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Apiological: mathematical speculations about bees (Part 2: Estimating nest volumes)

This is part 2 of a three-part series of mathematical speculations about bees. Part 1 looked at honeycomb geometry.

Honeybees scout for nesting sites in tree cavities and other nooks and crannies, and need to know whether a chamber is large enough to contain all the honey necessary to feed their colony throughout the winter. A volume of less than 10 litres would mean starvation for the whole colony, whereas 45 litres gives a high chance of survival. How are tiny honeybees able to estimate the capacity of these large enclosed spaces, which can be very irregular and have multiple chambers?

Apiological: mathematical speculations about bees (Part 1: Honeycomb geometry)

Bees have encouraged mathematical speculation for two millennia, since classical scholars tried to explain the geometrically appealing shape of honeycombs. How do bees tackle complex problems that humans would express mathematically? In this series we’ll explore three situations where understanding the maths could help explain the uncanny instincts of bees.

Honeycomb geometry

wild honeycomb

A curvy wild honeycomb.

Honeybees collect nectar from flowers and use it to produce honey, which they then store in honeycombs made of beeswax (in turn derived from honey). A question that has puzzled many inquiring minds across the ages is: why are honeycombs made of hexagonal cells?

The Roman scholar Varro, in his 1st century BC book-long poem De Agri Cultura (“On Agriculture”), briefly states

“Does not the chamber in the comb have six angles, the same number as the bee has feet? The geometricians prove that this hexagon inscribed in a circular figure encloses the greatest amount of space1.”

  1. Translation by Hooper and Ash in the Loeb. I’ve been told that ‘Hexagonon’ is in its singular form, and the only Greek word (also having Greek grammar) amongst this part of Varro’s Latin text. I would be happier that Varro understood what he was writing about if the text more explicitly described the construction, perhaps ‘Three hexagons encircling a point’, or ‘Six hexagons arranged around a seventh’. In translation, it could be viewed as falsely suggesting that the hexagon is the polygon with the greatest area that fits inside a circle. In his defense though, Varro also earlier suggests that orchards be arranged regularly in quincunxes, the arrangement of spots representing the number five on dice, to take up less room and give better quality produce. The centres of hexagons in a regular hexagonal tiling can be thought of as an elongated quincunx, repeated. As this is essentially the same result used in another context, I’ll give Varro the benefit of the doubt and defer to Varro’s poetic license. []

Crimewaves really are waves – but they can be stopped

Nothing puts your home insurance premium up like having been burgled in the past – because it means you’re more likely to be burgled again. Stanford researcher Nancy Rodríguez, with colleagues Henri Berestycki (who is first author, for the record) and Lenya Ryzhik, has developed a travelling waves model to explain this phenomenon – and, more importantly, how to stop it.

Crime, according to past research, tends to cluster in particular neighbourhoods – and even individual houses. Once a crime epicentre has been established, criminal activity tends to spread out in a wave pattern, gradually engulfing larger and larger areas.

Urgent review of Government calculations underway

According to an article behind the Times paywall which I haven’t read, an “urgent review is under way into the reliability of some of the Government’s most crucial calculations in the wake of the West Coast Main Line shambles“.

The part of the article above the paywall reports that checks for ensuring the accuracy of models for climate change, income distribution, benefit claims and farming subsidies are all included in the audit. Website PoliticsHome expands on the basic Times link, saying that “every Government department has been required to draw up a list of ‘business critical’ models that they rely on to do their jobs”.

If you subscribe to The Times, you can get the rest of the story on its website: Maths check across Whitehall after West Coast rail line fiasco.

Ready: reaction-diffusion simulator

Google Code, one of now approximately a million different websites which start with the word Google, is a sharing platform for developers to exchange open-source programs and nifty things they have made.

One such nifty thing is this Reaction-Diffusion package, based on our old friend Alan Turing’s famous equation. The reaction-diffusion equation, originally given in Turing’s 1952 paper The Chemical Basis of Morphogenesis, provides a model for how a mixture of chemicals, reacting with each other while moving under the action of diffusion, might result in the kind of patterns we see in animal print and elsewhere in nature.

Screenshot of Ready 0.3, showing a simulation of a reaction-diffusion system on a lion's fur

Model for processing language in context

New research looks at how language is used to convey information in context, something which is, according to its abstract “one of the most astonishing features of human language”. Apparently there have been “many” theories providing “informal accounts of communicative inference” but few have succeeded in making “precise, quantitative predictions about pragmatic reasoning”.