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?

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

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

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 ??

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.

The golden ratio, often represented by the Greek letter φ (phi), is an irrational number that holds a special mathematical property. It is defined as the ratio of two numbers, where the ratio of the sum of both numbers to the larger number is equal to the ratio of the larger number to the smaller number. Mathematically, it can be expressed as φ = (1 + √5) / 2.

To understand why the golden ratio is irrational, we need to consider its decimal representation. If a number is rational, it can be expressed as a fraction of two integers. However, irrational numbers cannot be represented in such a way and have non-repeating, non-terminating decimal expansions.

To prove that the golden ratio is irrational, we can use proof by contradiction. Suppose the golden ratio were rational. Then, it could be expressed as a fraction a/b, where a and b are integers with no common factors other than 1.

By squaring both sides of the equation φ = (1 + √5) / 2 and rearranging the terms, we get the equation φ^2 - φ - 1 = 0. If we substitute φ = a/b into this equation, we get a quadratic equation with integer coefficients and find that φ satisfies this equation.

However, the discriminant of this quadratic equation, √(5^2 - 4 * 1 * (-1)), which simplifies to √5, is irrational. Since the discriminant is √5, we reach a contradiction because the fraction a/b should possess the same irrationality. Thus, we conclude that the golden ratio must be irrational.

Now, what does this tell us about Penrose tilings? Penrose tilings are a special type of non-periodic tiling discovered by mathematician Roger Penrose. These tilings possess unique mathematical properties related to the golden ratio.

In Penrose tilings, the shapes that form the tiles are based on a set of specific polygons with angles that are related to the golden ratio. These shapes can be combined in various ways to cover a plane without any regular repetition, resulting in intricate and aesthetically appealing patterns.

The use of the golden ratio in Penrose tilings suggests a deeper connection between mathematics and art. It highlights the idea that mathematical concepts, such as irrational numbers, can inspire and be reflected in artistic creations. The presence of the golden ratio in Penrose tilings contributes to their visual appeal and mathematical complexity.