27 June 2010

Taking the Pythagorean Route

A few years ago, I was walking with someone, and he said that he's a very lazy person. For example, he travels light and does not carry much around with him. He just carried around a pencil, which was quite light, considering that he was a college student, and did not carry books, computers, paper, water, or lunch around with him. (This was during the days before mobile phones with browsers was common.)

One way that he liked to walk, he said, was that he liked to take the Pythagorean route. This was based on Pythagoras' Theorem, which stated that moving in a diagonal direction covered a shorter distance, than moving to the same point covered by the sum of the horizontal and vertical distances of a right-angled triangle. That certainly is a much more efficient way of walking. Of course, sometimes there are certain impediments that block people, especially in cities, which are structured mainly as grids and as rectangles. Still, the general idea that to move in a diagonal direction to where you would like, especially for long distances, can save some time, especially if there are not too many impediments by taking such a diagonal route.


After I wrote this, I just thought that taking the Pythagorean route was essentially walking in the most direct path to a destination. The reason that it seems that we are walking along a diagonal is because cities and towns are arranged as grids.
I also didn't realize when I 1st wrote this. There's a pun in the title as "route" sounds like "root."
Shortest distance is not always shortest time. Depends what you mean by "efficient." Sometimes, the "Manhattan" route is the best possible, especially when there are big buildings in the way. But the more general case is simply where the terrain alters, e.g., hills, marsh, etc. The quickest path is then no longer obvious; it's a minimization problem.
I note that the Pythagorean route only deals with one plane. Cities may be set up on grids, but tunnels and skyways represent another opportunity to reach one's destination using a more Pythagorean route.
U can generalize this to the concept of keeping the closest to the inside of a curve when driving a car, and here it also has significant environmental as well as personal benefits. I once calculated (possibly incorrectly :) that u could gain a roughly 0.5 percent fuel reduction on an "averagely" windy road. Even such a small percentage would be very useful to our Earth at the present time. The only problem is that some national states have laws asking for drivers to keep to the outside lane of the road if not overtaking, which virtually eliminate any fuel/time savings from switching lanes on a multi-lane highway. Maybe time to adapt our laws and ideas to a more pro-environment stance?
Actually both are probably examples of a more general principal?, e.g. "find the shortest route for the required period of travel from A to B anticipating the probable constraints"
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