Originally posted by joe shmo
Actually, im going to say that the shape of the path for 2 ants from opposite corners is kindof like a cubed root function. I think it is not a spiral.
And im thinking the meet in the center the same time that it would take one ant to walk from one vertex to the next of the square
Just in case my result is vague, here is what i did (not very mathematical)
I drew a square and then decided to move each ant an arbitrary distance (d) from its original position. the direction of any ant at any given time can be found by the direction of a line connecting the position of any 2 ants in sequence at that time. Now there are 4 lines at an angle to each side, next the ants travel equal distances on those lines, you connect those lines and repeat as desired...The intersections of the lines give the path traveled...which is like a curvey X
Actually just put the square on to the cartesian coordinates
thinking that the lines I mentioned are tangent to the path of the ants ( I guess they represent the velocity vectors)
then the slope of one of the lines is ( if the length of the square is b) is:
Let C stand for "change in"
slope = Cy/(b-Cx)
dy/dx = y/(b-x)]
and doing separation of variables and integrating I get
y = +/-e^(b-x) as the path function (with/without the appropriate constants).
ahhh... let me know if my differential equation solving is incorrect, im not entirely sure, I haven't studied them yet.