In a golden triangle (an isosceles triangle where the ratio of the side to the base equals the golden ratio , and whose angles measure 36°-72°-72°), removing a golden gnomon (an isosceles triangle whose side lengths are in the golden ratio relative to the longest side of the original triangle) results in another golden triangle. These steps can be repeated over and over, decomposing the original triangle into an infinite sequence of similar golden triangles, by fixing a direction and determining the intersection of the base angle bisector with the opposite side at each step. By drawing circular arcs with an angular width equal to the vertex angle of the gnomon, 108°, you obtain a golden logarithmic spiral.

This applet demonstrates the Fibonacci Squares and the Fibonacci Spiral without going through all construction steps. It is a variation of the constructing applet. http://www.geogebratube.org/material/show/id/6095 Click the Play button.

To do a hands-on construction, open http://www.geogebratube.org/material/show/id/6095 and follow the steps.