Truth and Myth in the Golden Ratio

The Fibonacci sequence is a marvel that has captivated mathematicians throughout history. But its allure extends beyond math, influencing fields like art, biology, architecture, music, botany, and finance.

Is this truly the world's most mesmerizing number sequence, or are we stretching our imagination to find patterns where none lie?

Join us as we delve deep into this mathematical wonder that has fascinated minds across disciplines and eras.

1. Does Architectural design reflect the golden ratio?

The golden ratio is often touted as appearing in many architectural marvels, both ancient and contemporary. Examples span from the Great Pyramid of Giza and the Parthenon to modern landmarks like the Eiffel Tower, Toronto’s CN Tower, and the United Nations Secretariat building.

It's debated whether all these structures consciously employ the golden ratio in their designs. The Parthenon, with its intricate layout, leaves us guessing about its architects' intentions regarding the golden ratio.

However, there seems to be a deliberate use of the golden ratio in the design of Toronto’s CN Tower. The proportion between its total height (553.33 meters) and the height to its observation deck (342 meters) is strikingly close to 1.618, the golden ratio.​​

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?

Indeed, there's more nuance than popular narratives suggest. The classic "golden spiral" is crafted using consecutive golden rectangles, resulting in a spiral that expands by the golden ratio for every quarter turn. However, the Nautilus shell's spiral isn't an exact match to this golden spiral.

The spiral constructed from a Golden Rectangle is NOT a Nautilus Spiral.

Here's the crux of the matter: The golden ratio can shape spirals in multiple ways. 

For instance, a spiral that widens according to the golden ratio after every 180-degree twist more closely mirrors the spirals found in numerous 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

I trust you grasp the distinction.

Here's another intriguing insight: The nautilus shell expands according to the number 108 (refer to "The Number 108"). This number resonates with the pentagram, which fundamentally operates on the principles of the golden ratio.

Thus, it's evident that the Nautilus spiral can indeed reflect proportions approaching Phi.

3. Did renaissance artists use the golden ratio in their paintings?

Absolutely. Leonardo Da Vinci's "Last Supper" prominently features golden ratios.

Inspired by him, Salvador Dalí, in 1955, crafted "The Sacrament of the Last Supper" adhering to golden ratio dimensions.

Similarly, in Michelangelo’s "Creation of Adam," the iconic touch between God’s and Adam’s fingers occurs exactly at the golden ratio point of the width within their frame.

These instances highlight just a fraction of the Renaissance artists' fascination with the golden ratio in their masterpieces.

4. Are the spirals seen in nature based on the golden ratio?

Spirals in nature are often linked to the golden ratio, and while it's not universally true, there are noteworthy instances.

Sunflower seeds, for instance, form interconnecting spirals based on Fibonacci numbers. This Fibonacci-driven pattern maximizes the number of seeds on a seed head.

Beyond sunflowers, the golden ratio influences the growth of leaves, branches, and petals, optimizing sunlight absorption as new leaves emerge.

Nature frequently showcases logarithmic spirals: from spiral galaxies and ram horns to hurricanes and whirlpools. The golden spiral is a subset of these logarithmic spirals, and while it may be present in the examples listed, not all natural logarithmic spirals are golden spirals.​

5. Is there a new algorithm based on the golden ratio that can predict spiritual experience?

It seems that there might be a connection between mathematics and spiritual experience. Sacco's 2017 study tentatively suggested that the dynamic effects observed at age 18 could potentially predict an elevation in spiritual experiences. This observation aligns with the predictions of the FLCM. This offers a tentative link between the relationship of the golden ratio and spiritual experience.
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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?