# Q.E.D.

40 41 the golden ratIo natures favourite number What does a rectangle look like that is not too slim and not too wide, a rectangle that looks just right For many artists and scientists this age old beauty contest has a clear winner the socalled golden rectangle below, left whose ratio of long to short side equals the golden ratio f Phi, of diagonal to side in a regular pentagon opposite, top. The golden ratio is present in many of natures designs such as leaf arrangements and spiral galaxies. For example, if we take away a square from a golden rectangle below, we find we are left with another golden rectangle since f 1f 1 opposite, top. Repeating this process yields a spiral of squares that hugs many naturally occurring spirals. Combining three golden rectangles at right angles center, opposite, their twelve corners become the corners of an icosahedron. To prove this, we only have to check that all the triangles in the middle picture are equilateral, or equivalently, that the essentially two different edges of these triangles all have equal length. This paves the way to a beautiful construction of an icosahedron from an octahedron lower, opposite where the twelve corners of the former divide the twelve edges of the latter in the golden ratio.