Dancing On The Head Of a Pin

May 23, 2008 – 12:15 am

Previously I discussed a letter from reader Scott about parts per billion, and we had plenty of fun discovering that the allowable level of lead in drinking water was equal to about a shot glass in an olympic size pool. But reader Scott had brought up parts per billion as part of a larger discussion on how many angels could dance on the head of a pin.

It's a bit difficult to quantify the number of angels on the head of a pin, because first we'd have to quantify the size of an angel. That can range from subatomic to supermassive to no size at all. And the day we can get the world to agree on the size of an angel, we'll have solved a number of problems of much greater importance than how many of them can dance on the head of a pin.

Of course, my answer is "all of them... if we set the pin in the path of a conga line populated by all the angels who each step on the pin as they dance by." All I was asked was "how many angels can dance on the head of a pin?" No one ever defined what "dance on" meant precisely, nor is this modified by any language requiring those angels to dance on it simultaneously, rather than in succession. And yes, I did turn down a scholarship to law school. Why do you ask?

But that doesn't provide any Rough Equivalents and it doesn't mathematically attack the problem of how many very small things can occupy the head of a pin. So I decided to take it seriously (for a very brief moment).

According to Wayne's Word (an encyclopedia of natural history), they use a 1.5 millimeter diameter for the size of the head of a pin whenever they use the head of a pin in comparisons, and that seems reasonable to me. That gives you a surface area on a flat, round pin head of 1.767 square millimeters.

I don't know how big an angel is, but I do know how big a flu virus is. Since we'll measure the flu virus in nanometers, we have to break down the pin head area into nanometers. Since a millimeter is a million nanometers on a side, it would represent a trillion square nanometers. So we've got 1.767 trillion square nanometers of area on the pin head.

With a 70 nanometer diameter, a perfectly round flu virus would take up 3848.5 square nanometers. But they're not perfectly round, and if they were, we couldn't cram them in so all their edges were touching anyway. We'd still have gaps. So, to keep things fair, I'm assigning each virus a square space, 70 nanometers on a side, meaning 4900 square nanometers.

When we divide 1.76 trillion square nanometers of space on the head of this pin by 4900 nanometers, we find out we can fit 360,612,245 influenza virus cells on the head of a pin. So if we concluded that an angel was the size of a flu virus, the guardian angels for the entire populations of the United States and Canada could fit on the head of a pin, with room to spare.

But let's expand this out. In a square inch, there are 645.16 square millimeters, or 645.16 trillion square nanometers. So, in a square inch, we could pack in 131.6 billion flu viruses. It would be a bit cramped, but it would be close to 20 flu virus cells for every man, woman, and child in the world... in a square inch.

But you know how I like to lay things out for distance. I've done a calculator to get a distance in pennies laid end to end or bologna slices laid end to end. What about influenza virus cells laid end to end? I'm approximately 25,714,260 flu virus cells tall.

Feel like telling the world your height in flu virus or your answer to how many angels can dance on the head of a pin? Post it in the comments below.