What are the known methods of drawing a golden rectangle with a ruler and compass?

Question

Googling and Wikipedia, gives only the following construction

A golden rectangle can be constructed with only a straightedge and compass in four simple steps:

1. Draw a square.
2. Draw a line from the midpoint of one side of the square to an opposite corner.
3. Use that line as the radius to draw an arc that defines the height of the rectangle.
4. Complete the golden rectangle.

Question: What are the other known methods of constructing a golden rectangle with a ruler and compass?

Answer 1

HINT.-Drawing $\sqrt5$, what it remains is easy and well known.

Draw a circle of radius $3$ (i.e. of diameter $5+1$). In this circle draw the perpendicular chord cutting the diameter in a point dividing it in a proportion $5$ to $1$. This chord has lenght $2x$ and it is quite known that $x^2=5\cdot1$ so do you have $x=\sqrt5$.

Answer 2

That method is called "exterior division", whereby your starting line segment is extended such that the extension is in golden ratio with the starting segment. There is another method, called "interior division", which focuses on splitting up a given line segment into golden ratio sections. It works as follows:

1. start with a segment $AB$ and draw a perpendicular line from point $A$ or point $B$ with length equal to $AB/2$. Label the endpoint $C$.

2. draw the hypotenuse $AC$ (or $BC$). Draw an arc centred at $C$ of radius $AB/2$. Label its intersection with the hypotenuse $D$.

3. with the compass centred at point $A$ (or point $B$ - whichever one you didn't draw the perpendicular from), draw an arc through point $D$. It will intersect the $AB$ at a point $I$; the resulting sections will be in the golden ratio with each other and the larger section will be in the golden ratio with the whole.

Now to address your question directly: aside from each of our methods, I know of no other way to construct objects in the golden ratio. This is because compass-and-straightedge permits only the construction of lines and circles and the square root of any non-square integer can only be constructed in this setup as the intersection between a line and a circle. In other words, no matter what you do, you'll be forced to use a circle in the manner described by your method and mine (which are really just the same, only dressed up differently).