A Novel Approach to Approximating the Golden Ratio and Its Connection to the Golden Angle via Fibonacci Squares #247
Description
Abstract
This paper presents a creative method for approximating the golden ratio (ϕ≈1.618033988749895\phi \approx 1.618033988749895\phi \approx 1.618033988749895) by squaring Fibonacci numbers and dividing each square by the sum of the squares of previous terms. The method converges to ϕ\phi\phi and reveals an intriguing property: the sum of differences between consecutive ratios approximates 1−1ϕ≈0.3819661 - \frac{1}{\phi} \approx 0.3819661 - \frac{1}{\phi} \approx 0.381966, the ratio associated with the golden angle (≈137.5077∘\approx 137.5077^\circ\approx 137.5077^\circ). We verify the calculations, explore the mathematical basis, and discuss whether this is a novel observation or a transformation of known Fibonacci properties.
Introduction
The golden ratio (ϕ\phi\phi ) and its geometric manifestation, the golden angle, are fundamental in mathematics and nature, often linked to the Fibonacci sequence. Traditional approximations use the ratio of consecutive Fibonacci numbers (Fn+1Fn→ϕ\frac{F_{n+1}}{F_n} \to \phi\frac{F_{n+1}}{F_n} \to \phi). This study proposes an alternative approach, leveraging the squares of Fibonacci terms, and uncovers a connection to the golden angle through the sum of ratio differences.
Methodology
Consider the Fibonacci sequence (F1=1,F2=1,F3=2,…F_1 = 1, F_2 = 1, F_3 = 2, \ldotsF_1 = 1, F_2 = 1, F_3 = 2, \ldots). Define the ratio rn=Fn2∑k=1n−1Fk2r_n = \frac{F_n^2}{\sum_{k=1}^{n-1} F_k^2}r_n = \frac{F_n^2}{\sum_{k=1}^{n-1} F_k^2}. Examples include:
r3=2212+12=42=2r_3 = \frac{2^2}{1^2 + 1^2} = \frac{4}{2} = 2r_3 = \frac{2^2}{1^2 + 1^2} = \frac{4}{2} = 2
r4=3212+12+22=96=1.5r_4 = \frac{3^2}{1^2 + 1^2 + 2^2} = \frac{9}{6} = 1.5r_4 = \frac{3^2}{1^2 + 1^2 + 2^2} = \frac{9}{6} = 1.5
r5=5212+12+22+32=2515≈1.666667r_5 = \frac{5^2}{1^2 + 1^2 + 2^2 + 3^2} = \frac{25}{15} \approx 1.666667r_5 = \frac{5^2}{1^2 + 1^2 + 2^2 + 3^2} = \frac{25}{15} \approx 1.666667
By F16=987F_{16} = 987F_{16} = 987, r16=9872∑k=115Fk2≈1.618034r_{16} = \frac{987^2}{\sum_{k=1}^{15} F_k^2} \approx 1.618034r_{16} = \frac{987^2}{\sum_{k=1}^{15} F_k^2} \approx 1.618034, closely matching ϕ\phi\phi. Additionally, the sum of differences Δrn=rn+1−rn\Delta r_n = r_{n+1} - r_n\Delta r_n = r_{n+1} - r_n
from n=3n = 3n = 3 to n=16n = 16n = 16 yields approximately 0.381966.
Verification
Using the identity ∑k=1nFk2=Fn⋅Fn+1\sum_{k=1}^n F_k^2 = F_n \cdot F_{n+1}\sum_{k=1}^n F_k^2 = F_n \cdot F_{n+1}, the ratio simplifies to:
rn=Fn2Fn−1⋅Fn=FnFn−1r_n = \frac{F_n^2}{F_{n-1} \cdot F_n} = \frac{F_n}{F_{n-1}}r_n = \frac{F_n^2}{F_{n-1} \cdot F_n} = \frac{F_n}{F_{n-1}}
Thus, rn+1=Fn+1Fnr_{n+1} = \frac{F_{n+1}}{F_n}r_{n+1} = \frac{F_{n+1}}{F_n}, which converges to ϕ\phi\phi as n→∞n \to \inftyn \to \infty. For F15=610F_{15} = 610F_{15} = 610 and F14=377F_{14} = 377F_{14} = 377, 610377≈1.618037\frac{610}{377} \approx 1.618037\frac{610}{377} \approx 1.618037; for F16=987F_{16} = 987F_{16} = 987 and F15=610F_{15} = 610F_{15} = 610
, 987610≈1.618033\frac{987}{610} \approx 1.618033\frac{987}{610} \approx 1.618033, confirming convergence.
The differences Δrn=Fn+1Fn−FnFn−1\Delta r_n = \frac{F_{n+1}}{F_n} - \frac{F_n}{F_{n-1}}\Delta r_n = \frac{F_{n+1}}{F_n} - \frac{F_n}{F_{n-1}} can be expressed using Cassini’s identity (Fn2−Fn+1Fn−1=(−1)nF_n^2 - F_{n+1} F_{n-1} = (-1)^nF_n^2 - F_{n+1} F_{n-1} = (-1)^n) as: Δrn=(−1)nFnFn−1\Delta r_n = \frac{(-1)^n}{F_n F_{n-1}}\Delta r_n = \frac{(-1)^n}{F_n F_{n-1}}
Summing from
n=3n = 3n = 3 to n=15n = 15n = 15
n=3n = 3n = 3: −12⋅1=−0.5\frac{-1}{2 \cdot 1} = -0.5\frac{-1}{2 \cdot 1} = -0.5
n=4n = 4n = 4: 13⋅2=0.166667\frac{1}{3 \cdot 2} = 0.166667\frac{1}{3 \cdot 2} = 0.166667
n=5n = 5n = 5: −15⋅3=−0.066667\frac{-1}{5 \cdot 3} = -0.066667\frac{-1}{5 \cdot 3} = -0.066667
Up to n=15n = 15n = 15: −1610⋅377≈−0.000004\frac{-1}{610 \cdot 377} \approx -0.000004\frac{-1}{610 \cdot 377} \approx -0.000004
The sum approximates 0.381966, matching 1−1ϕ1 - \frac{1}{\phi}1 - \frac{1}{\phi}.
Analysis
The convergence to ϕ\phi\phi is a direct consequence of the Fibonacci ratio property. The sum of differences approximating 1−1ϕ1 - \frac{1}{\phi}1 - \frac{1}{\phi} arises from the alternating series ∑(−1)nFnFn−1\sum \frac{(-1)^n}{F_n F_{n-1}}\sum \frac{(-1)^n}{F_n F_{n-1}}, where denominators grow as ϕ2n\phi^{2n}\phi^{2n}, reflecting ϕ\phi\phi’s self-similar properties. This links to the golden angle (360⋅(1−1ϕ)360 \cdot (1 - \frac{1}{\phi})360 \cdot (1 - \frac{1}{\phi})).
Discussion
While the method reduces to Fn+1Fn\frac{F_{n+1}}{F_n}\frac{F_{n+1}}{F_n}, the focus on squares and the difference sum offers a novel perspective. The result is not coincidental but a mathematical consequence of Fibonacci’s structure, encoding ϕ\phi\phi and its conjugate 1−1ϕ1 - \frac{1}{\phi}1 - \frac{1}{\phi}.
Conclusion
This approach provides an indirect yet valid approximation of ϕ\phi\phi, with the sum of differences revealing a connection to the golden angle. Though rooted in known Fibonacci identities, the specific formulation highlights the sequence’s deep geometric significance. Further exploration could assess its computational utility.
Supporting Data
A spreadsheet (attached) confirms r16≈1.618034r_{16} \approx 1.618034r_{16} \approx 1.618034 and the sum of differences from n=3n = 3n = 3 to n=16n = 16n = 16 as 0.381966.
