frustratingly specific yet non-committal at the same time. To help clear up one layer of this conundrum: judge by volume. If a recipe says to let the dough double in size, you should wait for it to double in size. The timing is usually given as a guideline, but can vary wildly in practice. Even a difference of a few degrees in the ambient temperature can change the rise time by 20% or more, and there’s often a marked difference in volume between 50, 60, and 70 minutes. The timing can be all over the place, so let volume be your guide (how to robustly measure volume will have to be a topic for a future post).

I remember carefully mixing some dough, setting it on the counter, and waiting. It should have started rising right? Doubling in volume sounds like a pretty dramatic transformation—and it is—so presumably I should have been seeing some initial progress. Something wasn’t quite right though: nothing was happening. Not what I expected. Come on, yeast, wake up and help me out here! Don’t worry, this particular memory ends with some decent bread, but why did the dough rise so unpredictably? What went wrong with my mental model?

Without thinking too hard about it, I had imagined a bunch of yeast getting introduced to some dough and starting to happily munch away on the sugars from the flour, producing precious carbon dioxide (rise) and flavor in the process. If you had asked me for a graph of the volume increase over time, this is what I would have intuitively drawn:

I would have guessed a linear behavior, or in other words: that the rate of change is constant. Lots of things in life behave linearly, or at least close enough for practical purposes: wages and salary, objects in motion, traveling from point A to point B (with some interesting exceptions), so it’s natural that I would have assumed the same for a humble batch of dough.

In the case of yeast, I was missing a crucial element to my mental model: the yeast are multiplying. With the vast majority of bread recipes, you start with a fairly tiny amount of yeast. Over time, that intrepid microorganism landing party spreads through the raw flour and water, constantly dividing and reproducing. If the population of yeast stayed constant over time, steadily eating away, we would see much more linear behavior. Instead, the yeast are constantly doubling—1, 2, 4, 8 billion cells and so on—and the population grows exponentially, so the rising dough behaves a lot more like this:

You might be familiar with exponential growth for things like compound interest, but did you expect to see it in bread? This far better explains why yeasted dough seems so inert toward the beginning, and why it rises so vigorously later on. Shifting our assumptions from linear to nonlinear (in this case, exponential) can make baking bread much more predictable.

Other times in the kitchen, I’ve been caught off guard by food browning suddenly and quickly while in the oven. When roasting a chicken, it might sit pale and unchanged for most of the cook time, then quickly threaten to turn to charcoal just as my attention lapses.

Why so sneaky, chicken? The answer is water, and more specifically: the energy it takes to turn it to steam. It takes 1 calorie (by definition) to raise the temperature of 1 gram of water by 1 °C, while it takes another 540 calories to then evaporate that same gram of water. Only after enough of the water evaporates does further browning really take off. This is the same phenomenon that causes the dreaded “stall” when cooking large cuts like brisket for barbecue.

Often, in a particularly difficult dance class, I feel like I can relate to that brisket. Not just because of how overheated I can feel, but also because I frequently feel like I’m stalling. All warmed up, starting to break a sweat, trying to commit the choreography to muscle memory, but nothing feels right. Maybe I even feel like I’m falling behind. Some of the best dancers I know have this uncanny ability to make everything look effortless. How am I supposed to do that when none of my limbs are cooperating and each movement feels awkward and forced? Yet, I try to push through. Three steps and a hand gesture coalesce into phrase. I internalize the jagged rhythm of a drum fill. A memory surfaces that carries the right emotional raw matter to inspire my character. I push through the stall.

10 minutes later, I feel stuck again. We made it through the verse but it turns out that the chorus is a completely different beast. Back at the bottom of a cliff. Somehow, a lot happens in the 60 to 90 minutes of a typical class that I take, and by the end I can almost always say that I’ve made some good progress. The path to get there, however, is almost never a straight line, nor is it a single hurdle to overcome. It is a path of many struggles:

I find that when I can anticipate this pattern, it makes it easier to push through those difficult moments, because I know it’s a natural part of the process that I’ve been through before. I know that learning isn’t a linear process, and feeling stuck now doesn’t mean I’ll feel stuck for the entire class.

Zooming out from the dance studio, we see the same interesting pattern in how species evolve. When we learn about evolution, we naturally imagine slow and steady change over time, and yes this does happen, but it’s not the whole story. Punctuated equilibrium is a theory where evolutionary change happens in rare but rapid events, such as catastrophic environment changes, separated by periods of relative stability. We’re more accustomed to thinking about the former, but what about the latter? Where in your life do you see rare but rapid change? Moving between cities or jobs? Starting a family? Moving in with someone? So much of how we grow happens sporadically rather than steadily.

Linearity is an extremely useful concept. If it takes me 20 minutes to walk a mile, I can be pretty certain that it will take me 40 minutes to walk two miles. If my hair grows an inch every two months, I can reasonably predict how long it would take to grow out a majestic mohawk. But linearity, or rather the assumption of linearity, can also set up some misleading expectations. Biological processes, like yeast in dough, are often exponential in nature. Sometimes there are hidden “walls” like the evaporation energy of water, or external factors like difficult choreography or environmental changes. Sometimes the world just doesn’t make sense at all, no matter how hard we try:

In any case, choosing the right mental model, linear or otherwise, can help set you up for fewer surprises, better choices, and hopefully some more success along the way—at least with bread and dance class.

Where else have you noticed something behaving in a surprisingly nonlinear way, and what ended up being the reason? I’d love to hear from you in comments or replies!