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\title{How many angels can dance on a potato head?}
\author{Anders Sandberg}
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\abstract{\noindent We derive upper bounds for the density of dancing angels on
a typucal tuber of the species {\em Solanum tuberosum}.}


\section{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.

\section{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 $k D M$ distinguishable
bits can exist, where $k=2.57686 \cdot 10^{43}$. This gives us a bound
of around $9 \cdot 10^{40}$ angels\footnote{If some angels have not decided yet, contrary to the theology of Thomas of Aquinas, then this bound will not be strict.}.

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 $0.0025 c^2 / 2 G m$ angels crowd together (here I ignore the
mass of the potato). For angels of human weight (80 kg), we get a
limit of just $2 \cdot 10^{22}$ 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.

\section{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. 

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