Want to draw a star?

Get paper, pencil, straight edge (ruler) and a compass

Decide how long you want the arms to be. Make a square (see black square) that length.

Bisect the square and draw a (see violet) diagonal. Use that diagonal to swing an arc down to the line of the arms, and where it intersects, the right arm starts. You have just created a golden section (see below). The width of the neck is in golden proportion to the arms. 

Set the compass to the length of the arms plus the width of the neck at its base (see green line and circle). That will be the distance between each point of the star. 

To find the top point, bisect the neck and swing the arc (green) from the left hand to the bisecting line, and that is the top of the head.

Now draw a line from that head point toward and beyond each foot point. To find the exact point, swing your green arc from each of the hands to meet those lines. 

Fill in the lines and a star is born. At last!

 

Why did I put this here? 

Just for fun. But mostly because it’s a recipe with a story. Like all good recipes, it starts with someone who wants to do something a bit tricky. Here’s my story…

When I was a teenager, I would on occasion, rest from homework and continue a pursuit to make (“construct”) a five-pointed star using the classic tools of geometry—the compass and straight-edge—without ruler-measured lengths or protractor-measured angles. It was a puzzle. Making the (108°) angle for the pentagon-heart of the star always eluded me. How could something as simple-looking and common as the “star” be so hard to make? Since then, I have learned why it was tricky. The pentagon is based on an elusive and exotic number called phi (pron. fee).

Phi is a relationship of two lengths where the short length is about 62% of the long one, as in a 3-by-5 or 5-by-8 card. The long and short lengths added together (e.g. 3+5=8) have the same relationship with the long as the long has with the short. Phi is a bit more than 1.618, but you can never write it perfectly as a number. It is one of those endlessly non-repeating sequences like the more famous pi. The proportion has classically been called “Golden” as in Golden Ratio or Golden Mean. You can get it into your calculator by taking the square root of 1.25 and adding .5 (1.6180339887…). If the arms of a 5-pointed star measure 1 unit, then the width of each arm at its base, where it connects to the body of the star, are a golden proportion of 1, i.e., 0.618033989.

A few years ago, I learned how to construct a golden proportion with a compass and straightedge. It has surprisingly few steps. With the proportion in place, it was easy to figure out the rest of the steps to make a star. My image, Starchitecture, is a golden rectangle with many more within it. Cutting a square from a golden rectangle gives back a smaller golden rectangle. I did this a few times.

This image also reveals the secret of constructing such golden proportions and a five-pointed star. It also presents the mysterious, elusive, embedded, saturating, holo-graphic nature of mathematical ideas and creation. Like this star, art is born, beauty is born, from realities hidden beneath the surface.