Four sided domains in hierarchical space dividing patterns

Area of Science:

• Materials Science
• Geometry
• Physics

Background:

• Cracks in ceramic glazes form complex networks that partition surfaces.
• Surface patterns often follow rules seen in Voronoi tessellations or soap froths, where domains average six sides.

Purpose of the Study:

• To analyze the geometric properties of crack networks in ceramic glazes.
• To understand the formation process leading to the observed domain structure.
• To introduce a new concept, "geometrical charge," for analyzing hierarchical patterns.

Main Methods:

• Analysis of crack patterns in ceramic glazes.
• Comparison with established geometric models like Voronoi tessellations.
• Introduction of a novel topological descriptor, "geometrical charge."

Main Results:

• Ceramic glaze crack networks result in domains with an average of four sides.
• This contrasts with the average of six sides found in Voronoi tessellations and soap froths.
• The four-sided domain characteristic is linked to a hierarchical, non-reorganizing domain division process.

Conclusions:

• The average four-sided domain structure in ceramic cracks is a result of hierarchical formation.
• This finding deviates from typical space-filling patterns governed by Euler's theorem.
• The "geometrical charge" is a conserved quantity in such hierarchical crack network formation.