| To construct a golden rectangle start with a double square i.e. a 1 by 2 rectangle. |
| The diagonal of this rectangle will have a length which is the square root of five times the short side of the rectangle. (Link to proof.) (pythagoras theorem) |
| Next if you draw a circle with a radius equal to the side of the square centred on the corner of the double square, you can extend the diagonal out to the circumference of the circle. This makes a line which is the square root of five plus one times the side of the square. |
| The next step would be to draw a circle with a pair of compasses centred on one end of the red line. |
| Finally to make the width of the golden rectangle extend the bottom line of your double square out to the circumference of the circle. For the height use the long side of the double square. Add the other two lines at right angles to the loose ends. This gives a rectangle whose width is the square root of five plus one times the length of the side of the small square and whose height is twice the length of the side of the small square. Which makes the width the square root of five plus one divided by two times the height. |
| Alternatively you can make a smaller golden rectangle by starting from this stage and drawing a circle centred on the opposite corner of the double square whose edge touches but does not cut the first circle. |
| Extend the width of the rectangle to the circumference of the last circle as shown by the yellow lines in the picture. This is a golden rectangle as two into root five minus one is the same as root five plus one into two. There are infinitely many other ways to do it. Sacred Geometry |
