Friday, 12 March 2010

Another golden pentagram construction

Here’s a nifty little way to draw up a set of golden pentagrams – which are, between themselves, proportional by 1:1.618. I’ve not seen this method elsewhere…and there’ll be loads of ways to construct these shapes but I find this one quite enjoyable to work through.

Where other methods I’ve found (have a look in the February posts) have begun with the golden rectangle, this one begins with a square. The key to drawing the five-points of the pentagram (five-point star) on a circle’s circumference is obviously in figuring out the split of the circumference by 5 (5 equidistant points after all). So we need to be able to chop a circle into 5 equal angles without using measuring tools.

Beginning with a square, we draw a circle whose centre is at the corner with a radius equal to the square’s length. Then we extend the square into the circle by making it a golden rectangle, as described in the second image below.

In the third image we have our special measurement (the dashed line) – one that will cut our circle into 5 equal parts. It happens to be the diagonal measurement inside the segment by which we extended the square. Continue the line, in a circle, to the edge of the main circle. At this point it doesn’t matter where we put our compass – you’ll notice the end result is a pentagram on its side so if you want one with a horizontal base, place the compass at the top of the circle’s circumference in the centre, instead of at the square’s corner.

We’ll make two new points on the circumference by continuing the line right round. These points mark where we’ll put the compass to make further guides. The fourth and fifth images show this process – and we get a beautiful shape at the end of it…a sort of bloated group of diminishing pentagrams and pentagons.

In the last image, the points are simply drawn together with straight edges. The conclusion is an image that harvests squares, triangles, circles, pentagrams and pentagons all in a number of sizes in proportion of the ratio 1:1.618. Further points can be joined, more lines extended and even more interesting patterns uncovered.

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