Designers are attracted to proportional systems with fixed ratios that offer harmonic ‘certainty’. The usual choices are very limited – the Fibonacci series, the square root of two (circle and square relationship) and the Golden Mean. There are however other geometric generators that create fixed ratios, albeit on a closed or finite progression. One of these is illustrated below.

The geometry sets the following constraints:

– Circle B repeats six times at regular intervals around circle A, forming a hexagon.

– Circles B and C are located tangentially to circle A and each other, determining their relative diameters.

Based on the above the ratios can be calculated as follows:

An interesting feature of this geometry is the complexity of the relative ratios, both in terms of one radius to another and as a progression of radii from the smallest (C) to the largest (A), which can be expressed as 1 : 1.732 : 3.732.

Proportional systems are only useful to the designer when they offer creative parameters to the design process rather than rigid constraints. If the ‘rules’ are too prescriptive the result is a vapid ‘design by numbers’. For this particular system the designer could simply adopt the relative ratios of the three circles as a proportional determinant, or also make use of the underlying hexagonal geometry. The examples below use both proportion and location, but with the use of squares located within three circles as a design variant:

This simple pattern can be elaborated by including squares outside as well as within the circles:

This variant on the initial geometry doubles the range of potential proportional relationships: The diagram below shows the relative ratios of the three squares within the circles (left), outside of the circles (middle) and combined (right):

The interlacing of one geometry with another system (the circle-square relationship in the above example) provides dynamism to an otherwise static progression of ratios, and underlies much of CODA Projects’ concept generation. Whether it results in truly harmonic forms is debatable, but this approach does succeed in liberating the designer from the banality of the Cartesian grid which by definition is additive rather than harmonic.