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2012
unknownj
Sometimes it's good fun to think in maths..

As people who stalk me on Google Earth are aware, my office is donut shaped. There are two paths around the office, one following the outside edge, one the inside edge. Along the way, there are paths connecting those two paths, radiating from the centre of the donut. Naturally, it's important to get around the office using the optimal route.

Now usually, the points I want to get to are both on the outer path. So the question is, do I go in towards the middle, take the (shorter) inner path around, and then back out again when I reach my destination? Or do I go right around the outside..

In the case where, imagining the office as a clock-face, I want to go from 12 to 3, then there's a simple solution - the outside path is best. Assuming that the outer path has radius 1, and the inner path has a variable radius 0-1, there are two trivial cases:

Trivial case 1: The radius of the inner path is zero. Ergo, my route is two sides of a square, as I go to the centre, and then out again. Total travel time: 2

Trivial case 2: The radius of the inner path is one. Ergo, my route is one quarter the circumference, being Pi/2.

Picturing it, you can already see how the latter is better - why go along two sides of a square when you can cut across with a curve? And, obviously, Pi/2 is less than 2. Logically, it can be deduced that for journeys making up a quarter (or less) of the circumference of the donut, it's always optimal to stick to the outside edge.

But what if I wanted to go halfway around the donut? Using the same logic, I have two more trivial cases:

T1: Radius of the inner path is zero - my journey is therefore 1 to the centre, and 1 out to the other side. Total travel time: 2

T2: Radius of the inner path is one. My route is half the circumference, being Pi.

In this case, the first instance gives the optimal route - therefore a route that uses an inner path will always be better than using the outer path. There is, of course, a point at which they are both equal. Anything with a smaller angle will mean the outer path is optimal, a larger angle means the inner path is optimal.

Using these trivial cases, my inner path journey length between two points will always be two. So it's the angle between two points on the outer path at which the length is two, for a circle of radius 1.

The total circumference is equal to 2Pi. Therefore the fraction of that circumference that is equal to 2 is 1/Pi, or two radians. Going back to the clock analogy, any journey that takes me from 12 to almost-4 or further is better done on the inner path, otherwise the outer path is best.

It doesn't actually matter where the inner path actually is - whether it is a circle of radius 0, radius 1, or somewhere in between (as it actually is) - the relationship still holds true. So given an angle of two radians, as long as every step you take is either directly towards the centre, or in a circle around the centre, and provided you never move away from your target, the distance will always be the same. It took a while for me to fully get my head around it, but once you do, it's neat..

And this is what I think about when I'm walking around the office to get to the drinks machine...