GOLDEN ratio.yml

[akhilsmokie7-cloud]

pi-golden-ratio/
├── main.py
├── requirements.txt
└── README.md

Main.py
import numpy as np
import matplotlib.pyplot as plt
import argparse

def golden_rectangle_plot(phi, pi_val, width=1):
height = width / phi
fig, ax = plt.subplots()
# Main rectangle
rect = plt.Rectangle((0, 0), width, height, fill=None, edgecolor='gold', linewidth=2)
ax.add_patch(rect)
# Square inside
square = plt.Rectangle((0, 0), height, height, fill=None, edgecolor='blue', linewidth=2)
ax.add_patch(square)
# Arc using pi
arc = plt.Arc((height, height), 2height, 2height, angle=0, theta1=0, theta2=180, edgecolor='red', linewidth=2)
ax.add_patch(arc)
ax.set_xlim(-0.2, width+0.2)
ax.set_ylim(-0.2, height+0.2)
ax.set_aspect('equal')
plt.title(f'Golden Rectangle (φ ≈ {phi:.5f}), Arc (π ≈ {pi_val:.5f})')
plt.show()

def fibonacci_spiral_plot(n_terms=10):
phi = (1 + np.sqrt(5)) / 2
a, b = 1, 1
x, y = 0, 0
angle = 0
pts = [(x, y)]
for _ in range(n_terms):
x += np.cos(np.deg2rad(angle)) * b
y += np.sin(np.deg2rad(angle)) * b
pts.append((x, y))
a, b = b, a + b
angle += 90
pts = np.array(pts)
plt.plot(pts[:,0], pts[:,1], marker='o')
plt.title("Fibonacci Spiral (approaches golden ratio)")
plt.axis('equal')
plt.show()

def main():
parser = argparse.ArgumentParser(description='π and Golden Ratio Visualizations')
parser.add_argument('--rectangle', action='store_true', help='Plot golden rectangle with arc using π')
parser.add_argument('--spiral', action='store_true', help='Plot Fibonacci spiral')
parser.add_argument('--phi', type=float, default=(1 + np.sqrt(5)) / 2, help='Golden ratio value (default φ)')
parser.add_argument('--pi', type=float, default=np.pi, help='Pi value (default π)')
parser.add_argument('--terms', type=int, default=10, help='Terms in Fibonacci spiral')
args = parser.parse_args()

if args.rectangle:
    golden_rectangle_plot(args.phi, args.pi)
if args.spiral:
    fibonacci_spiral_plot(args.terms)

if name == "main":
main()
numpy
matplotlib

π and Golden Ratio (φ) Exploration

This repository demonstrates mathematical and geometric relationships between π (pi ≈ 3.14159) and the golden ratio (φ ≈ 1.61803) using Python visualizations.

Features

• Golden Rectangle Visualization: Plots a golden rectangle and overlays an arc using π.
• Fibonacci Spiral Visualization: Plots a spiral built from Fibonacci numbers, which approximates the golden ratio.
• Command Line Input: Select which visualizations to run and customize constants.

Usage

1. Install requirements:
2. Plot a golden rectangle with a π arc:
3. Plot a Fibonacci spiral:
4. Customize parameters: ## Further Ideas

• Add number theory demonstrations
• Explore pentagons and other golden ratio shapes
• Animate transitions or combine both visualizations

License

MIT License
Here are resources with transcendence proofs:

• A concise exposition for π:
  Simple proofs: Pi is transcendental « Math Scholar

• Mathematical StackExchange discussion and references: Proof π is transcendental without symmetric function theory

• The Stacks Project on transcendence bases, including π:
  Section 9.26: Transcendence—The Stacks project

The golden ratio φ is not transcendental (it is algebraic: the root of x² − x − 1 = 0), so proofs of transcendence are only relevant for π and similar numbers.

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1. Simple proofs: Pi is transcendental « Math Scholar
2. Proof π is transcendental without symmetric function theory
3. Section 9.26: Transcendence—The Stacks project