Phi and the Fibonacci numbers

On the Fibonacci and Nature page we saw a graph which showed that the ratio of successive Fibonacci numbers gets closer and closer to Phi.

Here is the connection the other way round, where we can discover the Fibonacci numbers arising from the number Phi.

The graph on the right shows a line whose gradient is Phi, that is the line y = Phi x = 16180339.. x

Since Phi is not the ratio of any two integers, the graph will never go through any points of the form (i,j) where i and j are whole numbers - apart from one trivial exception - can you spot it? So we can ask

What are the nearest integer-coordinate points to the Phi line? Let's start at the origin and work up the line. The first is (0,0) of course, so here ARE two integers i=0 and j=0 making the point (i,j) exactly on the line! In fact ANY line y=kx will go through the origin, so that is why we will ignore this point as a "trivial exception" (as mathematicians like to put it). The next point close to the line looks like (0,1) although (1,2) is nearer still. The next nearest seems even closer: (2,3) and (3,5) even closer again. So far our sequence of "integer coordinate points close to the Phi line" is as follows: (0,1), (1,2), (2,3), (3,5) What is the next closest point? and the next? Surprised? The coordinates are successive Fibonacci numbers!

Let's call these the Fibonacci points. Notice that the ratio y/x for each Fibonacci point (x,y) gets closer and closer to Phi=1618... but the interesting point that we see on this graph is that

the Fibonacci points are the closest points to the Phi line.

1-61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More..

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