Apr 30, 2013
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.