This construction came from when I recorded how to get a golden rectangle from a horizontal line without extending the line (as shown in the post called Two Constructions). If you search for a golden rectangle construction method you’ll likely be told to create a square and extend it to a rectangle so I wanted to know how to do it in reverse as it were.
Anyway, the five-point star (pentagram) is the absolute necessity behind the use of the golden section throughout most of its history in the art world. Gothic architects devised a master diagram based largely on the five-point star, pentagon and decagon (the latter two are made by joining the five points/five points and points between).
My earlier pentagon construction is most often found in books on the subject but this one below is of my own making – I haven’t seen it elsewhere so far.
CONSTRUCTION: The first image shows the drawing I made to begin with. I’d learned my lesson from before with the circles so I checked this digitally straight away (and it works). So, as in the second image, you draw a golden rectangle any way you like and divide it in two. Then use the length of a half to extend lines from all edges. The end points, as in the third image, make the centre points of circles whose radii (weirdest plural form ever) will each touch two corners of the rectangle. Do this for every point and you get a pretty picture, as in the fourth image. I’ve marked in red the key points here which will inform the drawing of the pentagon (where lines overlap). The last image includes our pentagon. It’s relatively simple to draw when you know the following: each line in the star must be the same length; the first two lines should start at the top red mark and one should hit the the mark on the far left and the other, the mark on the far right; the cross-bar sits directly over the middle line of the guiding rectangle.
Et voila. This construction is a little different in that the rectangle you use to draw the star is not actually related by size. In more common constructions of the star, the length of the lines is equal to the length of the rectangle (they’re just pointed towards the centre so they look shorter). The star here is made of lines that, even when multiplied by 1.618 do not become equal to any measurement of the rectangle.