The Fibonacci numbers and ratio have been studied by mathematicians of all ages. It is also familiar to those in art, biology, architecture, music, botany, and finance.
Is this the most astounding number sequence in the world? Or are we distorting reality to see mathematics where none exists?
Let’s take a closer look at the mathematical phenomenon that has attracted the attention of thinkers from all disciplines and periods.
1. Does Architectural design reflect the golden ratio?
It has been claimed the golden ratio appears in several ancient and modern architectures: The great pyramid of Giza, Parthenon, Notre Dame Cathedral, Tajmahal, Eiffel Tower, Toronto’s CN tower, the United Nations Secretariat building, and more.
The golden ratio may or may not have been included intentionally among all of these buildings.
The architects of the Parthenon could have been aware of the golden ratio. Though the Parthenon has an intricate design and it is not clear that the golden ratio was intentionally represented.
However, it seems that the golden ratio was intentionally included in the design of Toronto’s CN tower. The ratio of the total height (553.33 meters) to the height of the observation deck (at 342 meters) is 1.618.
The CN Tower is a communications tower built in 1976. It was the world’s tallest free-standing structure at the time.
2. Is the spiral of the Nautilus shell based on the golden ratio?
Yes. But the truth is a bit different than what you often hear.
The traditional “golden spiral” is constructed from adjacent golden rectangles. This creates a spiral that increases in dimension by the golden ratio with every 90-degree turn of the spiral.
The spiral of the Nautilus shell does not match this golden spiral.
The spiral constructed from a Golden Rectangle is NOT a Nautilus Spiral.
But here is the crucial issue: There is more than one way to create a spiral with golden ratio proportions.
You can create a spiral that expands by the golden ratio with every 180-degree turn of the spiral. This spiral is a closer match for the spirals of many Nautilus shells.
A spiral expanding by the golden ratio at every 180-degree turn is a closer match to some Nautilus shells for the first few rotations
You see the difference, I hope.
Here is another scientific fact: The mathematical proportion of growth of the nautilus shell is the number 108 (see The Number 108). This is the number of the pentagram that is based on the logic of the golden ratio.
So we see that the Nautilus spiral can exhibit proportions close to Phi.
3. Did renaissance artists use the golden ratio in their paintings?
Yes. Golden ratios are quite easy to see in Leonardo Da Vinci’s “Last Supper”.
Following his lead, in 1955, Salvador Dalí painted The Sacrament of the Last Supper, with golden ratio proportions.
Michelangelo’s “Creation of Adam” also has God’s finger touching Adam’s finger at the precise golden ratio point of the width of the area in which they are framed.
Those are just a few examples of how Renaissance artists used the golden ratio in their paintings.
4. Are the spirals seen in nature based on the golden ratio?
You may have heard that spirals found in nature are based on the golden ratio. While that might not always be 100% accurate, it’s definitely true in some cases.
Sunflower seeds are well-known for being clustered in a pattern of interconnecting spirals based on Fibonacci numbers. Because patterning based on Fibonacci numbers allow for the highest number of seeds on a seed head.
This optimization behavior is not just found in sunflower seeds. Leaves, branches, and petals can grow in spirals based on the golden ratio. This allows the most sunlight to reach older leaves as new leaves grow.
Logarithmic spirals occur commonly in nature, for example in the arms of spiral galaxies, ram horns, hurricanes, whirlpools, and many more.
The golden spiral is one particular case of logarithmic spirals and can appear in the examples mentioned above.
Although, the logarithmic spirals that appear in nature do not have to be golden spirals necessarily.
5. Is there a new algorithm based on the golden ratio that can predict spiritual experience?
Yes, it appears that mathematics may have a role in spiritual experience. Sacco found in a 2017 study that the dynamical effects of age 18 predicted increased spiritual experience. This was as predicted by the FLCM. This supports the link between the relationship of the golden ratio and spiritual experience.
However, the dynamical effects of age 11 and 30 did not support the hypothesis of increased spiritual experience as predicted. This result required an alternative explanation.
It should be evident that children at age 11 may not be able to communicate the complexities of some spiritual experiences. And it’s entirely reasonable that spiritual experiences could be linked with the instability that is less likely at age 30 when personality becomes stabilized.
In the real world, there are many compelling examples of the golden ratio in natural phenomena. But not all phenomena in nature are based on the golden ratio.
You can find plenty of examples where the golden ratio is not found in nature as might be expected. But you can’t deny the power of data and evidence to prove the inherent power of the golden ratio in reality.
The critical question you need to ask yourself is: Why does the golden ratio show up so commonly in nature?