Sungchul Ji, Ph.D.
Emeritus Professor of Theoretical Cell Biology, Rutgers University

Most people have heard of the Fibonacci sequence. Many have also seen the Golden Spiral in seashells, flowers, galaxies, and even hurricanes. Because they often appear together, it is tempting to assume that the Golden Spiral is automatically produced by the Fibonacci sequence.

But this is not true.

A simple equation will generate the Fibonacci numbers and the Fibonacci rectangles, but it will not generate the Golden Spiral. Something essential is missing — something that explains why nature overwhelmingly prefers one specific spiral out of infinitely many possible spirals.

To understand what that missing ingredient is, we must take a journey from mathematics into physics, biology, and even philosophy.

1. The Three Mathematical Actors: FS, GR, and GS

To keep things clear, let’s distinguish among three very different entities:

1. Fibonacci Series (FS)

A list of numbers generated by a simple rule:

2. Golden Ratio (GR)

A number obtained as the limit of the ratio:

3. Golden Spiral (GS)

A geometric curve in two or three dimensions, defined by:

These three are often conflated, but they belong to entirely different categories:

This distinction is crucial for understanding how nature builds structure.

2. What Fibonacci Actually Produces

If you build squares whose sides are Fibonacci numbers and arrange them as in the well-known diagram, you get what mathematicians call Fibonacci rectangles.

But then you face a choice.

You can draw infinitely many curves connecting the opposite corners of these squares:

• circular arcs
• parabolic arcs
• sine-like curves
• straight-line segments
• distorted or noisy curves
• and, yes, the Golden Spiral

Nothing in the Fibonacci mathematics tells you which one to choose.

This is where our story becomes interesting.

3. The Missing Ingredient: Selection

The Golden Spiral is not simply computed — it is selected.

This is the core message of the IRVSE principle (see Figure 2 below [1]:

Iterative Reproduction with Variations and Selection by Environment

Steps (I)–(IV) of IRVSE generate the Fibonacci structure (i.e., a nested set of rectangles)
But Step (V) is different:

Step (V): Selection

“Form a spiral by connecting the opposite corners with a circular arc selected out of almost an infinite number of possible curves.”

This step requires what biologists call selection, what physicists call constraint, and what philosophers call telos or intentionality.

Without this step, no Golden Spiral emerges — only rectangles.

4. Peirce’s Semiotics: Why Selection = Thirdness

Charles Sanders Peirce described the world in terms of three modes of being:

1. Firstness — possibility, quality, raw form
   • corresponds to Fibonacci numbers
2. Secondness — actualization, brute facts, geometry
   • corresponds to rectangles
3. Thirdness — law, meaning, habit, selection
   • corresponds to choosing the Golden Spiral

The Golden Spiral is a Thirdness phenomenon.
It arises when possible forms become selected forms.

This is missing from the purely numerical Fibonacci mechanism.

5. The Geometry of Reality (GOR): XY vs XYZ

In the GOR framework [1], the universe is structured along three axes:

• X-axis: Energy/Matter
• Y-axis: Information/Mind
• Z-axis: Spirit/Consciousness/Selection

Using this language, we can summarize:

Golden Ratio (GR)

Lives in the XY plane — a realm of energy and information.

Golden Spiral (GS)

Lives in the XYZ volume — where selection, spirit, and telos operate.

Here is the key difference:

• FS → GR (automatic algebraic limit)
• FS → GS (requires Z-axis: selection)

This is why the Golden Spiral belongs to gnergitonic space (XYZ) [1] (see Figure 1 above), not merely gnergic space (XY).

6. Chris King’s SEC vs IRVSE

Chris King (author of Symbiotic Existential Cosmology, SEC) [2] correctly notes that:

• Fibonacci series automatically yields the Golden Ratio
• The Golden Ratio is deeply embedded in natural structures

Where we diverge is here:

Chris implicitly equates the Golden Ratio with the Golden Spiral.

But:

• A limit of ratios (GR) is not a curve (GS).
• A number is not geometry.
• Form (algebra) is not the same as Matter (geometry/topology).

This conflation collapses an essential distinction:
emergence vs selection.

IRVSE restores this missing step.

7. Why This Matters for Understanding Nature

The appearance of Golden Spirals in nature — shells, storms, galaxies, flowers — suggests that:

• nature is not random
• nature is not purely mechanical
• nature selects certain forms over others

This selection occurs because the Golden Spiral is:

• energetically efficient
• growth-compatible
• scale-invariant
• dynamically stable
• aesthetically symmetric

In other words:

The universe “chooses” the Golden Spiral because it works,
not because Fibonacci forces it.

Nature uses IRV (growth + variation)
and then applies SE (selection by environment).

Just like evolution.

8. The Deeper Implication: The Universe Has Telos

If the Golden Spiral requires selection, then selection is fundamental.

This means:

• geometry is not arbitrary
• mathematics alone does not dictate form
• nature exhibits intentionality or orientation
• the universe is not merely computational; it is teleological
• IRVSE may be a universal creative principle

This is consistent with:

• Peirce’s Thirdness
• Spinoza’s Nature/God
• modern theories of self-organization [3, 4]
• systems biology
• cosmological pattern formation

The Golden Spiral becomes a window into the mind of nature.

9. Final Thoughts

The Golden Ratio is a number.
The Golden Spiral is a geometric form.
The two are connected, but they are not identical.

Nature generates spirals not merely by computation,
but by selection.

The Fibonacci sequence provides the possibilities.
The universe chooses the realization.

And that choice — this act of selection —
is where meaning, mind, and spirit enter the story.

References:
[1] Ji, S. (2025). The Geometry of Reality (GOR): A Triadic Framework for Matter, Mind, and Spirit.
https://622622.substack.com/p/geometry-of-reality
[2] Chri King. Symbiotic Existential Cosmology. https://www.dhushara.com/cossym/SEC/SEC0.htm
[3] Ji, S. (2025). Self-Organization as the Ultimate Variational Principle of Nature
https://622622.substack.com/p/self-organization-as-the-ultimate
[4] Ji, S. (2018). The Gnergy Principle of Organization. In: The Cell Language Theory: Connecting Mind and Matter. Wo

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https://622622.substack.com/p/how-fibonacci-teleology-and-the-geometry