Logarithmic Spirals of the Nautilus Shell.

The spiral is a common element of Sacred Geometry as well as to all natural development. Spirals in nature tend to follow the Golden Ratio (Phi) or Fibonacci Sequence in their rates of expansion. The key to Sacred Geometry is the relationship between the progression of growth and proportion. Harmonic proportion and progression are the essence of the created universe and is consistent with nature around us. The natural progression follows a series that is popularized in the West as the “Fibonacci Series” where the first two numbers in the series are added to create the third number for a series of number that begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…and goes on ad infinitum. The ratio of the numbers gains great importance as the series continues. By dividing one number by the previous number, the answers result in or come closer to phi: 3/5 = 1.6666, 13/8 = 1.6250, 233/144 = 1.6180. These numbers can be demonstrated with the spiral of the Nautilus. 

The Golden Ratio (phi) is the unique ratio such that the ratio of the whole to the larger portion is the same as the ratio of the larger portion to the smaller portion. The ratio links each chamber of the nautilus to the new growth and symbolically, each new generation to its ancestors, preserving the continuity of relationship as the means for retracing its lineage. This geometry of the Nautilus can be found in the spiral patterns of cauliflower, the placement of the leaves on most plants, the arrangement of pattern on a pine cone. The ratios can be retrieved from the shape of our DNA and the measurement of distant galaxies as the Sacred Geometry demonstrates the blueprint of the sacred foundation of all things and the interconnectedness of all the various parts of the whole. 

Like standard jigsaw puzzles, this puzzle has only one solution, but instead of every piece being a different shape and approximately the same size, every piece is the same shape and a different size. The placement of the pieces is based on the golden angle (≈137.5º), and results in a pattern frequently found in nature, for example on pine cones or sunflowers. The puzzle has 8 spirals in one direction, and 13 in the other.

Fibonacci Series.

The Fibonacci series covers the simplest golden section sequence which can be expressed in whole-numbers (the golden section of 89 being 55, and that of 55 being 34, etc.):

2, 3, 5, 8, 13, 21, 34, 55, 89 …

In it each number equals the sum of the two preceding numbers (that is, 2+3 =5, 3+5=8, 5+8=13, etc.).

The sequence approaches nearer and nearer the proportion of the geometrical golden section i.e. the irrational key-number of the geometric mean: the square of every number is equal to the product of the numbers preceding and following it - with the difference of plus or minus 1.

The Fibonacci series embodies the low of natural growth. In the fir-cone starting from the centre, a system of spirals runs in the right and left directions, in which the number of spirals always result in the values of the Fibonacci sequence: 3, 5, 8 and 13 spirals.

A similar setting can be seen on the sunflower, pineapple, chamomile, dandelion, marguerite, cactus, likewise in the arrangement of leaves on the stem and in the horns of some ruminating animals.

Mathematician Leonardo Fibonacci discovered the universal pattern of life; by adding the sum of two previous numbers beginning with zero to infinity (0,1,1,2,3…). The Fibonacci sequence, the growth of all things is found in nature in the simplest and most lovely things that can be offered to our universe.

Happy Fibonacci Friday and never stop searching for the infinite patterns of life.

Golden ratio in phyllotaxis

The appearance of Fibonacci numbers and the golden ratio everywhere in nature is mostly a persistent myth. Famous (but most presumably accidental) examples are the ratios between the phalanges of your fingers, or the position of the belly button in the human body. One example where there is an explanation for the golden ratio’s involvement is phyllotaxis, the arrangement of leaves on a plant stem. In sunflowers, pineapples, romanesco, aloe plants, pine cones, artichokes and numerous other flowers, the number of leaves or seed spirals frequently equals a Fibonacci number.

Let’s consider a mathematical model. We define a “flower” starting from a central growing point, producing a new leaf or seed after each α turns (α being a fixed parameter) and constantly growing outward. For simple rational numbers we get the following patterns:

So we see this results in some radial spokes (as many as the denominator in α's irreducible fraction). For a flower this is very unappealing, since this arrangements waste a lot of space. Instead a plant wants to maximize its exposure to sunlight, dew or carbon dioxide.

More interesting patterns occur when we choose an irrational α:

You notice this arrangements fill space more evenly but also stagnate into spiraling patterns, resembling the rational case. These spirals correspond with the best rational approximations for α. Centrally in the left example you clearly notice three spirals, because 1/π ≈ 1/3. After a while they break apart into 22 spirals: 1/π ≈ 7/22. One can show that the “best rational approximations” or convergents of an irrational number are precisely the fractions resulting from keeping only a limited number of terms in the continued fraction expansion.

So if a flower wants to distribute its seeds optimally, it needs an α which is “hard to approximate” with fractions, and the golden ratio is essentially the hardest one because its continued fraction consists only of 1’s. The convergents of the golden ratio have only Fibonacci numbers as denominators (in lower terms), which helps explain their ubiquitous occurrence in nature.

Indeed, if we run the model with the inverse golden ratio, we get a marvelous uniform pattern, not exhibiting any obvious spirals, resembling for instance a sunflower’s face:

Finally, the parameter α appears to be rather sensitive: even a small deviation in the angle of rotation quickly spoils the delicate balance achieved by the golden ratio.

The Golden Mean was used in the design of sacred buildings in ancient architecture to produce spiritual energy that facilitated connectivity with spiritual realms through prayer. Our reality is very structured, and indeed Life is even more structured. This is reflected though Nature in the form of geometry. Geometry is the very basis of our reality, and hence we live in a coherent world governed by unseen laws.  These are always manifested in our world.  The Golden Mean governs the proportion of our world and it can be found even in the most seemingly proportion-less (active) living forms.