- The paper establishes, via an exhaustive algorithmic search, that no new convex pentagon tiling families exist beyond the known 15.
- The study employs combinatorial backtracking and geometric constraints, validating 371 angle condition sets for feasible tiling.
- The research confirms that convex pentagon tilings only enable periodic arrangements, decisively closing a longstanding tiling problem.
Exhaustive Search of Convex Pentagons that Tile the Plane
This paper, authored by Michaël Rao, addresses the mathematical problem of determining all the families of convex pentagons capable of tiling the plane. The problem is deeply rooted in the area of geometric tiling, wherein the goal is to explore which shapes can completely cover a plane without overlaps or gaps. Historically, this question has been resolved in cases involving triangles, quadrilaterals, and hexagons, but convex pentagons had remained an open problem. The paper asserts that no new families beyond the previously known 15 exist, marking a significant achievement in tiling theory.
Main Contributions
The primary contribution of this paper is conducting a complete and exhaustive search to identify all existing convex pentagon families capable of tiling the plane:
- Completion of an Exhaustive Search: The research carried out a systematic search, confirming that beyond the 15 well-documented families of pentagons, no additional families exist.
- Exclusion of Non-Periodic Tilings: The findings convincingly demonstrate that there is no convex pentagon that permits only non-periodic tilings.
- Algorithmic Approach: The analysis relies heavily on the use of combinatorial and algorithmic methods to delineate conditions under which convex pentagons tile the plane. The search methodology applies backtracking and constraints to explore potential tiling scenarios.
Theoretical Framework
The research builds on a rigorous mathematical framework:
- Notation and Definitions: The paper establishes specific notations, such as defining vertex types and introducing the concept of positive density for tilings.
- Good Sets and Compatibility: It introduces the concept of "good sets" of vertex vectors and compatible vectors, crucial in deducing the vertex types of a tiling.
- Geometric Constraints: It further utilizes geometric constraints and linear programming to manage and check possible configurations of tiles.
Numerical and Analytical Results
- Verification of Known Families: The exhaustive search validated all known families, including Reinhardt’s and Kershner’s pioneering work, and other subsequent discoveries by researchers like R. James, M. Rice, and R. Stein.
- 371 Angle Condition Sets: The research identified 371 complete angle condition sets, which were reduced through rigorous computation and mirrored permutation analysis to confirm those convex pentagons without discovering new ones.
Implications and Future Research
- Geometric Understanding: The paper enhances geometric understanding by affirming that any convex pentagon tiling scenario has been comprehensively covered with these families.
- Algorithmic Development: The methodologies and algorithms developed could be beneficial in other geometric and combinatorial problems in computational mathematics.
- Exploration Beyond Convex Polygons: Research could extend into non-convex polygons or higher dimensions, exploring further geometric tiling configurations.
This paper presents a robust mathematical approach to resolving the problem of tiling the plane with convex pentagons. While the exhaustive search adds no new families to the existing 15, it eliminates uncertainty, providing a definitive closure to the problem within its stipulated constraints. Further application of this research can benefit broader tiling theory and computational geometry analyses.
