There are also many manifestations of the Golden Ratio in Geometry and Trigonometry such as (but not limited to) the golden rectangle, the golden triangle, the golden spiral (a means of approximating the logarithmic spiral) and, as was briefly discussed above and loaded with the Golden Ratio, the pentagon and pentagram.
The golden rectangle is a rectangle constructed such that the ratios of width (w) to length (l) are: w/l=l/(w+l). This rectangle is supposedly the most universally "perfect" and visually appealing rectangle possible (Posamentier, 2007). To learn more about the golden rectangle including how to construct it, use the applet here and click on "Golden Rectangle". Here is an activity plan and a task sheet to go along with the applet.
There have been a few studies carried out to try to determine if people really do prefer the Golden Ratio to others. There were similar studies carried out by German psychologist Gustav Fechner in 1876 followed by Witmar in 1894, Lalo in 1908 and Thorndike in 1917 in which they surveyed individuals and had them choose the rectangle they found the most visually appealing. The results of these studies suggested that the golden rectangle is indeed preferred to the others offered, even the square. There are theories about why people find things in the Golden Ratio most appealing, having to do with the way our eyes and minds take in things we encounter. Our eyes want to be able to take in a shape without too much of a process (Huntley, 1970).
Doubts have since been cast on these studies by later ones such as those carried out by H.R. Schiffman of Rutgers University in 1966 who asked subjects to draw the most pleasing rectangles. His results suggested a ratio of about 1.9. Godkewitsch of the University of Toronto published in 1974 results of a study he did in which the golden rectangle was intermixed with different ranges of sizes of rectangles and his results seemed to indicate that the preference for a golden rectangle had more to do with where it was located in association with other rectangles. A matched pairs study carried out by British psychologist Chris McManus published in 1980 concluded that there was moderate evidence that we do prefer the golden rectangle, but the exact ratio of preference is harder to nail down (Livio, 2003)
The golden triangle is one in which the ratio of the side to the base = Φ. It is constructed by beginning with an isosceles triangle ΔABC whose vertex angle A is 36 degrees. Bisect angle B with segment AD (angle bisector). The new triangle ΔBCD is similar to ΔABC since they are both isosceles and have congruent vertex angles. If we let AD = x then (since our triangles are isosceles) we have BC = BD = AD = x and we get the equation 1/x = x/(1-x). This, again, yields the Golden Ratio (Posamentier, 2007).
The golden rectangle and in fact also the golden triangle can be used to approximate the logarithmic spiral. Here we will concentrate on the rectangle. To see how this is constructed look at the applet here scroll down and click on "Golden Spiral." This is the spiral that is found in nature in many ways from small to galaxy sized.