When is a line not a line?

Conventional wisdom says that the shortest distance between two points is a straight line.  But the subtleties inherent in that statement may not be immediately obvious.  So here’s a question…   which of the three paths below is the shortest route from point A to point B?

The answer (like a lot of optical illusions, although this certainly isn’t one) is that they are all the same!  But Graph 3 is clearly a straighter line…   or is it?  In fact, it isn’t a straight line at all, but made up of orthogonal segments just like the other two graphs, only the segments are smaller.  And therein lies the problem.

Consider the question this way… The traveler needs to get from point A on the grid to point B, and the one rule is that he can only travel on the grid.  He can choose any path he likes, but he’s at least smart enough to know that every move should always get him closer to point B (either right or up) and not further away (down or left) because that would certainly be counter-productive. 

We can easily calculate the distance of each segment (right or up) and the total number of segments to get the total distance traveled:

	Graph 1	Graph 2	Graph 3
# x & y grid segments	1	4	16
Grid square size	1/1 = 1 unit/seg	1/4 = 0.25 units/seg	1/16 = 0.0625 units/seg
Total Distance (x + y)	1*1 + 1*1 = 2 units	4*0.25 + 4*0.25 = 2 units	16*0.0625 + 16*0.0625 = 2 units

It seems that no matter how many turns our traveler makes, he always travels 2 units (1 unit right, plus 1 unit up).  In fact, if confined to on-the-grid motion, no matter how fine the grid is, and no matter how close he comes to what looks like a straight line, he has to travel a minimum of 1 unit to the right and 1 unit up to get from point A to point B, a total distance of 2 units.  That is the minimum total distance if he takes any one of the most efficient grid routes possible.  This holds true even if the grid had a billion-billion (or more) divisions between A and B on each axis.  So what might look like a near-perfect line actually isn’t any shorter than taking the “long way around”.

So is the “conventional wisdom” true or have we been befuddled into believing something totally irrational?  Actually I could say both, because the conventional wisdom IS true, and strangely, it depends on something that is “totally irrational” – irrational numbers!

In a sense, irrational numbers are what allows the traveler in the real world to take the short-cut, going “off-the-grid”, and enabling him to turn his body 45° and walk directly toward his target.  That special number √2 gives the traveler the magical capability to chop off about 30% of his total distance.  Some of you familiar with calculus might be tempted to say that the stair steps will turn into a line when the number of segments (n) on each axis approaches infinity, but that isn’t the problem in this case.  The problem is that the traveler is confined to movement in the direction of the axes.  That is to say, even if you were to calculate the distance for lim n→∞ , the answer would still be 2. 

Here’s a thought-provoking idea, though…  there is a concept in physics that space-time itself might be discrete – that is, there is a minimum distance beyond which the concept of distance measurement becomes meaningless.  According to the idea, it occurs at scales of the Planck Length (a concept related to the Planck Scale), the smallest distance that can have any physical meaning in our universe.  I’m sure there are hordes of physicists out there who will cringe at what I’m about to say and provide perfectly logical reasons why I’m way off the mark (hey remember, I’m not a real mathematician or physicist, and this blog is for high-school math and physics students), but here goes…   To me, the concept of being discrete implies that there has to be some kind of grid.  If this concept is true, then what does that mean in the context of the discussion above?  Is there really such thing as a straight line in our universe at all?  I suppose it depends on how you define a “straight line” at such small scales.