And it just occurred to me, maybe because of Christmas, that when I would wrap a present with a ribbon, I would instinctively wrap the ribbon in that spiral route and very vaguely remember someone mentioning that such was the way to save on ribbon.
That’s clever the shortest distance between two points is a “straight” line. 
I have been wondering if this spider’s choices were in compliance with James’ AO - specifically with his PHT - “Perception of Hope and Threat”.
So the spider sees the fly - gets a positive perception to pursue - but how to get there?
I guess it isn’t impossible that the spider was born with a deep understanding of geometric maths - but would that be necessary?
The spider would see that he only had certain beginning options - the 4 surrounding walls - and precisely where to meet them at their edges. How would a small spider decide that?
The spider could see the straight distance to the fly - and could see that it wasn’t an option (no perception of hope) so he has to decide which wall and where to meet the wall.
The spider could see that the fly was lower than him - so “lower” should be perceived as more hopeful - and that leaves 3 choices - the floor or either side wall at a lower position.
The spider should see that the floor edge is farther away from his preferred straight path to the fly (direct line of sight). By heading to a side wall and a little lower - he could perceive a hope of getting a little closer faster (than the floor option). The walls are closer and he can get there sooner before having to make another choice. By proceeding slightly downward - he slightly alleviates the need downward. That one decision could be made without regard to any future path decisions.
Once at the edge of either wall he perceives the line of sight distance to the fly and has only 3 options. Why he chose which wall would have to be something related to the specifics of the spider itself - orientation - best eye vision - last thought - a sore toe - whatever might influence the decision of which of two identical paths to take.
The line of sight path would seem most closely duplicated by again - a slightly downward move while advancing along that wall. That gives him 2 options - far edge and bottom edge. Again he could perceive a shorter path toward the bottom while keeping the line of sight in focus. And again he could make this decision without considering what path to take next - only that it immediately seems closer to the objective and along the line of sight.
So now he is at the bottom edge about half way across the room. And at that point he can only see one edge to pursue - the edge just beneath the fly.
And again by keeping the sight he wouldn’t see the best option to proceed to the point directly under the fly - it is a longer distance to travel before the next decision and veers further from the direct line of sight (although not by much). So he proceeds - still following his perception - to a bottom edge point below the fly but closer to the edge he just left.
At that point his perception is a straight line of sight path - so he takes it.
Ok - so what this is telling me is that the spider’s line of sight (best perception of hope) is what is guiding his decisions - not his university maths professor. And that technique of following his best line of sight perception happens (in this case) to be the best mathematical choice as well.
Sometime the stupid are smart enough. 
An interesting experiment might be to make the ceiling angle slightly downward toward the fly’s location and see if spiders detect that and have the foresight to change their choices - are they looking ahead to the next step - or just making each choice based on the immediate surroundings.
Of course today he would probably just have Amazon deliver it into his little clutches. 