Question

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?

Answer 1

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

Answer 2

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

Answer 3

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 ??

Answer 4

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.