Behavioral Ecology Vol. 12 No. 6: 655-658
© 2001 International Society for Behavioral Ecology
The accuracy of Buffon's needle: a rule of thumb used by ants to estimate area
Centre for Mathematical Biology and Department of Biology and Biochemistry, University of Bath, Bath BA2 7AY, UK
Address correspondence to E.B. Mallon, who is now at the Experimental Ecology Group, ETH Zurich, CH-8092 Zurich, Switzerland. E-mail: mallon{at}eco.umnw.ethz.ch . N.R. Franks is now at the School of Biological Sciences, University of Bristol, Bristol, UK.
Received 5 July 2000; revised 19 October 2000; accepted 20 November 2000.
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ABSTRACT |
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Colonies of the ant Leptothorax albipennis naturally inhabit flat rock crevices. Scouts can determine, before initiating an emigration, if a nest has sufficient area to house their colony. They do so with a rule of thumb: the Buffon's needle algorithm. Based on a derivation from the classical statistical geometry of Comte George de Buffon in the 18th century, it can be shown that it is possible to estimate the area of a plane from the frequency of intersections between two sets of randomly scattered lines of known lengths. Our earlier work has shown that individual ants use this Buffon's needle algorithm by laying individual-specific trail pheromones on a first visit to a potential nest site and by assessing the frequency at which they intersect that path on a second visit. Nest area would be inversely proportional to the intersection frequency. The simplest procedure would be for individual ants to keep their first-visit path-length constant regardless of the size of the nest they are visiting. Here we show, for the first time, that this is the case. We also determine the potential quality of information that individual ants might have at their disposal from their own path-laying and path-crossing activities. Hence, we can determine the potential accuracy of nest area estimation by individual ants. Our findings suggest that ants using the Buffon's needle rule of thumb might obtain remarkably accurate assessments of nest area.
Key words: ants, Buffon's needle algorithm, Leptothorax albipennis, nest measurement, pheromones.
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INTRODUCTION |
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The ability of animals to make decisions about where to live is crucial to their fitness (Lack, 1968
;
Seeley, 1985;
von Frisch, 1974). Many
animals have the ability to discriminate among nesting sites or among
territories on the basis of quality. Rendell and Verbeek
(1996) showed that the nest
cavity choice of the tree swallow is influenced by the amount of debris
remaining from previous inhabitants, as this affects both the size of the
cavity and the number of ectoparasites. For cliff swallows, the time it would
take to build a nest at a new site is the most important factor in site
selection (Gauthier and Thomas,
1993). Honeybees have been shown to discriminate between potential
nest cavities based on many different variables, including volume of the nest
cavity, presence or absence of comb, and height above the ground
(Seeley, 1977).
Although Seeley (1977)
proposed that volume estimation was related to how long honeybees walked
around the nest, none of the other reports proposed a mechanism by which the
individual could assess the quality of the site or the quantity of resources.
Few studies have attempted to elucidate such a mechanism (although see
Schmidt and Smith, 1986, for
their elegant study of host egg measurement by the parasitoid wasp,
Trichogramma minutum).
Leptothorax albipennis is a small (approximately 3 mm long)
monomorphic myrmicine ant. It lives in small colonies with fewer than 500
workers and a single queen. L. albipennis colonies in Britain nest in
crevices in rocks (Partridge et al.,
1997). These crevices can be replicated in the laboratory in
geometry and size by artificial nests made from microscope slides
(Sendova-Franks and Franks,
1993). Mallon and Franks
(2000) demonstrated that the
workers of this species are capable of measuring the floor area of artificial
nests and rejecting those that are too small. They provided evidence that
individual ants make measurements of the area of the new nest using a method
which they termed the "Buffon's needle algorithm."
Buffon's needle is a classic exercise in geometrical probability named
after the eighteenth-century mathematician Georges Louis Leclerc Comte de
Buffon (Kendall and Moran,
1963). An equation (Â =
2SL/
N) can be derived from Buffon's analysis which
relates the area of a plane (Â) to the
lengths of two sets of random lines (S and L) and the number
of intersections (N) between them (see
Mallon and Franks, 2000, for a
detailed explanation).
Mallon and Franks (2000)
showed that upon entering a nest each scout ant lays an individual-specific
trail pheromone inside the nest. She then leaves the nest, reenters it, and
estimates her frequency of intersection with her previous trail. This
intersection frequency is inversely proportional to the area of the nest.
Mallon and Franks (2000)
reduced this intersection frequency (between first- and second-visit paths) by
presenting ants with nests in which there was a floor-sheet that had holes
over half its area and by taking away this sheet between first and second
visits. Removal of this sheet should have removed approximately half the
trails laid on first visits and should also have reduced trail intersections
on second visits by a factor of about 2. This caused colonies to accept nests
that they would have otherwise rejected as too small.
What is the potential accuracy of this rule of thumb? The simplest way this algorithm could give an estimate of area would be if each scout ant lays the same length of trail on her first visit to a new nest, no matter what size the nest (see Discussion). Here we detail experiments to test this prediction and also present experiments to quantify the accuracy of the Buffon's needle algorithm.
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METHODS |
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Collection and marking of colonies
Colonies of Leptothorax albipennis (Curtis) were collected from Portland Bill, England, in March 1999. They were cultured tured in the laboratory by housing each colony in a nest made by sandwiching a piece of cardboard between two glass microscope slides (Sendova-Franks and Franks, 1993
). The standard nesting cavity within the cardboard was 35
x 25 x 0.8 mm. The single entrance to the cavity measured 2
x 2 x 0.8 mm. Each nest was placed in a petri dish, 100 x
100 x 17 mm, the walls of which were covered with Fluon to prevent the
ants from escaping. In each of three colonies, we marked every worker with a unique combination of four different colors of paint spots (from a palette of six possible colors): one spot on the head, one on the thorax, and two on the gaster.
Emigration
We induced the colonies to emigrate to new nests of standard (25 x 35
mm) and double-standard (35 x 50 mm) floor area. The old and new nests
were placed 10 cm apart (entrance to entrance) in a large (220 x 220 mm)
petri dish with Fluoncoated sides. The emigration was triggered by removing
the upper glass slide from the old nest so that the cavity was exposed. The
experiment was recorded using a Panasonic NVSX30B S-VHS camera, trained on the
new nest, recording directly onto a Panasonic TL700 time-lapse VCR in real
time. The color resolution of the camera did not permit sufficiently accurate
recognition of individuals from the videotape. Hence, we determined the
identity of the ants entering and leaving the nest by direct observation
during each experiment and recorded this information on the audio channel of
the tape. Using this setup, it was possible to record exactly when each ant
entered and left the nest. The filming stopped when most ants entering the
nest were involved in recruiting or transporting activities, usually within
1-2 h of the beginning of the emigration. Each colony emigrated four times,
twice to a standard nest and twice to a double standard-sized nest.
Measurement of visit distance
We recorded both first and second visits for 10 ants to both standard- and
double-sized nests. The paths of these visits were traced onto acetate sheets
directly from the monitor by advancing the videotape at 1-s intervals and
marking the position of the gaster (for first visits) or head (for second
visits). For first visits we recorded gaster position, as it would indicate
approximately the position of trail laying (for myrmicine ants most of the
trail-laying pheromones are produced by their hind regions;
Hölldobler
and Wilson, 1990). The head position during second visits would
indicate roughly where the ant was intersecting its first-visit path with its
antennae. Each point was connected to the next by a straight line and the
distance measured with a map measurer. We recorded the total distance covered
for each visit in millimeters. We estimated the number of intersections made
between the paths of first and second visits to the same nest by overlaying
the acetate sheets from the two visits and counting the number of
intersections between the two lines.
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RESULTS |
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The distribution of distances traveled for each first visit of scouts from one of the colonies was normalized by the square-root transformation. There was no significant difference between the distance an ant traveled inside the nest on her first visit to a standard-sized nest (raw data median: 486 mm, square-root-transformed data: mean ± SD = 21.64 ± 3.83) and the distance traveled on her first visit to a double standard-sized nest (600 mm, 25.57 ± 8.43; paired t test: t = 1.79, n = 9).
For the data pooled from the three colonies, there was also no significant difference between the visit duration (seconds; square-root transformed to normalize) of an ant on its first visit to a standard-sized nest (raw data median: 222 s, square-root-transformed data: mean ± SD = 15.801 ± 4.460) and on its first visit to the double standard-sized nest (182 s, 13.765 ± 4.482; paired t test: t = 1.65, n = 23).
The intersection frequency (intersections per millimeter) data, calculated for one colony, was also normalized by the square-root transformation. There was a significant difference between the intersection frequencies of an ant when it visited the standard nest (mean ± SD = 0.7193 ± 0.2004) and the double-standard nest (0.5205 ± 0.1660; paired t test: t = 2.96, n = 9, p =.016; Figure 1).
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We observed that when an enters a nest containing several other ants, she spends time making antennal contact with and grooming these ants. This might cause considerable variation in visit duration. For example, ants that enter the new nest early in the emigration may spend less time in the nest because there are fewer ants to groom. To test whether this was the case, a correlation was performed of the duration of first visits of ants of the colony against the average nest population of the new nest at the time of their visits. There was no significant correlation between average population and visit duration (Spearman rank correlation: rs =.168, n = 59, p =.203; Figure 2).
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Assuming these 10 ants were using Buffon's needle algorithm (see equation 2 in Discussion), we calculated the nest area they would have perceived on the basis of their own trail laying and intersection frequencies. Figure 3 compares the mean perceived area for each type of nest against its true value.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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We have shown that ants surveying a new nest cover a similar distance on their first visit to a nest regardless of the size of the nest cavity. They also spend the same amount of time in a nest on their first visit regardless of the size of the nest. First-visit durations are not correlated with the number of nest mates that are in the new nest at the time of the visit. The frequency at which an ant intersects its previous path is significantly greater when the ant scouts a standard-sized nest compared to a double-sized nest. Furthermore, there is likely to be only a small amount of variation in the perceived area for different ants when they visit the same-size nest.
The area of a plane can be calculated from the following equation (see
Introduction):
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from equation 2, we have
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The Buffon's needle algorithm shows that there can be a simple relationship
between the intersection frequency of two sets of random lines and area. One
remaining issue is the degree of accuracy of nest area estimation individual
ants could achieve given their own particular pattern of trail laying and rate
of path intersection. We can estimate this by putting the empirically
calculated intersection frequencies of each of the 10 ants and their
individual average first-visit lengths into equation 2. We are not, of course,
suggesting that ants know about or that they calculate distances in
millimeters; rather, we are using this equation to calculate relative
perceived areas to see how much variation there is in the estimation of nest
area by individual ants. Such variation is relatively small (see
Figure 3). Notice how close the
perceived areas are to the nests' true values. However, this is not really
important. What matters is that there is a close approximation to a 2:1 ratio
for perceived areas for doublesized nests versus standard-sized nests (the
calculated ratio is 1.96:1). That is, ants using Buffon's needle algorithm
would measure the double-standard nest as having twice the area of a
standard-sized nest. For other cases in which the accuracy of the rules of
thumb used by insects have been determined, see Cartwright and Collett
(1983) and
Müller and Wehner
(1988).
Mallon and Franks (2000)
introduced Buffon's needle algorithm as a possible method by which scout ants
could measure nest area. The present study has provided further evidence that
these ants are using such an algorithm. We also show the potential accuracy of
this method by feeding the ants' own (potential) data into the algorithm. It
may be the case that there is also collective decision making in this system
in which the assessments of many different scouts are combined to add yet
another level of robustness. This will be the subject of further work.
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ACKNOWLEDGEMENTS |
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We thank Jay Denny, Liz Langridge, Stephen Pratt, and Ana Sen-dova-Franks for useful discussions and encouragement. E.M is supported by a Natural Sciences Demonstratorship from the University of Bath.
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