Listen to your teacher, kids. Pay attention in math class, or you won’t be able to live your dreams, especially if they include the ability to depict the golden ratio at will. Your mileage may vary.
Anyhow, I’m a huge fan of phi. The Golden Ratio. Just love the damned thing. Always keeps showing up, too.
Start with a square. For ease of calculation, assign a value of 2 units to the length of each side. Draw a vertical line, halving this square. Using a compass (or thumb and index fingers), determine the distance from the midpoint on the bottom side of your original square (where the vertical line you just drew intersects with it) to the top right corner of your original square. If you were to open up a small can of Pythagoras, you’ll note that that distance equals the square root of five.
Anyhow, swing an arc centered on the square bottom halfway point from that upper right hand corner in a clockwise direction until it intersects with a horizontal line that is made by extending the bottom of the square to the right. From that point of intersection, draw a vertical line until it intersects with another horizontal line, one that is made by extending the top of the square to the right. Congratulations. You have constructed a golden rectangle. Draw a square within the smaller rectangle that you just created that is adjacent to your original square. The remaining area is another golden rectangle. Continue creating smaller squares in your newly created smaller golden rectangles, and you are able to form a golden spiral.
Owing to the role of “5″ in phi, a regular pentagon and a regular five-sided star will also yield phi. In the case of the latter, where the arms of a five pointed star intersect, they cut the line into shorter and longer segments whose ratio is phi.
Starting with a regular pentagon, draw an isosceles triangle using the base, with the top point of the pentagon as the third point of the triangle. A bit of geometry will reveal that this triangle is 72-72-36. Draw an angle bisector from one of the 72 degree angles to the opposing side. You’ve just bisected that side into segments whose ratio is phi. And you’ve created another, smaller, 72-72-36 triangle. Lather, rinse, repeat: serially creating these smaller triangular subdivisions will similarly determine a golden spiral.
One plus the square root of five, divided by two. I just love that crazy bastard, I really do.