Dogs Chasing One Another Part I: The Solution

 Over the summer I spent a week or so playing with a problem Steven Strogatz proposed on an episode of the 3B1B podcast, which goes as follows: 4 dogs sit on the corners of a square. At some instant, each dog chases the dog in the next clockwise corner, each with equal speed. The dogs chase each other in a spiral and eventually meet in the centre. How far does each dog travel before the collision? 

The value of the problem extends far beyond its solution and it has expanded to fill 3 posts. The first (which you are reading) shares some problems with similar solutions before giving the deceptively simple solution to the title problem. The second compares 2 tasks, a programming simulation of the problem and an analytic approach to explicitly finding the paths taken by the dogs. The third will hopefully be more philosophical, comparing the skills of proof writing and coding.

Before solving this it is helpful to think of another problem with a similar solution, which I encountered in Pólya's 'How To Solve It'. Imagine 2 ships sailing the high seas, of which we are given starting positions and constant velocities. We're asked how we might find the minimum distance between the ships. Of course an energetic teenager might parametrise both paths by time, find the pythagorean distance between the ships and fix the derivative of it equal to zero to find the time of closest approach, perhaps being so precocious as to neglect the square root. But my sagacity compels me simplify this problem with 2 arbitrary moving parts to a problem with only one, by attaching myself to one of the moving parts. In this case it pays to be a captain on a ship with the egocentric view that the world literally revolves around you, rather than having the 'objective' birds eye view: working in the reference frame of one of the ships, I can view my ship as stationary. Then the other ship is moving with velocity "theirs minus mine" and the problem reduces to finding the minimal distance between a point and a line, which occurs when the joining line is perpendicular.

Something similar occurs when studying Kelvin-Helmholtz instabilities: the situation where 2 layers of fluid are moving at different velocities. We find, using the Bernoulli principle at the interface of the fluids, that the fluids must have equal speeds in opposite directions: this proves no problem, as we can always swap to a reference frame moving at the average of the two velocities. The takeaway here is that, although you might have a favourite reference frame, the universe does not.

I hope it's become apparent that the answer will involve reference frames: if we imagine life from the perspective of dog 2, dog 1 is at all times taking the most direct path towards us. Regardless of the fact that we are moving relative to the ground, relative to us dog 1 is taking a straight path. Therefore it travels the initial distance between us, i.e. the side length of the square. [[aside: this argument only works because if dog 1's motion is radial then dog 2's is purely tangential. This doesn't work for other polygons]] Steven Strogatz gives his solution in The Guardian by saying 'Imagine a dog has a GoPro on its head. The footage of its pursuit of its neighbouring dog along the spiral path would be exactly the same, and take the same time as, the footage of the dog if all it did was run directly along the edge of the square to a stationary dog.' 

In the setting of general relativity, the tangent vector to dog 1's path is constant in the reference frame of dog 2 so obeys the geodesic equation.

There seem to be 2 contradictory moral statements here: the world revolves around me. In my reference frame, I stay completely still and I force the world to move under my feet. But such is true for everyone else: it's always helpful to view things from the perspective of others.

I hope you found this solution as pleasing as I did, but know that this was far from my first attempt. Initially I tried to crack this walnut with a sledgehammer, as shown in the next post.