*Picture taken from another nice blog post about swarming bees, within a nice blog about bees: http://thelazybfarm.com/why-do-honeybees-swarm*

**Seeley’s book**

I just read a very interesting book about bees, “Honeybee Democracy”, by Thomas Seeley. The part that I cannot get out of my mind is the end of Chapter 3, describing the way bees probably assess the dimensions of the possible new homes. Therefore I ‘ll not speak about it right away. However let me advertise a bit some really cool other parts from the book.

The whole book describes the dynamics of swarms of bees.

Most of the time bees will be going about their business of collecting nectar and pollen, growing, building up their nest and the reserves of honey, termoregulating, and similar “routine” stuff. This is everyday life has interesting stuff (e.g.: who decides the allocation of bees to different flower spots? how does the dance describing the direction and the distance of a given flower spot encode the information?.. the answers are somewhat surprising! they are answered “en passant” during the book) however the main focus is about the point where, in a moment of high resources, “the swarm splits” and part of the bees has to take off together with a daughter queen..

**New swarm
**

…Then the swarm will take a momentary position on a branch of a tree or similar, in the open. In this situation it will have to decide where to go settle. The bees at this point have their bellies full of honey reserves and once the weather becomes sunny some of the most active bees decide to explore the surroundings in search of some cavity where the whole swarm could settle (what are the factors discriminting the behavior? maybe it’s just the very full stomach giving her lots of energy? are some bees geneticaly predisposed?).

Then several scouts will go explore cavities around. Once they find something interesting, they will come back and “dance the location” to the other bees, more emphatically and repeatedly, or for fewer times and with less vigour, depending on the overall “quality” they assign to the site. Some of the bees will perhaps get convinced to explore the same cavity (note about the “convincing”: it’s not clear to me how far this was experimentally proved.. could it be just an educated guess? is it the same mechanism which is at work during the search for flowers?). If, after exploring, these new bees get “enthusiastic” themselves, they will “dance” again, if they are not they will not (fascinating fact: once a bee “said its say” about a location, it will “shut up”, leaving the deliberation to the group.. would that be a strategy which could help humans to deliberate too? what about the case where the kind of “deliberation” is the one about scientific truth, and takes place in scientific journals? Probably to some extent differences between bees and humans can be explained on an evolutionary/genetic basis, but that is a topic for another story).

Just when all the returning explorers agree on the same location will the whole swarm take off and go on to settle in that place (how can the whole swarm be directed by just the relatively few bees which really know the location? by the way, how many bees know the location in percentage? how do these swarms cover all the distance wihtout losing cohesion? how about not loosing the queen? ..there is a chapter with the answers).

**The crucial qualities of a nesting site**

Like a human home, a nesting site does not just have one single parameter to weigh on, rahter there are several, e.g.: 1) the height from the ground will help to avoid predators, e.g. bears; 2) the integrity of the walls will help in the winter months for thermoregulation; 3) a particularly damp site will be rejected too; 4) a wider entrance will allow more predators to enter and more heat to esape, therefore smaller openings are preferred, etc….

There are more factors, and they were tested through experiments, some of which are very clever. I left out one quality of nests, which is quite crucial: The volume of the cavity. It is very important for a swarm to have enough volume mainly because there must be enough space to store reserves of honey for the winter. The principal cause of death of a swarm during winter is precisely the depletion of the honey reserves, and the volume of the cavity is the main constraint this kind of resources (no scientific experiment on this is mentioned in the book, or better, it is said that in traditional honey hives the apicultors leave more space thant necessary on purpose, and it is all filled by honey reserves.. how much space is the limit? is the location playing no role on these matters?or is the location in nice flower-ful regions one of the factors in the nest choice too?).

**How do bees measure the volume of a cavity?**

You have to immagine that you are a bee, an animal which is optimized to wander around on sunny days and detect flowers using for orientation the sun… (by the way, in rainy periods do the bees always rest? What about just cloudy days? what is the precise factor? maybe the sun visibility? do they loose orientation if they try to fly in these days, or can they find other clues to orient?.. I don’t know now the answers to these questions.. one point which is more or less true is that on cloudy days several kinds of flowers close up, therefore surely they become less attractive..but again: how much? is that a complete answer?)

….and, through a small hole, you enter a very dark cavity which you never saw before. How do you go about measuring the volume? What we know are some scattered facts, and some analogies to other cases. One fact is that bees seem to be just walking around the walls, doing just some small “hopping” flight to move around. Another thing is that the more they walk, the bigger the cavity seems to be from their point of view: this was tested by an ingenious experiment by the author in 1977. We also know that the light inside the cavity is not so strong, basically a good approximation is to say that the cavity is really like total darkness. So let me describe how the book presents the “volume measurement procedure” and then let me give my own small hypothesis to contrast that.

**Parenthesis: the Buffon needle problem**

Buffon’s needle problem has the following formulation : consider a collection parallel lines in the plane disposed at distance . Then drop a needle of lenght at random in that plane. The expected/average number of crossings between the segment and the set of lines will be .(after you read this paragraph you’ll have the tools for proving this on your own)

If we take tow independent variables choosing two segments of lenght at random then we get the seame result, since expectations of independent variables sum. From there it is a small step to note that even is we impose that the end of the first segment coincides with the start of the second, we still get the same average number of intersections. This is because anyways, the identified points are themselves distributed uniformly at random. We use here the symmetry of the plane.

By iterating this reasoning, we obtain that any “random curve” of lenght will give the average number of intersections again like in the above formula. Note that there isn’t just one notion of “random curve”, and in this case we mean “a limit curve obtained as welding of more and more short independent segments” (another popular notion is that of a “Brownian path”, however I don’t know the correct framework for quantifying the quantity of crossings of a Brownian path of lenght with the set of parallel lines above; it feels like the usual notion of arclenght is not natural in the setting of Brownian motion).

**Random walks of bees measure area?**

Let’s return to the bees, and to the question of how they measure the volume of pontential nesting sites.

If you are a bee inside a dark cavity, the idea is that you are going to move at random. Then the hypothesis is that bees secrete some kind of pheromone, then check whether they passed through the same point by smelling it around with their antennas. The interesting “prnciple” described in Seeley’s book, and based on analogous ideas regarding scouting ants (where, i have to admit, it is a bit more plausible) by Dornhaus and Franks, is the following: *the area of a surface is proportional to the average number of crossings of two random paths of given length. *To get more details on this, type “Buffon needle algorithm” on google(.. but don’t expect to find any details or proofs.. at least I didn’t find any!)

I have to put quotation marks around the word *principle*, because it seems quite far-fetched, and it is probably *false* in general. But it would be very interesting if any version of this could be proved, i.e. if there were a version of Buffon’s result (i.e. a general formula for the numbers of crossings) where the parallel lines are replaced by another random object (this version is easier) or where the underlying plane is replaced by a general kind of surface (this version is really challenging!).

What I have big doubts about is the extent to which one can hope to find the *surface area of an object of unknown shape* by looking at *crossings of random lines*. It would be really cool to find some algorith like that!

**Once they know the area, how do bees measure volume?**

The next step once the bees calculated the surface of the cavity, is to calculate the volume: the hypothesis is that to do this the bees measure the *average mean free path length *during their short hopping flights. If you know area and average free path lenght then by a simple calculation you can compute a very good estimate of the volume.