Finally completed the game the legend of Zelda : twilight princess after 60 hours of gameplay.
Fantastic game even though its a gamecube port. rating : 9.5/10
I realized it has been a very long time since i did some proper maths... Today lets talk about one of the most important aspects of modern mathematics, and indeed all of mathematics, isomorphism.
The concept of isomorphism is simple. When we add up 30 oranges with 20 oranges, we do not really add up all the oranges, rather we assign the number 20 to the 20 oranges and the number 30 to the 30 oranges and add up 20 and 30 to get 50 and thus our result would be 50 oranges.
It applies internally in mathematics as well.
consider the pair of numbers (x,y)
where x is the integers (mod 2), and
where y is the integers (mod 3).
if we list out all (x,y), they will look like this.
- (0,0)
- (1,1)
- (0,2)
- (1,0)
- (0,1)
- (1,2) ............. and so on.
notice that the elements repeat themselves after the sixth one.
If we assign each element to a number like this, where the right hand side is the integers (mod 6),
- (0,0) -> 0
- (1,1) -> 1
- (0,2) -> 2
- (1,0) -> 3
- (0,1) -> 4
- (1,2) -> 5
note that (1,2) is assigned to 5 and (1,0) is assigned to 3. If we know that the two columms are isomorphic, we can just add 3 to 5 to get 8, which is simply 2 under mod 6. and the element assigned to 2 is just the answer, (0,2).
Using the concept of isomorphism, areas of mathematics that are new can be "mapped" into areas of mathematics that are well known and mathematicians can use all the developed tools of this familiar area of mathematics to attack the new ones.
Think about geometry and algebra, in algebra it is a fairly tedious thing to solve a cubic equation, however if we port the problem over to geometry, we can find the roots simply by plotting the graph of the cubic and measuring the distance from the origin. Admittedly when we draw it the answer is never perfect, but in the ideal world of mathematics, that distance is exactly the root.
