Core Principles
Recursion in λ-calculus tiling allows generation of self-similar patterns through fixed-point combinators. This research explores mathematical limits of pattern recurrence in both spatial and computational domains.
- Fixed-point combinators as pattern generators
- Bounded recursion with geometric constraints
- Computational limits of recursive tiling
Theoretical Foundation
Recursion in geometric tiling follows strict reduction rules while maintaining spatial coherence. These patterns rely on fixed-point combinators to achieve infinite generation within finite spaces.
Y = λf.(λx.f(x x)) (λx.f(x x))
pattern = Y(λself. λn. if n=0 then □ else rotate(45°) self(n-1))
Live Recursion
Formal Analysis of Recursion
Type Constraints
Recursive patterns must conform to geometric continuity proofs to ensure pattern coherence.
if n=0 then Square
else Rotate(Recursive(n-1))
Computational Limits
The halting condition for all recursive tiling expressions is mathematically provable through strong normalization proofs.
Key Theorem
The Curry-Howard correspondence in our recursive tiling system demonstrates how every valid λ-exression forms a spatial proof in the geometry of recursion.
Want to Experiment?
Our experimental sandbox environment allows researchers to test recursion limits with mathematical guarantees of spatial coherence.
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