Mathematics and Nature

Overview

This hobby project presents a captivating exploration of the intersection between mathematics and nature. Through a collection of Jupyter notebooks, it demonstrates how mathematical equations and physical principles can model, simulate, and visualize natural structures and phenomena — from the delicate geometry of flower petals to the wave-like behaviour of quantum particles.

Visualizations

🪷 Lotus Flower (Lotus_Plot.ipynb)

Models the structure of a lotus flower using three layered polar equations. The animation sequentially draws each petal layer with a tracing dot and guide line, revealing how simple trigonometric combinations produce organic, flower-like shapes.

• Equations: Three polar functions (r1, r2, r3) built from combinations of cos, abs, and rational expressions, each offset by a separation_factor to create concentric petal rings.
• Animation: 1500-frame FuncAnimation at 30 fps (~50 s), rendered as an HTML5 inline video optimized for 9:16 (portrait) aspect ratio.
• Dependencies: numpy, matplotlib, IPython.display
• Environment: Google Colab (uses google.colab.drive for drive mounting).

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🦋 Butterfly Curve (Butterfly_Equations.ipynb)

Visualizes the classic butterfly curve — a parametric equation whose graph resembles the wings of a butterfly. A white tracing dot follows the curve as it is drawn.

• Technique: matplotlib.animation.FuncAnimation renders the curve frame-by-frame, saving the result as an embedded HTML5 video (800 × 800 resolution).
• Known Issue: The notebook output contains MatplotlibDeprecationWarning messages related to set_data being called with non-sequence types (deprecated since matplotlib 3.7). This should be addressed in a future update to ensure forward compatibility.
• Dependencies: numpy, matplotlib, IPython.display
• Environment: Google Colab.

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🌹 3D Rose (Equations_of_Rose.ipynb)

Constructs a photorealistic 3D rose using a parametric surface equation. The rose "grows" petal-by-petal via an animated rotation, rendered against a black background with a custom RdPu (red-purple) colormap and gold edge highlights.

• Equation: A single rose(x, theta) function returning (X, Y, Z) coordinates, built from exponential decay (phi), polynomial shaping, and angular modulation.
• Parameters: Fully configurable global constants — THETA_POINTS, GRID_FRAMES, EXTRA_ROTATION_FRAMES, ROTATION_SPEED, FPS, DPI, BITRATE, etc.
• Output: Saved directly to rose_animation.mp4 via FFMpegWriter.
• Dependencies: numpy, matplotlib (including mpl_toolkits.mplot3d, matplotlib.animation, matplotlib.colors)
• System Requirement: ffmpeg must be installed and available on the system PATH.
• Environment: Google Colab.

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🔬 Young's Double-Slit Experiment (Youngs_Double_Slit.ipynb)

Simulates the quantum-mechanical double-slit experiment for a single electron, showing how a localized wave-packet diffracts through two narrow slits and produces the iconic interference pattern.

• Physics: Uses the qmsolve library to build a Hamiltonian for a single particle (m = m_e) in a 2D potential well containing a double slit, then runs a split-step time simulation.
• Simulation Parameters (configurable):

  Parameter	Value
  Slit separation	2.0 Å
  Slit width	0.7 Å
  Slit depth	0.5 Å
  Initial wavefunction σ	1.0 Å
  Initial velocity	80 Å/fs
  Total simulation time	0.7 fs
  Grid size (N)	512
  Stored steps	801

• Visualization: Each frame maps the complex wavefunction Ψ(x, y, t) to RGBA colour via HSV conversion, overlaid on a log-scaled potential image. Saved as double_slit_simulation_reel.mp4 (9:16 portrait, 300 dpi, 30 fps).
• Dependencies: numpy, matplotlib, qmsolve
• System Requirement: ffmpeg must be installed and available on the system PATH.
• Environment: Google Colab (install qmsolve via !pip install qmsolve).

Getting Started

Prerequisites

• Python 3.8+
• Google Colab (recommended) — all notebooks are authored and tested in Colab.
• Alternatively, run locally with the following installed:
  • numpy
  • matplotlib
  • ffmpeg (system package — required by Equations_of_Rose.ipynb and Youngs_Double_Slit.ipynb)
  • qmsolve (required only by Youngs_Double_Slit.ipynb; install via pip install qmsolve)

Running

1. Open any .ipynb file in Google Colab (or Jupyter).
2. Run all cells sequentially.
3. Animations will either render inline (HTML5 video) or be saved as .mp4 files in the working directory.

Repository Structure

Mathematics_And_Nature/
├── Butterfly_Equations.ipynb   # Butterfly curve animation
├── Butterfly_Equations.jpeg    # Example output image
├── Equations_of_Rose.ipynb     # 3D rose surface animation
├── Lotus_Plot.ipynb            # Lotus flower polar-plot animation
├── Lotus_Plot.jpeg             # Example output image
├── Youngs_Double_Slit.ipynb    # Quantum double-slit simulation
├── LICENSE                     # MIT License
├── .gitignore                  # Standard Python/Jupyter gitignore
└── README.md                   # This file

Contributing

Contributions to enhance the Mathematics and Nature visualizations are welcome. Whether it's improving the equations, refining the animations, or fixing bugs, your input is valuable. Please follow these steps to contribute:

1. Fork the repository.
2. Create a new branch for your feature (git checkout -b feature/AmazingFeature).
3. Commit your changes (git commit -m 'Add some AmazingFeature').
4. Push to the branch (git push origin feature/AmazingFeature).
5. Open a pull request.

License

Distributed under the MIT License. See LICENSE for more information.