Hair Length: Large-Scale Models

This is the third part of the series on hair length.

You’ve hopefully read Part 2 and seen the 10-day/5-hair graph, so I’ll just dive right into this!

Here’s a 20-day/5-hair graph:

And a 30-day/5-hair graph:

And a 100-day/10-hair graph:

Notice: for each graph, there’s a beginning region of positive change in average hair length, followed by a region that oscillates around a fixed value. This value represents the point at which hair seems to “know” when to stop growing.

Still not convinced? Take a look at this super-large-scale graph, done with five trials:

For some conclusions, move on to the next section!

<< Part 2: An Example

Part 4: Conclusions >>

Hair Length: An Example

This is the second part of the series on hair length.

Suppose I had a patch of five hairs, or grass, or something similar. The graph below (made with Matplotlib and pyplot) summarizes what would happen to the hairs over time if none were cut off:

Notice how the individual hairs grow at an average rate of 1mm/day, and so does the average hair length. But what if we let some fall? The graph below gives a typical example:

Click onward to see more conclusions for this model… applied at even LARGER scales!

<< Part 1: The Model

Part 3: Large-Scale Modeling >>

Hair Length: The Model

This is the first part of the series on hair length.

So we’re assuming that all hair grows at the same rate, and that the body does not know how long hair is.

Then how does hair “know” when to stop growing? How come hair grows fast when you shave it, but slowly reaches maximum length when you trim only a fraction of it? How can hair reach a stable average length even if some hairs are shorter and longer than others? And how come there’s the occasional hair that’s significantly longer than the rest?

The explanation is that hair falls out at a predictable rate. What we see when we try to determine our hair length is our average hair length; although the average hair length may increase at different rates over time, each individual hair grows at the same rate as its neighbor. When hair falls out, the average length decreases.

For an demonstration of the model, proceed to Part 2: An Example!

<< Hair Length: An Investigation

Part 2: An Example >>

Hair Length: An Investigation

What if you learned one day that everything you thought you knew about hair growth was wrong?

Why does hair stop growing after a certain point? It’s a legitimate question, and there are many explanations that may or may not be wrong to varying degrees. But  there are two commonly heard justifications for why hair “knows” when to stop growing:

1. Hair cannot grow after a certain length. According to this theory, hair needs nutrients to grow. The body produces these nutrients and sends them up the length of the hair, causing the hair to grow longer. However, when hair grows too long, the nutrients would not reach the top, and the hair stops growing.
2. The body is able to detect how long hair is, and changes the rate of growth based on hair length. There are various mechanisms for this; for example, there may be specialized cells that measure the force a hair exerts as it hangs off the skin. Other proposed mechanisms involve cells that measure the frequency of dust collisions (more hair = fewer collisions = less growth) or follicles that use some form of physics calculation to tell how long the attached hair is.

These two explanations are very reasonable, and they may even be true. But they seem too involved, and there’s probably a better explanation out there. A preferable, less involved explanation has the following premises:

• All hair grows at the same rate – Under this assumption, you don’t have to deal with complex hair-growth mechanisms that, in reality, would be too hard to implement.
• The body has no way of knowing how long hair is – It seems pretty difficult for the body to find a mechanism to “know” how long hair is. It is possible, but very hard to implement.

Given these assumptions, how can a simpler model be made? Follow to the next post to find out!

Hair Length: An Investigation

Part 1: The Model >>

An Introduction

Math is cool. Yay. Give yourself a chance to see math applied to, like, anything and everything. And then you’ll know why this blog exists.

Math is a tool to understand nature. But no one said it can’t be fun!