I worried that all of my geometrical ‘constructions’ may be misinterpreted so the following examples of geometry used in design may put things into perspective. The point of making those constructions, like the golden circles, pentagrams and so on, is to find methods of drawing proportionally-related shapes without measuring anything (cheating?). Doing that will make it easier to understand how proportion works and also allows you to work out interesting layouts or even grids.
I have, in fact, stolen the examples below from a book about geometry in design (by K. Elam, which I’ve posted about before). They may be a little aged but the two were done 50 years apart and geometric proportion will never be irrelevant to graphic design. The posters are ‘Folies-Bergère’ by Jules Chéret and ‘Negerkunst’ by Max Bill, respectively. I think I’ve added to Elam’s analysis of Bill’s work here because she credits much of it on root 2 proportions – I found that the golden section works on it quite strongly.
In ‘Folies-Bergère’, then, we can see from an immediate use of rectangles which are smaller than the frame by phi (golden section=1.618) that the three characters are divided by these measurements. The second image shows a pentagram (five-point star) within a pentagon within a circle. The whole object is central in the frame. The edges of the pentagon are of a measurement which is equal to the frame breadth/1.618 (phi). In other words, the artist, presumably, used a pentagon whose sides relate to the frame breadth by the golden section. The pentagon works as the master guide – marking where the text will sit (above and below), and informing the angle of the dancer’s legs and the position of the elbow.
Bill ’s poster is a little more complicated but the images I have made here should speak for themselves. If you divide the non-golden rectangular frame by 1.618 you get rectangle A. It isn’t always necessary to work inside an area relative to the golden section but probably an advantage. I’ve wondered why Bill chose to position the shape where it is and can only suggest it is the mid-point of rectangle A when turned on its side. Turning it this way will not affect the proportions in use.
Divide rectangle A by 1.618 and you get the measurement used by Bill to draw his text box. The text box is actually square but this is still proportional to the rest because a square is a phi rectangle with a phi² rectangle next to it. (Phi² is 2.618, as opposed to 1.618 and therefore makes for a slimmer rectangle). Squares mean ‘1’ and are therefore neutral in terms of proportion. The text box is aligned by its left side to the centre of the circle above and is therefore a little cushioned from the edge. Bill may have chosen to move it up from the lower edge also – to give it some ‘breathing space’.
The size of the circles was a little unclear to me at first. The shape in the poster is basically two big circles placed in such a way that a circle of half their size fits into the area where they overlap. The large circle is, as shown, the difference between the corners of rectangles A and B. The smaller circle, as mentioned, is half the size. Why Bill chose to make the smaller circle half the size is a slight mystery. Halfs, in proportion, are known as ’static’ proportions because they fail to give aesthetically pleasing results. Phi, meanwhile, is a ’dynamic’ number because it gives proportions which are both systematic and beautiful.
The reason Bill chose to use a static proportion for the small circle may be because the exhibition was about art which used these proportions (and his geometric shape is a representation) or because using phi may make the shape conflict with the text box. I may re-design it by using only dynamic proportions, just to see what happens….