## Tuesday, February 26, 2008

### Terminal Length

It has been a thought in the past that there must be some limiting hair length beyond which it would be infeasible for a person to be able to grow their hair. Because your hair has random breakages and can get pulled out by its roots and of course cut. Lets see if we can make our model predictive of some difficult to measure thing. A reasonable model of the hair length of someone who never cuts their hair (me for instance) is probably a good place to start. For what follows let L represent the average hair length and l represent the length a strand coming from a particular follicle. To begin with though we should put down some assumptions to simplify the model first the hair has a constant growth rate meaning d2l/dt2 = 0, which I think is a pretty reasonable assumption. Secondly lets assume that d/dr * d/dt * l = 0 meaning that the rate of growth is the same over the entire head, admittedly this is simply not true but it is still not a horrible assumption. The solution to our system without any breakage or shortening mechanism is simply linear growth both for l and L. I proposed three modes of shortening, breakage, pulling, and cutting. Cutting is an uninteresting case as it is merely setting the initial length distribution of the hair and so the models I develop for the other two cases can easily be applied to cutting to obtain solutions. Breakage is easier to model than cutting so I will begin there. A breakage model of hair will look something like (dl/dt)*t - E*t*(dl/dt) = l. I've broken it into two terms to make it clear what I am doing the first term is the growth term and represents the total length in the absence of breakage the second term is the breakage term where E is a randomly determined factor between 0 and 1. The equation in a more compact form is t*(1-E)(dl/dt) = l. Obviously E must be a monotonically increasing function to make sense. Otherwise breakage would sometimes make the hair grow longer. Though there are many ways one might structure E to have the right kinds of properties I am going to make it a randomly increasing recursive function. in a unit time E has a certain probability M of increasing. if the current value is E0 the value of E when it increases will be E1 = E0 + (1-M)(1-E) admittedly this isn't a terribly accurate model of breakage since the amount of breakage each time is fixed but it does have some nice properties for instance the more probable we take the breakage to be the smaller the breakage and the actual length of hair likely to be lost increases with increasing length (though not directly as a virtue of our function of E) I haven't worked out the function L as a consequence of the model yet and it doesn't look pretty more to come in future.