To construct the "golden ratio", divide a unit square into two equal rectangles with sides 1 and 1/2.  Add the length of the diagonal in such a rectangle to its short side.  The new length is the "golden ratio"; the add-on rectangle as well as it plus the square are both "golden".

When you add a square along the long side of a golden rectangle, the new rectangle is again golden, and the next one too, and so on forever.  If you draw a quarter circle into each successive square so that each arc begins where the prior one ends, you obtain a "logarithmic" spiral of self- replicating growth.

The golden ratio yields also two golden triangles, with side lengths 1, phi, and phi for the slender one, and 1, 1, and phi for the obtuse one.  These fit alongside each other to reproduce again their identical shapes.  When you connect the pointed ends of each successive obtuse triangle with an arc around its apex, you obtain again a logarithmic spiral.

Adding two obtuse golden triangles with their broad base along the equal sides of the slender one produces the pentagon, a rich source of golden ratios.  Its diagonals are phi times its side length and divide each other in the same phi proportion to form a new pentagon.