A typical if not prototypical question: How many angels can dance on the head of a pin?

Let’s see if we can make any sense of it…

Firstly, we can use Sorbonna’s authority:

“The sixth, that an angel can at the same moment be in different places and can be omnipresent if he chooses. We condemn this error, for we firmly believe that an angel is in one definite place; so that, if he is here, he is not elsewhere at the same moment; for it is impossible that be should be omnipresent, for this is peculiar to God alone.”

Sorbonna, Condemnation of Errors, 1241

On angels’s topology:

Secondly, we hold these truths to be self-evident (hmmm, just kidding):

• Tenet 1: that an angel is in one definite place at any specific moment in time (see Sorbonna, Condemnation of Errors, #6 above).
• Tenet 2: that angels cannot share the same place at the same moment in time (sctrictly speaking, Sorbonna did not say anything about it, but I am sure they would support it. )
• Tenet 3: that the footprint of an angel on any surface (e.g. on the head of a pin, which is homeomorphic to a circle) is not 2D as it would be in case of a usual 3D creature, but has a lower dimensionality. I maintain that the aforementioned footprint is (obviously!) homeomorphic to a cross, but if I am mistaken we can review other options as well. For example, it may be fractal with dimensionality between 1 and 2.

Therefore, the problem “How many angels can dance on the head of a pin?” is reduced to the following question:

What is the max cardinality of number of non-intersecting crosses of arbitrary, but finite size can be placed on a circle?"

Solution:

Let’s take any sequence of positive numbers approaching 0, e.g. 1, 1/2, 1/4, 1/8, 1/16, … - would do.

For a particular cross let’s call its “size” the length of the smallest of the two crossbars.

Let’s split our crosses into caregories: 1) with the size greater or equal to 1, 2) with the size less than 1 and greater or equal to 1/2, 3) with the size less than 1/2 and greater or equal to 1/4, etc.

Let’s take one single category (e.g. with the size less than 1/8 and greater or equal to 1/16): all crosses there have the size at least 1/16, that is their crossbars have the length of, at least, 1/16. Well, it’s obvious that two crosses of the size at least 1/16 (or any finite size) cannot be placed with their centers (intersection points) too close together without intersecting. As an exercise for the reader: the distance between the centers of two non-intersecting crosses of size L cannot be less than L/2.

Thus, there can only be a finite number of crosses of each category on the circle.

We have a countable number of categories. Therefore, the total number of crosses is also no more than countable ( ℵ₀ see).

Notes:

1. Assuming that the dimensionality of the footprint in fractal (between 1 and 2) will not change the result.

2. Obviously, if the footprint is drastically different (a point, or homeomorphic to a line), the cardinality may become non-countable. I would assume that they may be only true for even higher divine creations, such as archangels.

3. “… But if we laugh with derision, we will never understand. Human intellectual capacity has not altered for thousands of years so far as we can tell. If intelligent people invested intense energy in issues that now seem foolish to us, then the failure lies in our understanding of their world, not in their distorted perceptions. Even the standard example of ancient nonsense ― the debate about angels on pinheads ― makes sense once you realize that theologians were not discussing whether five or eighteen would fit, but whether a pin could house a finite or an infinite number.”

  ― S. J. Gould, “Wide Hats and Narrow Minds”

(Just in case: all of this was a joke :) )