Fibonacci_Golden_Ratio_Research_Poster.pptx

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1. Applications of Fibonacci Sequences and the Golden Ratio A Mathematical and Research Perspective Presented by: [Your Name(s)] B.M.S. College of Engineering, Department of Computer Applications

2. Introduction • The Fibonacci sequence follows a recursive formula: Fn = Fn-1 + Fn-2. • The Golden Ratio (φ ≈ 1.618) naturally emerges from Fibonacci numbers. • These principles appear in nature, art, finance, and computational sciences.

3. Objectives • Analyze Fibonacci patterns in nature, architecture, and financial systems. • Explore the mathematical properties linking Fibonacci and the Golden Ratio. • Investigate real-world applications in various scientific disciplines.

4. Mathematical Foundation • Fibonacci Sequence Formula: Fn = Fn-1 + Fn-2 • Golden Ratio: φ = (1 + √5) / 2 • Fibonacci Approximation: Fn ≈ φⁿ / √5 • Fibonacci retracement levels: 23.6%, 38.2%, 61.8% (used in finance).

5. Applications in Nature • Fibonacci spirals in sunflowers, pinecones, and snail shells. • Phyllotaxis: Leaf arrangements in plants optimize sunlight exposure. • DNA proportions and animal body structures align with φ.

6. Applications in Architecture & Art • The Parthenon and Taj Mahal follow the Golden Ratio. • Da Vinci’s Mona Lisa and the Vitruvian Man exhibit Fibonacci proportions. • The rule of thirds in photography is based on Fibonacci numbers.

7. Fibonacci in Finance • Stock market analysis uses Fibonacci retracement to predict price movements. • Key retracement levels (23.6%, 38.2%, 61.8%) identify support/resistance levels. • Traders use Fibonacci time zones to forecast market trends.

8. Applications in Computing • Fibonacci heaps improve efficiency in algorithmic data structures. • Neural networks use Fibonacci-based weight optimization. • Cryptographic systems leverage Fibonacci sequences for secure key generation.

9. Conclusion • Fibonacci sequences and the Golden Ratio govern patterns in various fields. • Their mathematical properties enhance problem-solving in nature, finance, and AI. • Future research may uncover deeper applications in advanced computing and machine learning.