Relating the included middle (a logical-philosophical concept) to the golden ratio (a mathematical-geometric concept) is a fascinating intellectual exercise. Although they belong to different domains (the logic of becoming and the mathematics of forms), both point to a common mystery: the harmonic resolution of opposition and the tension toward a higher whole.

At first glance, they might seem at odds: classical logic (of the excluded middle) divides reality into two, whereas the golden ratio divides a segment into two unequal parts. However, if we delve deeper into the philosophy of mathematics and aesthetics, we find profound connections.

Here is a detailed look at how these two concepts relate:

1. The Triadic Structure (The "Third" as a Mathematical Mediator)

Binary logic divides reality into A and B (black/white, true/false). It is a logic of "either/or" disjunction.

The golden ratio, on the contrary, requires three terms to exist. The golden formula states that "the whole is to the greater part as the greater part is to the lesser part."

• The Whole (a+b)
• The Greater Part (a)
• The Lesser Part (b)

In this sense, the "Whole" acts as that "included middle." It is not merely the arithmetic sum of the parts, but the transcendent reality that unifies the relationship between the macrocosm (the greater part) and the microcosm (the lesser part). The golden ratio is the mathematical demonstration that the relationship between two opposites (greater/lesser) is only resolved if a transcendent third element (the Whole) is included.

2. The Reconciliation of Opposites (Harmonic Asymmetry)

Historically, philosophy has viewed the relationship between opposites in two ways:

• Pure symmetry (dualism): Two equal and opposite forces that cancel each other out or stagnate (like +1 and -1).
• Included middle (synthesis): A tension that is resolved at a higher level.

The golden ratio is the ultimate mathematical expression of harmonic asymmetry. It does not divide a line in half (static symmetry), but divides it unequally yet proportionally. This asymmetrical tension generates dynamism. In Nicholas of Cusa's coincidentia oppositorum, Infinity is the third that unites opposites; in geometry, the golden ratio is the "law" that unites difference (the greater and the lesser) without annulling it, creating beauty and continuous growth.

3. The "Divine Proportion" and the Trinity

This connection became explicit in the Renaissance, precisely during the time of Nicholas of Cusa. In 1509, the friar and mathematician Luca Pacioli published his famous treatise De Divina Proportione (illustrated by Leonardo da Vinci).

Pacioli called the golden ratio "Divine" for reasons that connect directly with the theological included middle:

• It is unique and indivisible: Like God.
• It defines the Trinity: Pacioli associated the three terms of the golden ratio (the Whole, the Greater Part, and the Lesser Part) with the three persons of the Holy Trinity (Father, Son, and Holy Spirit). In theology, the Holy Spirit is the "bond" (the included middle) that unites the Father and the Son. In golden mathematics, the proportional relationship is the "bond" that unites the whole with the parts.

4. Continuous Becoming: Dialectics and the Spiral

The included middle in Hegel's dialectic (Thesis-Antithesis-Synthesis) is dynamic; the synthesis becomes a new thesis, creating a spiral of constant evolution.

The golden ratio is the geometric translation of this dialectical becoming. If you take a golden rectangle and remove a square from it, the remaining rectangle is once again a golden rectangle. If you repeat this process indefinitely and draw an arc through the intersections, you obtain the golden logarithmic spiral (the shape of the nautilus, galaxies, or sunflowers).

It is a process where:

• Thesis (the extracted square)
• Antithesis (the remaining rectangle)
• Synthesis (the new golden rectangle, which is simultaneously the starting point for the next iteration).

The golden ratio generates a progression where the "third" (the synthesis) is always implicit in the relationship of the first two, advancing toward infinity.

5. The Vision of Basarab Nicolescu

The physicist and philosopher Basarab Nicolescu, creator of the modern theory of the "Included Middle" and transdisciplinarity, uses concepts very close to golden geometry. For Nicolescu, the included middle occurs when two contradictory terms on one "level of reality" are resolved on a higher level.

The golden ratio operates exactly like this:

• Level 1 (linear and finite): You have a segment divided into two unequal parts. If you look at them in isolation, they are in tension, they are opposites (one dominates the other).
• Level 2 (proportional and transcendent): By applying the golden formula, you access a higher level of understanding where the tension disappears. The inequality is transformed into universal harmony.

Summary

The golden ratio and the included middle are two sides of the same cosmic coin. While the included middle is the logic that explains how opposites can coexist and synthesize on a higher level of reality, the golden ratio is the mathematics and geometry that materializes that synthesis in physical space, biological growth, and aesthetic harmony. Both teach us that reality is not a simplistic dichotomy (A or B), but a triadic dance where the Whole is always greater than the sum of its parts.