Generating the P3 Tiling

summary

Penrose P3 tilings, which are aperiodic patterns using two rhombii, can be generated through a recursive subdivision process using Robinson triangles. By defining specific subdivision rules and implementing them as a term rewriting system, these patterns can be created programmatically. The process is further refined by using a coordinate system based on the golden ratio and complex roots of unity to avoid floating-point errors. This approach ensures mathematical precision while maintaining the structural properties of the tiling.

key points

1. The P3 tiling is created by recursively subdividing four types of Robinson triangles, which eventually pair up to form the required rhombii.
2. The subdivision rules can be modeled as a term rewriting system where triangles are treated as function symbols and new vertices are calculated using the golden ratio.
3. A coordinate system using integer coefficients of a four-vector basis allows for precise vertex representation and avoids the imprecision of floating-point arithmetic.

who should read it

Programmers and mathematics enthusiasts interested in computational geometry, tiling patterns, or the implementation of recursive algorithms.

discussion positive