Fundamental conical defects: The d-cone, its e-cone, and its p-cone

Area of Science:

• Geometry
• Topology
• Materials Science

Background:

• Surface disclinations are topological defects in materials.
• Understanding their geometry is crucial for material properties.
• Existing models often focus on equilibrium states.

Purpose of the Study:

• To develop a geometric method for analyzing surface disclinations.
• To relate angular deficits to fold line rotations.
• To distinguish primitive disclinations from composite ones.

Main Methods:

• Using paper card models to create discrete surface disclinations.
• Applying Gauss's mapping to describe vertex geometry.
• Analyzing vector diagrams of fold line rotations.

Main Results:

• A method to quantify angular excess/deficit using fold line rotations.
• Demonstrated a relationship between rotation area and angular deficit.
• Identified primitive disclinations and composite types like d-cones.

Conclusions:

• The geometric approach provides kinematic insights without equilibrium assumptions.
• Some disclinations are fundamental (primitive), while others are combinations.
• This method simplifies the understanding of complex disclination structures.