## Thursday, March 18, 2010

### Beard Combinatorics

 Abe knew how to rock the beard.
If you’ve ever shaved off a full beard before, you may have been tempted to try out a few different facial hair configurations in the process.  Maybe you start by shaving it into some pork chop sideburns, then reduce it to a goatee, then shave a little more until it’s a fu manchu before finally trying out the Charlie Chaplin look.  In principle, how many different beard combinations are there?

To solve this, we need to first know how many hairs are in a beard.  Obviously, this will vary from person to person.  Robin Williams’ bird nest will have a lot more hairs than Sidney Crosby’s pathetic excuse for a playoff beard.  I’ll assume a typical beard covers a total area of 30 cm (~1.0 ft) by 10 cm (~ 4.0 in) or, roughly, 0.030 m2.  On my face, individual hairs are separated by about 0.5 mm on average, meaning there are 0.25 mm2 per hair.  From this data, we can estimate that there are roughly 120,000 hairs on a reasonably large beard.

To simplify things, I’ll say that a specific beard is defined by which hairs are shaved and which are not.1  That said, a hair can only be in one of two states: shaved or not shaved.  With this overly simplified definition, we can easily calculate the maximum number of possible beards:

# of possible beards = 2120,000 = 4.0×1036,123

That is a huge number2.  Just writing it out would take about 15 pages.

[1] In principle, you could differentiate beards not only by which hairs are shaved and not shaved but also by how long each hair is.  If any of you readers want to try enumerating this, I’d be interested to see what you get.
[2] You might be tempted to say that this number is way too big.  You’re right.  We have a very strict definition for what constitutes different beards.  After all, if you pluck one hair out of a goatee, it’s still a goatee.  There’s a pretty deep question here: how many hairs do you have to remove before it’s no longer a goatee.  This question applies not only to hairs forming beards, but also to molecules forming complex structures.  In physics, we try to define something called an “order parameter” that can tell what kind of structure a bunch of atoms are forming, but these definitions are always somewhat arbitrary.  I’d be curious to see if anyone could define a reasonably good beard order parameter.