Golden ratio

One of the most prominent forms through which the golden ratio expresses itself is the golden spiral, where the arc of the spiral is determined by phi. The scale can range from microscopic to super-galactic, but the arc remains constant at every stage of progression and level of existence. Not all spirals in nature are golden proportioned, but the golden spiral is prominently featured.


The golden spiral can be found in the tiniest sub-atomic particles of the microcosm to spiral galaxies of the macrocosm

Flower Patterns and Fibonacci Numbers


Why is it that the number of petals in a flower is often one of the following numbers: 3, 5, 8, 13, 21, 34 or 55? For example, the lily has three petals, buttercups have five of them, the chicory has 21 of them, the daisy has often 34 or 55 petals, etc. Furthermore, when one observes the heads of sunflowers, one notices two series of curves, one winding in one sense and one in another; the number of spirals not being the same in each sense. Why is the number of spirals in general either 21 and 34, either 34 and 55, either 55 and 89, or 89 and 144? The same for pinecones : why do they have either 8 spirals from one side and 13 from the other, or either 5 spirals from one side and 8 from the other? Finally, why is the number of diagonals of a pineapple also 8 in one direction and 13 in the other?


Are these numbers the product of chance? No! They all belong to the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. (where each number is obtained from the sum of the two preceding). A more abstract way of putting it is that the Fibonacci numbers fn are given by the formula f1 = 1, f2 = 2, f3 = 3, f4 = 5 and generally f n+2 = fn+1 + fn . For a long time, it had been noticed that these numbers were important in nature, but only relatively recently that one understands why. It is a question of efficiency during the growth process of plants.


The explanation is linked to another famous number, the golden mean, itself intimately linked to the spiral form of certain types of shell. Let's mention also that in the case of the sunflower, the pineapple and of the pinecone, the correspondence with the Fibonacci numbers is very exact, while in the case of the number of flower petals, it is only verified on average (and in certain cases, the number is doubled since the petals are arranged on two levels.

"Sacred Geometry is a concise approach to the understanding of the way in which the universe is manifest, ordered and sustained."
-- Karen Prior


The orbits of certain planets are phi related: the distance from Mercury to Venus is approximately 1.618 times the distance from the Sun to Mercury. The distance from Earth to Mars is approximately 1.618 times the distance from Venus to Earth.


The orbital relationships between certain planets are also golden proportional. For example, the relationship between Jupiter and Saturn is such that over a sixty year period Jupiter makes five revolutions around the Sun. By maing the position of Saturn at the end of each of Jupiter’s orbital cycles, a golden-proportioned pentagon and pentagram are traced out in space.

According to Vitruvius, the distance from fingertip to fingertip should be the same as that from head to toe. The sacred mean can be seen in the ratios of body parts. In the arm of the Vitruvian man for example, we can see that the ratio of A is to B is the same as that of B to C. The same rules apply throughout the human body.

"The workings of the human body are an analogy for the workings of the universe." -- Leonardo da Vinci (1452 - 1519)