How do animals ‘do’ geometry?

Individuals of some species, under some conditions, can use geometry to determine their orientation, to identify landmarks or to find a place (for example Cheng 1986; Vallortigara et al. 1990; Hermer & Spelke 1994; Spetch et al. 1996; review by Cheng & Spetch 1998). In this context, use of geometry would mean that an individual identifies a landmark not by its appearance, but by its spatial relationships to other landmarks, for example using information equivalent to ‘at the apex of the triangular array’. Another example of use of geometry would be the identification of a goal location as ‘in the centre of the landmark array’ rather than ‘1 m south of landmark A and 1 m north of landmark B’. These two accounts can be distinguished only by transforming the array, as they predict the same search location if subjects are trained and tested with the same array. So what transformations of the array are needed to test for the use of geometry? In a recent study, Kamil & Jones (1997) used just such a two-landmark array. They trained Clark’s nutcrackers, Nucifraga columbiana, to search for a food reward midway between two landmarks, using distances between the landmarks of 20–120 cm, in 20-cm increments (see Fig. 1a). Kamil & Jones transformed the landmark array for testing in two ways. First, they tested the nutcrackers with the landmarks from 30 to 110 cm from each other, at 20-cm increments. The nutcrackers searched at the midway point even with these unfamiliar distances. Second, they placed the landmarks at 100 cm from each other and rotated that array by 15, 45 or 90). At 15 and 45) the birds’ search patterns covered both the midpoint and a single vector from one of the landmarks; at 90) they used only the single vector. Is there sufficient evidence to show that these birds abstracted from their training a rule that food was hidden halfway between the landmarks, irrespective of the distance separating the landmarks? Use of such a geometric description makes intuitive sense, as it is the kind of thing humans do (Spetch et al. 1996, 1997), but Kamil & Jones carefully refrained from committing themselves to such a claim. This caution is justified, as there are at least two alternative search strategies that lead to a point halfway between two landmarks without having any concept of a halfway point. For strategy 1, assume that an animal can learn one vector from each landmark to the goal. The lengths of these vectors may be the mean of the landmark–goal distances experienced during training. To locate the goal when the landmarks are separated by an unfamiliar distance, the animal would average these two vectors. As long as the two landmark–goal vectors are the same length and are given the same weight, the vector average, and thus searching, will always be at the midpoint between the two landmarks (Fig. 1). This holds true for novel landmark–goal distances and even for distances outside the range experienced during training. An animal using this strategy needs to know nothing about the relationship between the two landmarks nor that the goal is to be found halfway between them. The two spatial relationships determining the final point of search are treated entirely independently. Bees and pigeons can locate goals using a system like this (Cheng 1988, 1990; Collett & Baron 1995; Collett et al. 1996; Cheng & Spetch 1998; Menzel et al. 1998). However, we can exclude this strategy as an adequate account for Kamil & Jones’ results, even though it predicts the same search locations as found in their study, because they reported quantitative data on error patterns that are incompatible with this strategy. Their birds made errors, along the line between the landmarks, roughly proportional to landmark distance, that is, with a constant coefficient of variation, following Weber’s law (see also Cheng 1990). Cheng (1988) found that vector averaging increases error. We suggest that this is not due to inaccuracy in the averaging process, but results from a systematic widening of the search pattern. When two Correspondence: R. Biegler, Department of Psychology, University of Newcastle, Newcastle upon Tyne NE1 7RU, U.K. (email robert.biegler@ncl.ac.uk).