Quantum Estimates: Where Angels Fear to Tread

This close to the holidays, it’s hard to stay focused on work.

I should be writing a post about useful estimation tools for quality statistics. But all those yuletide carols about hosts of angels singing from on high have distracted me.

Alas, I’ve fallen into the clutches of one of the world’s oldest estimation problems, posed centuries ago by medieval scholars:

Just how many heavenly angels can dance simultaneously on the point of a pin?

The answer to this question assumes that you believe in the existence of pins, of course.

Estimation in the Middle Ages: Ask Your Doctor

Over the centuries, a variety of estimation methods have been used to estimate the number of angels that can shake it on a pin.

One of the first concrete estimates can be found in a 14th century German work, Schwester Katrei. The book states that, according to doctors, 1,000 angels can comfortably fit on the point of a needle at one time.

How did medieval doctors know this? Presumably, the same way they knew that you could change husks of wheat into live mice by placing the husks in an open jar with a pair of sweaty underwear and waiting for about 21 days.

Spontaneous generation was all the rage back then. Perhaps the estimate itself was spontaneously generated from a pair of underwear.

At any rate, without more details concerning methodology, we have no idea whether this estimate was based on a random sample of angels from the entire population of heaven.

Be skeptical.

A Simple Solution Based on Superstring Theory

From the early 1300s to the late 1900s, the human race was crazy busy. It was a time of rapid progress and revolutionary advances that saw the invention of the wooden bathing suit, the parachute hat, and the detachable moustache guard. As a result, we made little headway into the angel-on-a-pin problem.

Thankfully, the advent of theoretical physics changed that. In 1995, Dr. Phil Schewe of the American Institute of Physics took another shot at the estimate, using basic concepts of superstring physics.

According to superstring physics, space cannot be infinitely divided. Ultimately, even the breakdown of space breaks down. This happens when the distance scale reaches a value of 10-35 meters. (You can verify this yourself with a ruler at home.)

Shewe assumed that the size of the pinpoint was equivalent to a single atom, which is 1 angstrom, or 10-10 meters. So to estimate the  maximum number of angels that could fit on the pinpoint, he simply divided the size of the pinpoint by the smallest possible breakdown of space possible:

10-10/ 10-35 = 1025

This simple, elegant approach results in an estimate between one septillion (1024) and one octillion (1027) angels. The estimate does not account for bumping wings, however, making it vulnerable to subsequent attack.

Quantum Gravitation: Factoring in Black Holes

Eschewing Schewe, Anders Sandberg of the Royal Institute of Technology tackled the problem in another way.

Instead of applying the seldom-used KISS principle (Keep It Simple, Stupid), Anders opted for the ever-popular MUCK principle (Manufacture Unfathomable Complexities, Knucklehead).

Sandberg notes that Schewe's estimate assumes that there is no overlap of angels on the pinpoint. But this defies basic laws of quantum mechanics, because "when packed at quantum gravity densities, the uncertainty relation will cause their wave function to overlap significantly even if there is a strong degeneracy pressure."

That goes without saying.

Using the upper limit of entropy, the Bekenstein bound, along with the assumption that angels have no mass, Sandberg comes up with an estimate 2.448*105 angels--about a quarter million angels. (To get a more concrete sense of this, imagine everyone from Fort Wayne, Indiana dancing the cha-cha on the tip of an IBM scanning tunneling microscope.)

If  the angels have mass, however, it's a whole 'nother quantum ballgame. In that case, "each angel contributes enough mass-energy to allow the information of an extra angel to move in."  Which means, paradoxically, that more angels can then pile onto the pinhead. Until they reach a critical mass that causes the pin to collapse into a black hole, scattering feathers all across the galaxy.

Based on these constraints, the upper bound of angels can be estimated at 8.6766*1049. That's somewhere between 1 quinquadecillion and 1 sexdicillion angels.

Note: This estimate does not hold true for all types of dances. Also, the angels must dance with speeds close to the velocity of light, which rules out a foxtrot. For more quantum caveats, click here.

A Better Idea

If all these quantum estimates leave you feeling woozy and confused, there's a better option. Open a copy of Minitab Release 6.17*1078 and choose Stat > Pinpoint > Count Angels. Under Methods, choose Quantum: With Overlap. Click OK.

Then have yourself a wonderful holiday season and a great New Year.

Attributions: Black hole image by Ute Kraus, licensed under Creative Commons.