Math Puzzle - Spider Crawl

[jyl]

Here is a question someone posed to me. I do not know the answer, but thought I'd give you guys equal opportunity to be frustrated.

There is a square, centered on the origin. At each corner of the square is a spider. The spiders are "D" distance apart. Each spider will always crawl directly toward the next spider, "next" meaning counterclockwise, wherever that next spider is located at that instant. All spiders crawl at the same speed "S", and all start crawling at the same time. Will any spider ever reach the origin? If yes, how far will he crawl to get there?

I am told there is a messy way to solve this (calculus), and at least two simple/elegant ways.

[Tyson Schmidt]

Instinct tells me they move in a spiral fashion and all meet in the center.

[jyl]

Is it asymptotic (converge on origin but never get there) is the dilemma, if not how then far do they travel.

[krystar]

create wormholes between spider A to B, B to C, C to D and D to origin.
all spiders go towards the origin. done

imagination and no limits problem solving. aint it grand

[jyl]

Here's what I have so far.

Since all spiders move identically, they remain in a square formation. The square rotates counter clock wise and shrinks in size, always centered at origin. Since each spider's direction of crawl is toward the next vertex of the square (the next spider), the direction of crawl is 45 degrees from the line between spider and origin. So now you can ignore the multiple spiders in a rotating square, just imagine a single spider crawling in a direction 45 degrees from the direction to the origin, which will be a spiral arc. I started to set up the equation for the spider's position, but using cartesian coordinates it got messy and ugly.

I always figure that when a math problem gets messy/ugly, you're most likely doing something the wrong or hard way. I'm told the simple method is to do this using polar coordinates, and the solution is something elegant involving logarithms. I haven't gotten that far yet.