Penrose tilings are non-periodic tilings discovered by mathematician and physicist Sir Roger Penrose in the 1970s. Unlike regular tilings (like square or hexagonal grids), Penrose tilings never repeat in a translational sense but exhibit local fivefold rotational symmetry and quasiperiodic order.

Penrose developed different types of tilings, with kite and dart tiling being one of the earliest and most visually iconic forms.

These tilings became important in mathematics, physics, and crystallography (e.g., explaining structures of quasicrystals), and were also a major influence in art (e.g., M.C. Escher's later work).

📖 Reference: R. Penrose, "Pentaplexity: A class of non-periodic tilings of the plane", The Mathematical Intelligencer, 1979. DOI: 10.1007/BF03026814

This version of the program generates Penrose tilings using iterative subdivision of acute (thin) and obtuse (thick) golden triangles based on Penrose’s and Robinson’s substitution rules. It is mathematically equivalent to recursive implementations, but uses an iterative stack-based approach to avoid Python recursion limits.

• Fixed Initial Pattern: Uses a symmetrical 10-triangle star (decagonal) as the seed.
• Triangle-Based Only: All tiles are golden triangles — either acute or obtuse.
• Iterative Subdivision: Substitution rules are applied using a controlled iteration stack (avoiding recursion depth errors).
• Implicit Matching Rules: No decorative arcs needed — aperiodicity and matching logic are encoded in the subdivision rules themselves.

🔁 Note:
While Penrose triangle tilings are defined recursively, this implementation uses a stack to simulate recursion.
This method is widely used in computational geometry and does not alter the mathematical correctness of the tiling.

• Color Modes:
  • mono: Grayscale mode (all triangles the same).
  • type: Colors based on triangle type (acute or obtuse).
  • color: Colors based on triangle orientation (reveals 5-fold symmetry).
• Recursion Depth Control: Set recursion depth (3–6 recommended).
• Export Options: Save the resulting tiling as .png or .svg.

Upon running, you’ll be prompted to:

1. Enter recursion depth (e.g., 4).
2. Select a color mode (mono/type/color).
3. Optionally save the output (e.g., tiling.png).

📝 This version offers a cleaner geometric representation and is particularly useful for studying the underlying substitution logic of Penrose tilings without the visual clutter of explicit matching rule enforcement.