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Did you know that the origins of the number pi can be traced back at least as far as ancient Egypt?

This mathematically magical number defines the ratio of the circumference of a circle to its diameter and, as Marcus du Sautoy describes in a special Inside Oxford Science podcast celebrating Pi Day (14 March), one of the very first recorded estimates of its value appears in the Egyptian Rhind Papyrus.

Marcus tells us that this papyrus, now housed in the British Museum, ‘is full of fantastic mathematics including ideas of how to use binary numbers to do multiplication 3000 years before the German mathematician Leibniz would reveal their potential…’

The papyrus describes how the Egyptian scribe Ahmes attempted to estimate the area of a circular field whose diameter is 9 units across.

Because the area of a circle is pi times the radius squared, if we know the area and we know the radius we can calculate pi. The Rhind papyrus states that a circular field with a diameter of 9 units is equal in area to a square with sides of 8 – but where did this idea come from?

‘My favourite theory sees the answer in the ancient game of Mancala. Mancala boards were very popular during this period and were even found carved on the roofs of temples,’ Marcus reveals.

Here, it’s the Mancala board rather than the game itself that are important as filling the circular holes with stones could inspire pi-tastic thoughts:

‘The player might have gone on to experiment with making larger circles and discovered that 64 stones can be arranged to make a large circle with diameter 9 stones. But 64 stones can also be rearranged into an 8 by 8 square.’

‘By rearranging the stones the circle has been approximated by a square whose area is 64 units. Recall that the area of a circle is pi times the radius squared. The radius in this case is 4 and a half. So Ahmes’s calculation gives the first estimate for pi as the area 64 divided by the radius squared 4.5 squared, which comes out at 256 over 81 or approximately 3.16. Not bad for a first estimate.’

Marcus goes on to explain how the Indian mathematician Aryabhata used a very accurate approximation for pi, (3.1416) to estimate the Earth’s circumference to within an accuracy of 70 miles. And how, in the film Pi, the central character would be just as likely to find the ASCI code for the novel Moby Dick alongside messages from God in the number’s decimal expansion.

What would the value of pi be if we all had fingers like Homer Simpson? You’ll just have to listen and find out…

John Pannell of Oxford's Department of Plant Sciences has been studying the mysteries of plant sexuality using Annual Mercury [Mercurialis annua].

A report of his research is published in Current Biology: I asked him about the reproductive choices made by plants, Darwin's views on hermaphoditism and how shifting sexes impact on conservation...

OxSciBlog: What are the evolutionary advantages for plants of being male/female or being hermaphrodite?
John Pannell: Most plants are hermaphrodites. Common intuition is that hermaphrodites benefit from being able to self-fertilise, thus circumventing the need to find a mating partner. This has obvious advantages for sessile [non-mobile] organisms such as plants. However, the vast majority of hermaphroditic plants do everything they can to avoid self-fertilisation because of the deleterious effects of inbreeding on their progeny.

One solution to the problem of selfing is to become a male or a female. While only about 5 per cent of plant species have fully separate sexes, hermaphroditism has been abandoned a great many times in different families, probably often driven by natural selection to avoid inbreeding.

OSB: How did Darwin think shifts between separate sexes and hermaphroditism (or vice-versa) within plant populations occur?
JP: Darwin wrote three books on plant reproduction, one devoted to the problems of self-fertilisation, one to the exquisite adaptations that hermaphrodites have evolved to avoid selfing, and one about plants that have individuals with different types of flowers, including those that are male or female.

He was puzzled about why hermaphroditism should ever have evolved towards separate sexes but realised that hermaphrodites might benefit by becoming specialists in one sexual function or the other. Because reproduction is expensive, specialisation in the production of seeds, say, would cause plants to divert resources away from pollen production. Darwin then saw the evolution of fully separate sexes as the outcome of incremental adjustments in the relative allocation of resources to one sex from the other.

OSB: Why is the herb Mercurialis annua a good plant to use in a study of these shifts?
JP: Mercurialis annua is a wonderful workhorse for studying shifts in the sexual system of plants, for a number of reasons. First, it displays remarkable within-species variation in its sexual system, with some populations being hermaphrodites and others having fully separate sexes.

Even more interesting is the occurrence of populations with an intermediate sexual system, where pure males mate with hermaphrodites. This is a rare sexual system in plants and animals, but its occurrence in Mercurialis annua allowed use to use males to exert a selection pressure in artificial populations on the allocation by hermaphrodites to sexual reproduction.

Because Mercurialis annua is also an annual plant, selections experiments over several generations within a single research project are feasible.

OSB: What has you research revealed about such shifts?
JP: Our research essentially confirmed the prediction that the presence of males in a population of hermaphrodites would cause selection to favour those that biased their allocation to their female function. Similarly, if you remove males from a population that formerly had them, the hermaphrodites are selected to become more male in their allocation.

The experiments thus not only confirm Darwin’s intuition that the sex expression of hermaphrodites responds to natural selection incrementally, but it also provides an illustration of how transitions between hermaphroditism and fully separate sexes can occur.

OSB: What is the significance of shifting sexual patterns to the long-term survival/conservation of plant species?
JP: The research highlights the fact that natural plant populations are in a dynamic balance between changing environments (in this case, whether males occur in the population or not) and responses to natural selection.

We chose for our experiments plant populations that were known to harbour a lot of genetic variation for pollen production, so the fact that hermaphrodites responded to selection is not so surprising. However, many populations of Mercurialis annua, particularly those in the north of the range, have very little genetic variation, and a study of ours published last year in Science showed that these northern population cannot respond to selection nearly as quickly as those in the south.

So, whereas some populations can respond to selection, others may not. Importantly, it is often the marginal populations of a species that are most threatened by environmental change, yet these are precisely the ones that might be less responsive to natural selection because of depleted genetic variation.

Dr John Pannell is based at Oxford's Department of Plant Sciences.

Owen Lewis of Oxford's Department of Zoology is investigating the decline of the Small Tortoiseshell butterfly (Aglais urticae).

Ahead of a report on his research project airing on BBC South's Inside Out [on 11 March], I asked Owen what might be behind the decline and how volunteers can help track the cuplrit...

OxSciBlog: How concerned should we be about the health of Britain's butterfly populations?
Owen Lewis: We should be very concerned. Butterfly populations respond rapidly to environmental changes and are a sensitive bellwether for what is happening to biodiversity more widely.

We have a fairly small butterfly fauna compared with mainland Europe – about 55 resident species – but many of these have declined dramatically over the last few decades. We know this because there is a long history of butterfly survey work in Britain, almost all of it done by amateurs (for example, the volunteers working for the charity Butterfly Conservation).

In general the specialised species have fared most badly: ones where the caterpillars have exacting requirements in terms of microclimate and food plants. Butterflies like the beautiful High Brown Fritillary used to occur in almost every wood in the southern half of Britain. Now they are confined to a few carefully managed localities in the south and west.

Fortunately, butterfly ecologists have worked out how to manage habitats to maintain the right conditions for these threatened species. Restoring them to their original distributions will be far more challenging: the modern agricultural landscape just isn’t suitable breeding habitat for most butterfly species, and the remaining islands of good habitat are scattered in a sea of hostile terrain.

OSB: Why is the Small Tortoiseshell decline a particular concern?
OL: Their caterpillars feed on stinging nettles and are (or at least were) one of the most widespread, abundant and widely-recognised butterfly species in the country. Most people will have seen the adult butterflies feeding on Buddleia bushes or Sedums in their gardens, or will remember keeping the spiky black caterpillars in a jar as children.

Unlike the habitat specialist species, we had thought that Small Tortoiseshell populations were holding up well. However, in the last 10 years there has been a huge slump in population sizes, particularly in the south of Britain. Overall, numbers in the south have been reduced by 50 per cent, but the situation is much worse in some areas, where a complete absence of sightings has been reported during the summers of 2007 and 2008.

The slump coincided with the arrival from the continent of a new species of parasitic fly called Sturmia bella in the late 1990s. The fly lays its eggs on nettle leaves, and the Small Tortoiseshell caterpillars consume the eggs unwittingly along with the leaves. The fly eggs hatch in the caterpillar’s gut, and the fly maggots then develop within the caterpillars, literally eating them alive and eventually killing them. The southern parts of Britain where Sturmia bella has been recorded are also the areas where the Small Tortoiseshell is faring most poorly. We want to work out if the parasite is to blame, or whether this is just a coincidence.

OSB: How can volunteers help in the study of this decline?
OL: We want to track the spread of Sturmia bella through the UK and establish how it is in affecting populations of the Small Tortoiseshell and its close relative the Peacock. We are asking volunteers to look out for the caterpillars on nettle plants this summer, and to collect some to grow up at home. Any butterflies that result can be released, and any flies or wasps that develop from them can be sent for us for identification, along with a complete datasheet with information on the numbers collected and where they were found. Full details on how to join in are available through Butterfly Conservation.

OSB: We seem to be seeing an increase in parasitism and disease across many insect populations, what factors may be to blame? Is climate change implicated?
OL: I’m not sure if parasitism and disease are on the rise in general, but climate change may certainly bring new threats. No-one knows how Sturmia bella got here from the continent, but many species are spreading northwards as the climate warms – and this includes new pests and diseases as well as their hosts.

Another theory about the decline in the Small Tortoiseshell is that it is linked directly to climatic changes: caterpillars seem to do less well in dry summers, like the ones experienced in the first few years of this century. It’ll be interesting to see if the wet summer of 2008 leads to a resurgence of Small Tortoiseshells in 2009. Let’s hope so!

Dr Owen Lewis is based at Oxford's Department of Zoology.

Image: Small Tortoiseshell butterfly. Credit: Jim Asher

Software that gives robots a sense of ‘déjà vu’ is the key to them operating effectively in unfamiliar environments, as New Scientist reports in an article on the work of Oxford engineers.

For decades engineers have wanted robots to do jobs that are too dangerous for humans – such as entering disaster zones or exploring other planets. Yet, all too often, once these robots leave the confines of the lab or factory floor they run into one big problem: they get lost.

‘To start with it’s just a small error, say turning 89 degrees instead of 90 degrees, but pretty soon this small error is compounded until the robot is nowhere near where it believes itself to be’ said Mark Cummins of the Oxford Mobile Robotics Group, who has been researching this navigation problem with Paul Newman, who leads the group.

‘Humans too can make these kind of errors but, unlike robots, we can spot when we have been somewhere before and readjust our mental map accordingly. We are giving robots this same sense of ‘déjà vu’ so that, just by taking cues from their environment, they can readjust their sense of where they are and correct their own ‘mental’ maps.’

It may sound a simple task but in fact this ‘where am I?’ question has proved one of the most intractable problems in robotics. At present many autonomous robots rely on pre-produced maps or GPS to find their way around, but GPS isn't available indoors, near tall buildings, under foliage, underwater, underground, or on other planets such a Mars – all places we might want a robot to operate.

Oxford engineers have spent years addressing a key part of the ‘where am I?’ question – figuring out when a vehicle has returned to a previously visited place (known as the "loop closing problem"). To tackle it they've created The FABMAP algorithm that, through a combination of machine learning and probabilistic inference, is able to compare a current view of a scene with impressions of all the places it has been before.

Crucially it does this with great precision and rapidly – fast enough for a robot to realise it is retracing its steps and adjust its route [see images above and below: the green and red circles show parts that were matched/unmatched between the two images].

‘At the moment it can recognise and label different elements of its surroundings – making distinctions between, for instance, gravel paths and roads, stone walls and doorways, even different building types,’ Paul tells us. ‘This sort of ‘semantic exploration’ is the first step towards not just mapping its surroundings but starting to understand them as a human would.’

Mark adds: ‘Another motivation is that this kind of vision research is building towards robots that have some richer understanding of their environment, rather than the bare position information you get from GPS.’

‘In the future we want people just to be thinking about the task they want a robot to perform, and how it can help, rather than worrying about how it finds its way around or gathers useful information about its surroundings. ‘Where am I?’ is an important question, and being able to answer it accurately is central to the future of robot technology.’

Dr Paul Newman and Mark Cummins are based at Oxford's Department of Engineering Science.

Stuart West, who recently joined Oxford's Department of Zoology, has just published in Current Biology about his research into how bacteria cells interact.

I asked Stuart about what bacteria working together or cheating each other means for infection in humans:

OxSciBlog: What are the social interactions that go on between bacterial cells?
Stuart West: Bacteria cooperate to perform a wide range of functions. They secrete a number of factors out of the cell that then provide benefits to the local group - for example to scavenge nutrients, aid movement, overcome their host, kill and degrade prey, kill competitors and degrade antibiotics. They join together to form 'slime cities' (biofilms). Many of these behaviours are controlled by a social signalling system that has been termed 'quorum sensing' and which appears to switch on cooperative behaviours when they are most beneficial, which is when population densities are high.

OSB: How do these interactions affect the overall virulence of an infection?
SW: Hugely. These cooperative behaviours are crucial to the growth of bacteria and the damage that they do to their host. Indeed, many of these traits are also termed 'virulence factors'.

OSB: What is 'quorum sensing' and why is it important to understand it?
SW: Quorum sensing (QS) is the process where bacteria use small molecules diffusable molecules as signals which control other behaviours. The signal molecules are secreted out of the cell, and can then be taken up by the same cell or other cells nearby. This uptake has two effects.

First, it stimulates the production of many products that are released out into the environment, and which are 'public goods' that benefit the local bacteria. Second, it leads to an increase in production of the signalling molecules themselves. At high cell densities this leads to a positive feedback that markedly increases the production of factors released by the cell. The idea here is that QS turns on the production of these extracellular public goods when it is most useful to do so: at high densities.

From a pure science perspective, QS is interesting, because signalling and communication can be hard to explain from an evolutionary perspective, because they are exploitable by individuals that lie and cheat. From an applied perspective, QS is fundamental to the success and virulence of pathogenic bacteria.

OSB: How might your findings help in the search for new ways to combat infection?
SW: One way is that cheats that do not perform cooperative behaviours such as QS could be introduced into hosts to outcompete wild-type cooperators. As well as directly reducing virulence, this could drive down the bacterial population size, which may benefit other intervention strategies such as treatment with antibiotics. Another way is that other beneficial traits, such as antibiotic susceptibility, could also be hitch-hiked into infections by such cheats who do not perform QS or other social behaviours.

Stuart West is Professor of Evolutionary Biology at Oxford's Department of Zoology.