GOLDEN POLYGONS

The golden ratio is the number (1 + Ö5/)2. A golden rectangle is a rectangle with width w and length d such that w/d or d/w is equal to (1+ Ö5)/2.
As pointed by Neil, Eric and Geoff (St Peters), the golden rectangle is an aesthetically pleasing shape and has been used by architects and artists to enhance the natural beauty
of their work. However, there are many more “golden shapes” which enhance nature’s beauty, for instance golden hexagons are being used by plastic surgeons to create the “perfect face”.
Below we will show you how to construct some of these shapes.
You will need a pencil, ruler and compass.

 Draw a line M containing two points - A and B. The points A and B are a distance of 1 unit apart. At the point A, draw a Line L perpendicular to M. (Diagram 1) With compass point at A, draw an arc through point B. The point of intersection of the arc with line L, is labelled C. With compass point at C, draw an arc through point A. The point of intersection of the arc with line L is labelled D. With compass point at B, draw an arc through point D. The point of intersection of the arc with line M, is labelled E. (Diagram 2) Note that the right-angled triangle DAB, has side AB of length 1 unit, side AD of length 2 units and so side DB is of length Ö5, and AE is of length 1 + Ö5 units. (Diagram 3) With compass point at E, draw an arc of radius the length of AD, and with the compass point at D draw an arc of radius the length of AE. Label the intersection of these two arcs F. (Diagram 4) Draw a line segment from D to F and a line segment from E to F. Now AEFD is a golden rectangle. (Diagram 5) Return to Diagram 4. Draw two intersecting arcs of radius the length of AE. One with compass point at point A and the other with compass point at point D. Label the point of intersection G. The triangle ADG is a golden triangle; it is an isosceles triangle with the ratio of the longer side (1 + Ö5 units) to the shorter side (2 units) equalling the golden ratio. (Diagram 6). Alternatively you may construct a golden triangle by drawing two arcs of radius length AD, one centred at point A and the other centred at point E. Label the point of intersection of the arcs H. (Diagram 7) Golden Pentagons To construct a golden pentagon we place a copy of the triangle AHE on the longer sides of the triangle AGD. (Diagram 8)