## [數學][文章]Fibonacci Numbers and Nature ### [數學][文章]Fibonacci Numbers and Nature

Fibonacci Numbers and Nature

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html yll SCTT yll scsnake

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=1·618... but the interesting point that we see on this graph is that the Fibonacci points are the closest points to the Phi line

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Cycles in the Fibonacci numbers
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Here are some patterns people have already noticed:

There is a cycle in the units column - the cycle of units digits (0,1,1,2,3,5,8,13,21,34,55,...) repeats from n=60 and again every 60 values.

There is also a cycle in the last two digits, repeating (00, 01, 01, 02, 03, 05, 08, 13, ...) from n=300 with a cycle of length 300.

For the last three digits, the cycle length is 1,500
for the last four digits,the cycle length is 15,000 and
for the last five digits the cycle length is 150,000

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Initial digit frequence in the Fibonacci numbers
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Digit: 1 2 3 4 5 6 7 8 9
Frequency: 30 18 13 9 8 6 5 7 4 100 values
Percent: 30 18 13 9 8 6 5 7 4
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Digit: 1 2 3 4 5 6 7 8 9
Percentage: 30 18 13 10 8 7 6 5 5

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