Step 1. Begin with a square sheet of paper and fold in half as shown along EF.

We take the side length of the square paper to be one unit, so rectangle AEFD has width 1/2 and length 1.

Step 2. Then fold along the line AF to create a right triangle AEF.

The lengths of the sides of triangle AEF are AE = 1/2, EF = 1, and FA = (1/2) sqrt 5.

Step 3. Now fold the angle bisector of angle BAF by folding AF to the edge AB. This fold intersects EF at G and the edge BC at H.

Since AH is the angle bisector, the ratio FG/GE equals the ratio of the legs AF/AE = sqrt 5. If GE = x, then FG = x sqrt 5.

Since FG + GE = 1, GE = x = 1/(1+sqrt 5).

Step 4. Now fold HI perpendicular to the edge. Triangles AEG and ABH are similar, with scaling ratio 2, so BH = 2/(1+sqrt 5).

But this is 1/g, where g is the golden ratio.

Thus the rectangle ABHI has width 1/g and length 1, It is similar to a rectangle of width 1 and length g, so it is a golden rectangle.