The Geometry and Calculus of Patience

A couple weeks ago, I wrote about How I Destroyed Everything by Lacking Patience, in that article I talked about a few things I destroyed in my life, and gave the story of Frank Gehry and his nearly 40 year wait until he found the ability to do the work he had been waiting for his whole life. It’s a story that resonates with many people. While I’ve always looked to stories to inspire, I also look to rational analysis to confirm what stories are telling me. I’m on a continual hunt for the truth after all, so this is an analytical look at the concept of patience.

Doing important work, it’s easy to get caught up in the short-term, tracking revenues, followers, likes, subscribers, engagement, customers, skill development or any other metric, however, over a short term, they nearly always look like linear growth. In the longer term, these things look like an exponential growth function until they start to reach their ceiling. As a result of these two facts, I’ll discuss two geometric shapes, the triangle described by linear growth, and the curve described by the exponential function that describes compounding returns.

A linear growth could happen for something like subscribers for a writer in a niche. They may grow by 10 subscribers per week (or month) for example at least early on when they are “pushing”. So, below their subscribers would look like this.

Using the concept of center of mass, or where you would need to put your finger under the triangle made with the graph above in order to balance it, it can be calculated where the halfway position is in the results of total essays read. If this graph was subscriber growth for a writer by week. If every week the writer is producing one essay that is read by all of his subscribers, then the center of mass would represent the point where he reached half of his essay reads.

The center of mass falls at 2/3rds of the length of the triangle, so week 80 out of 120 would be where half of the total essays read happened. The same numbers of essays were read in the 40 weeks from week 80 to 120. Half the time and the same number of reads! If the writer kept at this work for an additional 60 weeks with the same growth, he would double his results from the first 120 weeks. Same results in half the time!

This is a conservative case in many applications. People that stick with their work, hone their skills, and work to put it out into the world for people to consume, likely see something better than this linear growth curve. Growth for most things tends to look like an exponential growth curve on a scale for anywhere from 10-40 years. Depending on your market, it looks more like this:

Everyone is looking for the end of that curve, but most aren’t patient enough to get there. If that linear growth graph is measured in weeks, this chart is measured in months. At 120 weeks on the linear growth chart, we’re at month 30 on this chart. The growth up to month 30 is almost linear like the previous graph because the exponential gains haven’t compounded enough to distinguish much from a straight line yet. This is where most people quit. They feel they are putting in more effort than it’s worth. Little do they realize the gains are about to start compounding significantly if they can push through.

The most interesting thing about the exponential growth curve is the center of mass, the halfway point on results. This is more complex to figure the center of mass of, so I went to WolframAlpha which I used to use when I was in college to perform some complex calculus.

The result in the image is the equation for the location of the exponential curves center of mass of the area under the curve. A quick review of the variables:

  • P is the principal. The starting money, subscribers or any base metric from where you decide to start counting.
  • r is the growth rate as a decimal. r = 5% = 0.05.
  • n is how often the compounding is happening. I could have left it out as I was using 1 compound per period anyway.
  • x is the time in this case.

After going through this exercise, I was going to write out the equation for how fast things double in comparison to the triangle, but then the explanation just wasn’t clear to explain because it continually shifts based on the time scale, the growth rate, the principal started with. I instead decided to make a piece of art that shows the same truths.

Each cut on the top brass piece represents 10% of the results, it’s obvious that the cuts get closer and closer together, while at the same time, it seems like a long time to get to that first 10%. The piece is mounted on a pedestal at the center of mass of the curve, and the pedestal itself mirrors the proportions of the curve. The inlaid brass represents 4 positions. As shown, it represents results over time, essentially the exponential growth curve shown earlier. turn it 90 degrees it represents the assumed effort over time, another 90 degrees represents the negativity/mental resistance/doubt people go through. Starting highly negative and critical of their work, then as results grow, that negativity lowers, though likely never disappears. The last and final turn represents the effort to grow further. At small size, it’s easy to grow, as you get bigger results, the effort to grow further increases as well.

As you can also imagine, continuing to work and letting growth happen for twice as long, ends with a center of mass much closer to the right side of the exponential growth curve. Eventually, doubling your previous results starts happening at shorter and shorter intervals.

It’s my hope that this article, the math, and the rendering of my future art piece are enough to convince you to stick with whatever challenge you’re losing hope on, whether that’s a career, or ultimately a small business. Patience requires faith and time. The mathematics are in your favor, but only when you keep going. Not everything requires full-time dedication, but everything important does require consistency and patience.

P.S. For an ironic story, I was originally going to put a hand-blown glass dome over the art piece, but my wife, who believes my ideas about art like this are too abstract and conceptual, said, “I think that would separate it from the room too much.” I said, “After hearing that, you’re right. This piece has to be something you live with. It has to be something that you draw inspiration from daily, putting a glass dome over it is terrible symbolism, it makes separate from you. I can’t believe I was ever going to do that.” She just rolled her eyes and walked away.

None of this is Right is written by me, Brandon Donnelly. I believe that small business is the backbone of a healthy economy and democracy. Small business encourages competition through generosity, creativity, and skill. Small business provides more opportunity for workers to find a job that works for them. If you believe in small business, generosity, or creativity, my writings are for you.

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