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I believe that would be displacement?

Good thing he doesn't know topology

I remember when I first learned Manhattan distance. What a "fancy" term for something so simple.

Wow... I used to think about stuff like that

allthe time, but I never knew they actually defined such a thing!When I was a kid, I would think... hmmmm, if I bike along a set of streets that form a zig-zag, then maybe it will be faster, cuz it's kinda like a hypotenuse? But then I realized that the zig-zag would still be the same exact distance, and that the shortest street-distance between two points is rarely a straight line. :-(

Then again, like Randall Munroe, I have always worried about the efficiency of my path...

http://xkcd.com/85/

In a discrete space, they are the same distance.

Taxi math.

As long as the roads are square, it doesn't matter. When there are diagonals, though...

But the taxi is being paid by the minute, not by the shortest route.

So, he wants to maximize the distance. Where is the maximin saddle point?

In this case he would actually want to maximize time, that may or may not be taking airport road. This thus entirely depends on the amount of traffic on each route (slow moving traffic jam vs blazing through the side routes).

Yes, I got rushed and mistyped... Max time v. min distance?

I think this may be refering to something a greek philiospher said to a student who was a prince who wanted a shortcut to learning geometry: "In geometry there are no shorcuts even for princes." (or something similar)

The correct response for the lady in this comic is:

"Ok, then take me by the airport road".

My vague high school memory serves me that Pythagoras says otherwise. Even if it was the same distance, it would not be faster. Taking airport road would be one direction of acceleration/inertia/braking. Having to accelerate then brake to turn 90 degrees then accelerate again then brake would inevitably take longer to travel.

it's pythagorean theorem, isn't it? it's the same distance, not the faster choice....

It's called the taxicab norm. The euclidian norm takes the root of the sum of each coordinate squared (i.e. the magnitude of a vector) whereas the taxicab norm takes the absolute value of each coordinate.

It is not the same distance. It is, however, the same final displacement.

The taxi driver is getting crazy because we all know that for all triangles that exist, the sum of any pair of sides will be longer than the one you left out. In this situation we can use algebra and the Pythagorean Theorem to find out that airport road is shorter by 2ab where a and b are the lengths of Main Street and first avenue. LOL. Hats off to the passenger!