Saturday, August 13, 2011

Structural genetics: the Golden Ratio Phi ( φ )

The Golden Ratio Phi, represented with the Greek letter φ in honor the Greek sculptor Fidias, is a mathematical concept that makes reference to a peculiar proportion between straight line segments that can be observed in the nature (flowers, leaves, branches, roots, molluscs, starfishes, structures of chorale, rock crystals, snowflakes, ideal proportion between the parts of the human body, etc…) and in certain geometric figures (triangle, circle, squaring, rectangle, star, ellipse, rhombus, cube, sphere, pentagon, tetrahedron, octahedron, icosahedron, dodecahedron, etc.). It is an irrational algebraic number, a non-periodic infinite decimal, that cannot be reduced nor divided.

It was described and studied for the first time by Euclides towards 300 year B.C. In the Middle Ages an aesthetic and divine character was attributed to it considering perfect the proportions that followed the principles of the golden ratio. Its mathematical formula is φ = 1 + √5/2 = 1´6180339887 ....... until the infinite. The squared Root of 5 is also an irrational algebraic number with an infinite value.

φ = AC / BC = 1´61803.
In this flower of Potentilla reptans is very easy to find the golden ratio uniting with straight lines the recesses of each of its five petals drawing a regular pentagon. Soon it is enough to draw up another straight line uniting the recesses of two noncontiguous petals and we both obtain values to calculate the golden ratio. Sometimes the obtained number is not exact although very approximated. In order to sharpen plus the result the values of each one of the five angles of the pentagon can be obtained and to remove the average, and so a golden ratio is obtained practically exact.

φ = AB / BC = 1´61803.
In this flower of Eruca sativa is also very easy to find two values to calculate the golden ratio. When having four identical petals are sufficient with drawing a straight line uniting each two petals and so is obtained a perfect squaring. Soon the four sides of the squared one with two lines are united that are crossed in center of the flower and the four squaring identical smallest ones are obtained. Drawing up a line in diagonal that goes from the corner of squaring to the opposed corner of contiguous squaring it obtains first value AB. The other value BC is obtained drawing a diagonal line connecting  two opposed corners of a same one squaring. 

φ = AB / BC = 1´61803.
This small flower of Ophrys speculum has the mystical beauty attributed by the mathematicians to the forms that fulfill the proportions of the golden ratio. It was very simple to find two values to calculate it with an amazing exactitude.

The properties and possible applications of this euclidian number profusely have been studied by the mathematicians from the century eighteen to the present time, giving rise to diverse theorems, mathematical equations and formulas, like the famous Theorem of Kolmogórov-Arnold-Moser or theorem KAM.

It was found one close relation between the golden ratio and the Sequence of Fibonacci. Both mathematical concepts are widely represented in the nature.

φ = AB / CD = 1´61803.
This small flower almost albino of Solenopsis balearica, endemic to Majorca, it also has a structure in the form and distribution of its five petals that seem designed by a mathematician. A simple outline uniting the vertices of both lateral major petals gives the  first value AB and another outline uniting the vertices of both smaller petals gives to the other value CD. The simplicity and the beauty of the design are amazing. The obtained golden ratio is surprising exact.

φ = AB / AC = 1´61803. 
φ = AB / BC = 1´61803. 
This flower of marine lily, Pancratium maritimum, of an immaculate target and a perfect symmetry account with a design that facilitates much to find two values to calculate the golden ratio. First value AB is obtained uniting with a straight line the vertices of two opposite petals. In order to find the second value two lines draw up that go from the vertex of each one of the two previous petals to the vertex of a noncontiguous petal. These two lines are crossed in point C and form a cross with two long arms and two short arms. Each one of the long arms are the second value.
φ = AB / CD = 1´61803.
φ = CD / EF = 1´61803. 
 In the leaves it is something more complicated to find two values that allow to obtain an exact golden ratio. Nevertheless in this leaf of canary blackberry, Rubus palmensis, the proportions of their five leaflets have decreasing values that when dividing between them give a surprisingly accurate result.

φ = AC / AB = 1´61803.
 And finally another example of a canary plant, the Hedera canariensis, photographed in the gorgeous Bosque de Los Tiles of the Island of La Palma. Their cordate leaves of a perfect symmetry allow to find with facility two diagonal lines that when being divided between them give the golden ratio, the magic number of the divine beauty and the perfection.

Throughout the centuries this mathematical proportion with so abundant examples in the alive beings and rocks has been used by the painters, sculptors and architects to realise their more beautiful works in an eagerness to shape the ideal of perfection, symmetry and balance that so wisely the nature designs.