A question of many Benders

A recent episode of Futurama featured the lovable alcoholic robot Bender creating duplicates of himself that are 60% his size.  Later in the episode the duplicates continue to replicate at 60% size, until the sheer number of sub-atomic Benders start to overwhelm the world and eat away at it.  This is an actual end-of-world scenario called, as it is in this episode, the “grey goo,” but the episode also makes a point of talking about the generations of Benders and how their population increases without bound.

The question that popped into my mind is what generation of Bender would be so small as to influence matter at the subatomic level, as they do in the episode.  In particular, the diminutive Benders are shown altering the molecular structure of water, and so we shall use that as a frame of reference.  The size of a water molecule is 0.942 angstrom, about 94.2 picometers (1/10th a nanometer or 1.0 x 10^-10 meters).  The Benders pulling apart the molecules appear to be roughly three times that size, so we will assume they are roughly 300 picometers in height.

A full sized Bender seems to be the same height as a typical human, which we will assume to be 1.73 meters.  From this, we can derive the formula h(g) = 1.73(.6)^g where g is the number of generations and h(g) is the height of the g-th generation.  If we want to know at what generation the height will reach 300 x 10^-10 meters, we can easily substitute it in and solve for g (I’ll leave that for you to calculate).  According to my math, I get very near 35 generations, which we will use as our number.

So there were 35 generations of Benders out there to get to the sub-atomic sized Benders we saw in the show.  How many Benders does this mean?  We see that each time a Bender replicates himself, two copies are created.  If we assume a Bender only copies himself one time, that means there are 2^35 = 34.36 billion Benders in the 35th generation!  With all the Benders in all the other generations, this works out to be 2^0 + 2^1 + 2^2 + … + 2^35 = 68.72 billion Benders on the entire planet!  And that’s if a Bender only copies itself one time!  If each Bender copies itself twice, we would wind up with 4.7 x 10^21 (4.7 quintillion) Benders!  It seems that Bender’s call for each of his descendants to perform “1 quntillionth” of a task was not too far off!

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