Triangle Inequality
As my life now has been boring beyond belief, this journal has been left to collect dust and spider-webs. Time to inject a bit of life here to remind everyone that I'm still well and alive :D but since there's nothing particularly interesting to blog about from my life, I decided to write about something that I've pondered over. It's not exactly a question because I've made some sort of conclusions of my own already, but just to share it and see if anyone has any responses (other than what I've already said... like how my life is indeed very boring).
First, I must brikng your attention to this mathematical concept known as Triangle Inequality, which in essence states that for a triangle, any side will be shorter than the sum of the remaining two sides.
Using this right-angled triangle ABC as an example, the hypothenuse AB is shorter than the sum of AC and BC. It's the same for other 2 sides (AC < AB + BC and BC < AB + AC) as well, but for our discussion's sake let's just focus on AB.
Applying this to a real-life scenario, it means that if I'm standing at B and I need to get to A, it would be faster for me to walk along BA than to walk BC then CA (assuming, of course, I walk with the same constant speed and there are no obstructions or other factors affecting my journey).
But in real life, we seldom have the benefit of a completely cleared "shortest path" as assumed previously. More often than not, there are existing structures that block our path and force us to walk around them. Well, not everyone can be as lucky as Mr Bean...
Suppose we're supposed to travel through a city, where excellent city planning means that all roads are aligned at right angles.
The shortest route would not be possible as you can't walk through buildings (and walls), and walking diagonally on public roads is highly discouraged. We can only walk along the proper pedestrian walkways, which would make the "shorest diagonal route" end up something like this...
And you may have realised that the route BA is now exactly the same length as the BC-then-CA route. (In fact, in this case BA goes through more traffic junctions which may actually further increase the time required to travel via the route. But we shall not go into that, and simply assume that we are so lucky that all traffic lights are in our favour, or that the roads in the middle are all minor roads where the junctions are not traffic light-controlled, and there are no cars around so we can just walk non-stop.)
Which kinda contradicts the initial idea that walking directly towards your target endpoint is the fastest way.
Therefore whenever exploring unfamiliar areas using only a map, I tend to choose the seemingly longer way (BC-CA) especially if it's along main roads, as opposed to the zig-zag BA, which may entail weaving through minor roads, shophouses or crowded building walkways. And of course, there will be people who object to my decision, or would agree that my route is easier but still significantly longer. Hmm, unless there's something else to this problem that I've not noticed?
First, I must brikng your attention to this mathematical concept known as Triangle Inequality, which in essence states that for a triangle, any side will be shorter than the sum of the remaining two sides.
Using this right-angled triangle ABC as an example, the hypothenuse AB is shorter than the sum of AC and BC. It's the same for other 2 sides (AC < AB + BC and BC < AB + AC) as well, but for our discussion's sake let's just focus on AB.
Applying this to a real-life scenario, it means that if I'm standing at B and I need to get to A, it would be faster for me to walk along BA than to walk BC then CA (assuming, of course, I walk with the same constant speed and there are no obstructions or other factors affecting my journey).
But in real life, we seldom have the benefit of a completely cleared "shortest path" as assumed previously. More often than not, there are existing structures that block our path and force us to walk around them. Well, not everyone can be as lucky as Mr Bean...
Click to viewSuppose we're supposed to travel through a city, where excellent city planning means that all roads are aligned at right angles.
The shortest route would not be possible as you can't walk through buildings (and walls), and walking diagonally on public roads is highly discouraged. We can only walk along the proper pedestrian walkways, which would make the "shorest diagonal route" end up something like this...
And you may have realised that the route BA is now exactly the same length as the BC-then-CA route. (In fact, in this case BA goes through more traffic junctions which may actually further increase the time required to travel via the route. But we shall not go into that, and simply assume that we are so lucky that all traffic lights are in our favour, or that the roads in the middle are all minor roads where the junctions are not traffic light-controlled, and there are no cars around so we can just walk non-stop.)
Which kinda contradicts the initial idea that walking directly towards your target endpoint is the fastest way.
Therefore whenever exploring unfamiliar areas using only a map, I tend to choose the seemingly longer way (BC-CA) especially if it's along main roads, as opposed to the zig-zag BA, which may entail weaving through minor roads, shophouses or crowded building walkways. And of course, there will be people who object to my decision, or would agree that my route is easier but still significantly longer. Hmm, unless there's something else to this problem that I've not noticed?