Easy B13 Course (Part 1): What is the Fibonacci Sequence?

Easy B13 Course (Part 1): What is the Fibonacci Sequence?

🌀 What is the Fibonacci Sequence?

The Fibonacci sequence is defined as follows:

$$F0 = 0,\quad F1 = 1,\quad Fn = F{n-1} + F_{n-2}$$

In short, it is generated by a simple rule: add the previous two numbers to create the next one.
The sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

🌿 Fibonacci Structures in Nature

This sequence appears surprisingly often in the natural world:

• The spirals of sunflower seeds — The counts of clockwise and counter-clockwise spirals are Fibonacci numbers.
• The arrangement of scales on a pinecone.
• Romanesco (a type of cauliflower) — It displays a self-similar structure where the same shape appears even when magnified.
  These are examples of "fractal structures," where the parts resemble the whole.

🔁 Self-Similarity and Fractal Structures

• Self-similarity is the property of a shape or structure where it maintains the same form even when scaled up or down.
• A fractal is a figure with self-similarity, where the same structure appears no matter how finely it is subdivided.

The Fibonacci sequence has the property of converging to the golden ratio (approx. 1.618). When this appears in spirals and branching structures, it creates self-similar patterns.

📐 Fibonacci Spirals and Geometric Fractals

By arranging squares using the Fibonacci sequence and drawing 90° arcs in each square, you create a "Fibonacci spiral."
This spiral has fractal-like properties because the same shape appears regardless of the scale.

Furthermore, when these are scaled, rotated, and tiled, they form an infinitely extendable self-similar structure. Your figure (the 4-scale spiral) is a visualization of exactly this property.

1️⃣ 13 is the "First Sufficiently Large Fibonacci Ratio," Where Convergence to the Golden Ratio Stabilizes

The Fibonacci ratio (F{n+1}/Fn) converges to the golden ratio (\phi), but upon reaching 13 (F₇), the ratio becomes nearly stable, and the error in self-similarity decreases sharply.

Looking at the changes in the ratio:

n	Fₙ	Fₙ₊₁/Fₙ
5	5	1.6
6	8	1.625
7	13	1.61538…
8	21	1.61904…

At 13, the error is already around 0.003.
In other words, by using the structure up to 13, the golden ratio scale self-similarity is almost perfectly established.

→ The set from 1 to 13 is the smallest set with sufficient precision to serve as a base for a fractal.

2️⃣ 13 Creates the "First Large Square," Completing the Rotationally Symmetric Fractal

As shown in the initial figure,
when you arrange Fibonacci squares to draw a spiral, 13 squares are the first to "close the overall structure."

Reason:

• Up to 1, 1, 2, 3, 5, 8, it is local growth.
• The moment a 13th square is added, the spiral possesses a "full cycle structure."
• This enables 90° rotationally symmetric self-similarity.
• In short, 13 becomes the first "closed unit cell" of a self-similar fractal.

3️⃣ Fibonacci Numbers up to 13 Form the "Minimal Self-Similar Block"

To construct a fractal, you need a "minimal block" where the same shape emerges even when the scale is transformed.

In the case of Fibonacci:

• 1 → Point
• 1 → Line segment
• 2 → Small cell
• 3 → Medium cell
• 5 → Large cell
• 8 → Larger cell
• 13 → The first "complete block" containing all of these.
  In short, it is only by reaching 13 that the "full hierarchy of self-similarity" is complete.

4️⃣ 13 Creates the "Fractal Period"

When you scale down a Fibonacci spiral,
the following scales naturally emerge:
1/13, 1/169, 1/2197 …

This becomes the starting point of exponential self-similarity where:

$$13 = F7 \quad\Rightarrow\quad 13^k = F7^k$$

The "1/13," "1/169," and "1/2193" found in your figure are exactly this.

→ By making 13 the node, the scale is naturally hierarchized.
→ Fractal calculations become most simple.

---

References

Fusion Research Case Study on Electronics and Information Mathematical Engineering: How to Conceive New Algorithms in Electronic Circuits https://kobaweb.ei.st.gunma-u.ac.jp/lecture/20171201fibonacci.pdf