Geometry | Proportion | Use of Symmetry


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.

Chambered Nautilus


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”.

Calla Lily

Video by Cristobal Vila

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.

Snowflakes for example, utilize the geometry of the Hexagon, (proportions of the Hexagon involving the lengths of the sides and the radius of the enclosing circle contain the divine proportion). Snowflakes develope millions of individual patterns based on the molecular structure of water droplets that minimizes the (heat) energy in a stable crystal.It is both the material and the proportional characteristics of the hexagon that produce the patterns.
There is an intimate interrelationship between the material and  geometry of the patterns it forms, and the associated states of minimum energy of the water crystal(“snowflake”). Geometry & (heat)Energy  limit form.
Symmetries occur by the reflection of patterns on either side of an axis.There are 6 major axes in the snowflake patterns.
 Even at a molecular scale it is the GEOMETRY of atomic structure which determines the characteristics and interactions of the molecules of life, the RNA and DNA.
The stable configurations of nuclei sharing electrons and forming bonding patterns.
The majority of living organisms have evolved using bi-lateral symmetries to achieve a balance of functions and structure.

Bi-lateral Symmetry is a very powerful pattern generating geometry  that has been used historically by many cultures to denote importance,  imply perfection, create formality, and encourage movement along an axis.

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