According to legend, Hippasus was drowned by the Pythagoreans for revealing irrational numbers that seemed to violate their vision of the world’s harmony. Such numbers cannot be expressed as a ratio of integers — yet they can arise from simple geometric relationships. Examples include the diagonal of a unit square (√2) and the Golden Ratio, φ.
Most strikingly, two irrational quantities can combine to yield a rational integer in Binet-type expressions:
Lₙ = φⁿ + (−φ)⁻ⁿ
Here Ln is Lucas number, similar to Fibonacci’s. This relation can be adapted to express how we create a rational happiness from irrational Purpose or Ambition (A) and Essence or Detail (B):
Flourishing (Lₙ) = A × φⁿ + B × (−φ)⁻ⁿ
Flourishing materializes when it hits a rational value. Striking a balance (A = B) yields high Flourishing due to high Lₙ, while leaning to either side leads to stumbling due to an irrational result:
A > B represents an arrogant, insensitive person.
A < B represents a sensitive, yet impotent person.
A = B represents a peaceful warrior guided by his Heart.
Increasing n (evolution) exponentially increases the Ambition/Purpose term (Aφⁿ), while decreasing the Essence/Detail term (B(−φ)⁻ⁿ). Although the two terms have identical absolute magnitudes on the logarithmic scale (|log φⁿ| = |log φ⁻ⁿ|), the diminishing Essence/Detail term (B(−φ)⁻ⁿ) plays the leading role: The smaller the detail we take care of, the higher the purpose we can manifest!
What Do Mystics Say?
“Does this path have a heart?” — Carlos Castaneda, The Teachings of Don Juan
In other words, our true value depends more on what we nurture and care about than on what we imagine and seek. Lao Tzu expressed it like this:
“A man with outward courage dares to die; a man with inner courage dares to live.”
Outward courage is ambition coefficient (φⁿ), inward courage is essence coefficient (φ⁻ⁿ). When the tiny sentiment leads, then we dare to Live — and ambition follows. But when ambition leads, then the sentiment disappears — and we dare to Die. Hence, our true value lies in the smallest things we nurture — not in the largest things we pursue.
Golden Spiral vs Dialectic Synthesis
Lucas numbers, like Fibonacci numbers, can be represented through the Golden Spiral. The latter can be naturally compared with dialectical synthesis on the complex plane, driven by conscious adaptability (z) and subconscious inertia (δ). Equating the two expressions suggests that balanced development requires a particular relationship between z, δ, and the perspective angle θ (with n = 2θ/π).
When δ ≫ θ, this relationship approaches δ/z ≈ 3.26 ≈ π. Too little adaptability (δ/z ≫ π) leaves us making “empty circles,” while too much (δ/z ≪ π) collapses the spiral toward a straight line. In reality, adaptability is rarely excessive, as we tend to carry excessive subconscious burden (δ/z > π), reflecting an exaggeration of either ambition or sentimentality.