The ratio that gives the division of a line into smaller and larger segments such that the smaller segment is to the larger as the larger is to the whole line is known as “ The divine proportion”, or the “Golden Mean” (1 plus the square root of 5 divided by 2). This ratio was seen to pervade the natural world of plants and animals as a gnomon of growth. You cannot calculate the ratio exactly…. as it is an irrational number involving the square root of 5, repeating decimal places to infinity.
A proportional limit on any given pattern will produce many variations of the pattern that are intimately related, but slightly different. It is proportional limits that give Coleridge’s “unity in variety”. This is the paradox and power of limits; limits create variety within order.
The golden mean ratio has the property that makes it an ideal proportional system for growth patterns, as it is a ratio that is simultaneously arithmatic (additive) and geometric; controlling surface area and volume with a logarithmic spiral. This both facilitates and limits the growth of the organism in “proportional symmetry”.
The physical world can transcend the limitations of mathematical models describing the universe. In the physical world the compass arc used in Geometry, or the actual growth pattern produced by an organism defines the limit, even though we can’t calculate it exactly using mathematics. Here the reality of scale imposes the limits, at the atomic and molecular scale a point in “space”may occur where there is no “matter”, or it has a very low probability of being there as a proton, neutron, or electron.