Boolean Tile Basis

GitHub Pages site: https://pstdenis.github.io/Boolean-Tile/

The 16 Boolean functions of two variables — the "Boolean tiles" — form a complete algebraic, geometric, and physical system. This project traces them from truth tables through Łukasiewicz logic, iterated function system (IFS) fractals, Bloch-sphere quantum states, 2-qubit unitary gates, and Ising spin Hamiltonians.

Repository structure

• docs/ — GitHub Pages site with the full paper and interactive visualizations
• docs/Boolean Tile Basis.html — The working paper
• docs/Polar IFS.html — 2D polar IFS attractor visualization (greyscale annular rings)
• docs/Bloch.html — 3D Bloch-sphere spherical visualization of the 16 tiles
• docs/ — Additional interactive HTML demos

Key idea

Each 2×2 truth table (tile) defines a carry-free bitwise logic that, when iterated to infinite depth, produces a self-similar fractal attractor. Under $H^{\otimes 2}$ conjugation, the diagonal operator $D_f = \operatorname{diag}(f(00), f(01), f(10), f(11))$ becomes a real orthogonal 4×4 matrix $U_f^{(X)}$ — a genuine 2-qubit quantum gate whose generating Hamiltonian is a physical transverse Ising model:

$$U_f^{(X)} = \exp!\bigl(-i,(\tfrac{\pi}{2}I - \tfrac{\pi}{2}\sum_{S\neq\emptyset} \hat f(S) X_S)\bigr)$$

Chain of layers

1. Boolean — 16 truth tables
2. Continuous — Łukasiewicz (MV-algebra) extension
3. Complex — Phase-preserving near-MV-algebra on the disk
4. Fractal — IFS attractor (chaos game)
5. Quantum — $H^{\otimes2}$-conjugated unitary gates
6. Temporal — Depth iteration as discrete time
7. Many-worlds — Branching measure interpretation
8. Multi-qubit — $n$-register depth process
9. Physical — Ising spin Hamiltonians

Related work

The IFS/liear chaos-game approach to logical systems originates with Grim, Mar, and St. Denis (The Philosophical Computer, MIT Press 1998). This project extends their framework to the Bloch sphere, to quantum gate synthesis, and to physical Hamiltonian realizations.

License

All materials are provided for research and educational purposes.