One of my favourite websites:
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html
It would take you weeks to follow all the links and side-interests of this wonderful webpage
It starts with the Fibonacci numbers, which are very closely linked with the Golden Ratio
btw, the "golden ratio" = 1 + √5 : 2
Can someone xplain what the golden ratio is and how we know that it MUST be irrational. What does this tell us about our Penrose tilings?
4 answers
after you find the golden ratio, take its reciprocal on your calculator. Notice the decimal sequence does not change.
No other number has that property
No other number has that property
list the first 20 or so Fibonacci numbers.
From left to right, divide one number by the number before it, noting the result.
What do you notice about the answers ??
From left to right, divide one number by the number before it, noting the result.
What do you notice about the answers ??
take a line segment of length 1 , and divide it into a larger and a smaller segment , so that the ratio of the whole line to the larger = the ratio of the larger segment to the smaller.
Let the larger segment be x
then the smaller is 1 - x
1/x = x/(1-x)
x^2 = 1 - x
x^2 + x - 1 = 0
x = (-1 ± √(1 - 4(1)(-1))/2
= (-1 + √5)/2 , since the segment can't be negative
let's evaluate 1/x
= 1/((-1 + √5)/2)
= 2/(√5 - 1) * (√5+1)/ (√5+1)
= 2(√5+1)/4
= (√5 + 1)/2 <----- the golden ratio.
Let the larger segment be x
then the smaller is 1 - x
1/x = x/(1-x)
x^2 = 1 - x
x^2 + x - 1 = 0
x = (-1 ± √(1 - 4(1)(-1))/2
= (-1 + √5)/2 , since the segment can't be negative
let's evaluate 1/x
= 1/((-1 + √5)/2)
= 2/(√5 - 1) * (√5+1)/ (√5+1)
= 2(√5+1)/4
= (√5 + 1)/2 <----- the golden ratio.
2 answers