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With an infinite amount of time:
- start with a square
- join a square to one side
- Join a square along the edge where the two squares join (it has a side length twice the original squares side length)
- join another square to the side where this last added square joins the original shape.
- repeat step 4 (forever)
The final (limiting) rectangle will be a golden rectangle with sides in the ratio of 1 : φ.
The side lengths of the rectangles as created will be in the ratio:
1 : 1 (the initial square)
1 : 2
2 : 3 → 1 : 1.5
3 : 5 → 1 : 1.666...
5 : 8 → 1 : 1.6
8 : 13 → 1 : 1.625
13 : 21 → 1 : 61538....
21 : 34 → 1 : 1.619047....
34 : 55 → 1 : 1.617647....
55 : 89 → 1 : 1.6181818....
89 : 144 → 1.61797752.....
144 : 233 → 1.6180555....
233 : 377 → 1.6180257....
377 : 610 → 1.6180371....
610 : 987 → 1.6180327....
987 : 1597 → 1 : 61803444...
1597 : 2584 → 1 : 1.6180338...
2584 : 4181 → 1 : 1.6180340....
4181 : 6765 → 1 : 1.6180339....
6765 : 10946 → 1 : 1.6180339.....
The golden ratio φ = 1.6180339.....
So if you start with a 1mm square, when you get to a 55mm by 144mm rectangle it is approximately a golden rectangle. By 144 mm by 233 mm it is closer still (and probably as accurate as you can measure).
You may recognise the first column as the Fibonacci sequence - the ratio of each term to the next term gets closer to the Golden Ratio as the sequence proceeds.
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What does a golden rectangle have to do with phi?
phi is incorperated into the golden rectangle, because if you divide the longer side of the golden rectangle by the shorter sid, the answer will be phi.(1.168...)
Where are golden rectangles used in architecture?
The Golden Rectangle was believed to be founded by Pythagoras. The Golden Rectangle was used for many Greek Buildings such as the Parthenon, and the Villa Stein.
In a golden rectangle if the top and bottom lines of the rectangle are each 3 and one half inches how long would the side lines of the rectangle be?
To make it a golden rectangle the sides should be in 1:0.618 ratio. Lets say your width is made of a + b. a and b are in golden ratio. THis gives a + b = 3.5 <---- equ 1 b = .618 a (because they are in golden ratio) substitute to equ 1 1.618a = 3.5 a = 3.5/1.618 = 2.163 b = 1.336 now you can construct your sides with a = 2.163 to have a golden rectangle
What type of quadrilateral is formed when you connect the midpoints of a rectangle?
A rhombus is formed.
Why is the golden ratio golden?
You know the golden rectangle? Well it is in lots of parts of nature. From sea shells to galaxies. It is also a favorite in art and style.
