Posted by: Alexandre Borovik | August 23, 2008

Dragonflies and mathematics

This post is inspired by wonderful photographs of dragonflies at Susan Beckhardt‘s blog:

Behaviour of dragonflies involves some non-trivial mathematics, and my post is about how dragonflies manage to win over us, humans, in highly mathematical mental games.

I cannibalise here a section from my book Mathematics under the Microscope. In the subsequent text, I will frequently quote papers by my colleague Paul Glendinning [1,2].

Dragonflies, elegant creatures much beloved by poets and children, are consummate predators. As so frequently happens with beautiful predators, their real sophistication is reached not in delightful air acrobatics but in the ways they hunt and fight.

In their fights for territory, dragonflies, when they pursue the enemy with the aim of a sudden attack, use a remarkable and inventive concealment strategy. A dragonfly camouflages its approach so that the foe believes it to be stationary. This conclusion is the result of the reconstruction of stereo camera images carried out by Mizutani, Chahl and Srinivasen [3]. There are already several studies suggesting possible guidance mechanisms used by dragonflies [4,5]; however, the underlying mathematical problem is not that sophisticated. Indeed, I quote Paul Glendinning:

This is not as hard as it sounds. Even when moving, most animals have a good sense of the direction to a given fixed object at any time, and expect to see it on that line. If the aggressor
moves so that at each moment it is on the line between the target and a given fixed point, which could be its initial position, then its relative motion in the eyes of the target is the same as that of the stationary reference point. The only way that the target can know that it is not stationary is to notice the change in size of the aggressor as it approaches. Mizutani, Chahl and Srinivasen [3] extrapolate the lines between the aggressor and the target at several different times and show that to a good approximation these all meet at a point, the fixed reference point or initial condition of the aggressor.

In a development highly offensive to humans’ pride, it seems that we too can be duped by dragonflies and hoverflies: this is confirmed by experimental studies [6] and resolves the mystery of dragonfly flight which puzzled me when I was a child. When you see a dragonfly over a river or meadow, it frequently appears to be hanging in the air motionless, as if it were glued to the sky, and then suddenly jumps at you, whizzing by a few inches from your face — you feel on your skin the air stream from its wings. Only upon reading Glendinning’s paper did I realize that the dragonfly was not motionless — it was approaching me on a reconnaissance flight. If you consider for a second how such an astonishing defeat in a mathematical game against insects could ever happen, it becomes apparent that, although human vision is exceptionally good at detecting even the tiniest relative changes in the position of an object, we are not good at detecting gradual increases in the relative size of an object. You would probably agree with me if you have ever looked at a distant approaching train, seen as a spot of light at the vanishing point of the rail tracks. It is very hard to judge whether the train is stationary or is approaching the platform. It is even harder in modern Britain, where it has become a non-trivial proposition.

Humans use similar strategies, sometimes developed individually, using trial and error at a semi-conscious level, or perhaps learned as part of professional training. For example, baseball players apparently catch high balls by running along such paths on the field — and with varying speed — that the ball (which they keep permanently in view) is perceived as hanging motionless in the sky.

Another example comes from seafaring practice, as a criterion used in sailing to detect boats on a collision course. I again quote Glendinning:

If a boat appears to be stationary with respect to some distant reference point or has the same compass bearing from your boat over a period of time then it is on a collision course with you [7].

This is equivalent to active motion camouflage with the reference point at infinity.

Notice: geometric infinity appeared on the scene. Of course, from a geometric perspective, the reference point at infinity is the same as the vanishing point. Unlike the verbal infinity of counting, it requires much longer and more arduous training.

A classical problem on collisions from Littlewood’s Miscellany [8]:

4 ships A, B, C, D are sailing in fog with constant and different speeds and constant and different courses. The five pairs A and B, B and C, C and A, B and D, C and D have each had near collisions; call them ‘collisions’. Most people find unexpected the mathematical consequence that A and D necessarily ‘collide’.

Prove that! The problem can be classified as belonging to projective geometry, a mathematical discipline (and the class of mathematical structures) with historic origins in the study of geometric perspective (and vanishing points at infinity).


[1] P. Glendinning, , View from the Pennines: Non-trivial pursuits, Mathematics Today, 43 no. 4 (August 2003) 118-120.

[2] P. Glendinning, The mathematics of motion camouflage, Proc. Roy. Soc. (London) Series B 271 (2004) 477-481.

[3] A. Mizutani, J. S. Chahl and M. V. Srinivasan, Motion camouflage in dragonflies, Nature 423 (2003) 604.

[4] A. J. Anderson and P. W. McOwan, Model of a predatory stealth behaviour camouflaging motion, Proc.
Roy. Soc. (London) B 270 (2003) 189–195.

[5] M. V. Srinivasan and M. Davey, Strategies for active camouflage of motion. Proc. Roy. Soc. (London) B 259 (1995) 19–25.

[6] A. J. Anderson and P. W. McOwan, Humans deceived by predatory stealth strategy camouflaging motion. Proc. Roy. Soc. (London) B (Suppl.) Biology Letters 03b10042.S1–03b10042.S3.

[7] Cal Sailing Club, Introductory Handbook for Sailing Boats.

[8] J. E. Littlewood, A Mathematician’s Miscellany, Methuen & Co. Ltd., London, 1953.


  1. Very interesting! I worked for a short time with biology professor Robert Olberg, who studies these problems from the dragonfly’s end–what happens in the dragonfly’s brain when it detects and captures prey.

  2. по-моему, в задаче с лодками надо еще сказать, что курсы не только разные, но даже не параллельные, иначе можно построить пример с параллельными курсами A и D, удовлетворяющий условию задачи.

  3. I suspect that “constant and different courses” already mean that they are not parallel.

  4. Maybe.
    My english is not so good to grasp it.

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