A digital distance on the kisrhombille tiling.

Area of Science:

• Geometry
• Tessellation Theory
• Discrete Geometry

Background:

• The kisrhombille tiling is a semi-regular tessellation dual.
• It comprises right-angled triangle tiles with 12 orientations.
• Understanding neighbor relationships and paths is crucial for grid analysis.

Purpose of the Study:

• To define a coordinate system for the kisrhombille tiling.
• To introduce and prove a digital distance formula for tiles.
• To analyze the relationship between digital and hexagonal distances.

Main Methods:

• Defined a coordinate system for formal grid description.
• Established neighbor criteria based on shared boundaries.
• Constructed minimum paths to define digital distance.
• Derived and proved the digital distance formula.

Main Results:

• A formal coordinate system for the kisrhombille grid was established.
• The concept of digital distance between tiles was defined.
• The distance formula was proven through path construction.
• Digital distance is approximately twice the hexagonal distance of embedding hexagons.

Conclusions:

• The defined coordinate system facilitates formal analysis of the kisrhombille tiling.
• The digital distance provides a novel metric for tile proximity.
• The relationship between digital and hexagonal distances offers insights into the tiling's geometric properties.