On the Exact Algorithmic Extraction of Finite Tesselations Through Prime Extraction of Minimal Representative Forms

TL;DR

This paper introduces an exact, hierarchical algorithm for detecting and extracting finite tessellations in discrete grids, enabling precise pattern recognition in symbolic and computational domains.

Contribution

It presents a novel deterministic method combining composite discovery, normalization, and prime extraction to efficiently identify exact rectangular tessellations in finite grids.

Findings

Processing time under 1ms for simple tiles

Exponential growth in complexity for complex patterns

Scalable to grid sizes up to 32x32

Abstract

The identification of repeating patterns in discrete grids is rudimentary within symbolic reasoning, algorithm synthesis and structural optimization across diverse computational domains. Although statistical approaches targeting noisy data can approximately recognize patterns, symbolic analysis utilizing deterministic extraction of periodic structures is underdeveloped. This paper aims to fill this gap by employing a hierarchical algorithm that discovers exact tessellations in finite planar grids, addressing the problem where multiple independent patterns may coexist within a hierarchical structure. The proposed method utilizes composite discovery (dual inspection and breadth-first pruning) for identifying rectangular regions with internal repetition, normalization to a minimal representative form, and prime extraction (selective duplication and hierarchical memoization) to account for irregular dimensions and to achieve efficient computation time. We evaluate scalability on grid sizes from 2x2 to 32x32, showing overlap detection on simple repeating tiles exhibits processing time under 1ms, while complex patterns which require exhaustive search and systematic exploration shows exponential growth. This algorithm provides deterministic behavior for exact, axis-aligned, rectangular tessellations, addressing a critical gap in symbolic grid analysis techniques, applicable to puzzle solving reasoning tasks and identification of exact repeating structures in discrete symbolic domains.