How many angels can dance on a potato head?

Anders Sandberg November 17, 2000

2

Chapter 1

Basic Treatment

1.1 Introduction

The problem of the number of angels that can dance on the head of a pin has been a major theological question since the Middle Ages. Here we generalise it to the case of angels dancing on a potato, and show that unless angles have mass less than 10^{−20} kg the dance is mass limited. Mechanical considerations suggest that angular momentum will also play a role in the maximum angelic density.

1.2 Analysis

Assuming each angel contains at least one bit of information (fallen / not fallen), that the potato is a sphere of diameter of 5 centimeters and weighs around 70 grams, we can use the Bekenstein bound on information to calculate an upper bound on the angel density. In a system of diameter D and mass M, less than kDM distinguishable bits can exist, where k = 2.57686 · 10^{43}. This gives us a bound of around 9 · 10^{40} angels.

Note that this doesn’t take the mass of angels into account. If angels have mass m, then the potato will collapse into a black hole if more than 0.0025c^{2}/2Gm angels crowd together (here I ignore the mass of the potato). For angels of human weight (80 kg), we get a limit of just 2 · 10^{22} angels. Hence it seems that if angels have a mass larger than 10^{−20} kg, then their dance is mass limited rather than limited by quantum mechanics.

However, if they dance very quickly, then the angular momentum could lead to a situation like the Kerr metric, where no event horizon forms. Hence the number of dancing angels that can crowd together is likely much higher than the number of stationary angels. However, at these speeds the friction caused by their interaction with the potato is likely to vaporise it, or at least break it apart.

1.3 Discussion

Angel physics has so far mainly employed theological methods, but as this paper shows modern information physics, quantum gravity and classical relativity theory provide powerful tools for exploring the dynamics and statics of angels.