Digital distances on the 4-fold pentille tessellation

Area of Science:

• Geometry
• Tessellation Theory
• Computational Geometry

Background:

• The Cairo pattern is a specific type of pentagonal tessellation with unique neighbor properties.
• Previous research calculated path-based distances using complex six-tuple coordinates for side neighbors.
• A need exists for a simplified grid description and alternative distance metrics.

Purpose of the Study:

• To develop a simpler coordinate system for the Cairo pattern using coordinate pairs.
• To compute path-based distances on the Cairo pattern considering corner neighbors.
• To contribute to the understanding of periodic tessellations with convex pentagons.

Main Methods:

• Developing a novel coordinate pair system for addressing tiles in the Cairo pattern.
• Implementing algorithms to calculate path-based distances based on corner-neighbor criteria.
• Analyzing the properties of two specific classes of 8 periodic pentagonal tessellations.

Main Results:

• A simplified coordinate pair system for the Cairo pattern grid has been established.
• New distance calculations based on corner neighbors have been successfully computed.
• The findings offer a more accessible framework for analyzing this tessellation.

Conclusions:

• The simplified coordinate system enhances the usability of the Cairo pattern for analysis.
• The corner-neighbor distance metric provides a new perspective on spatial relationships within the tessellation.
• This work advances the study of pentagonal tessellations and their digital distance properties.