Symmetry Information
Solid: Tetrahedron
Symmetry Group: Td
Faces: 4
Edges: 6
Vertices: 4
Gaussian Curvature: Positive
What is Holonomy?
When you parallel-transport a vector around a closed loop on a curved surface, it may not return to its original orientation. The angle between the start and end vectors is called the holonomy.
Holonomy = ∫loop K dA
Where K is Gaussian curvature and A is area. On positively curved surfaces (like spheres), vectors rotate clockwise. On negatively curved surfaces (like saddles), they rotate counterclockwise.
Key Concepts:
- Parallel Transport: Moving a vector while keeping it parallel to itself
- Closed Loop: A path that returns to its starting point
- Holonomy Gap: The difference between start and end orientations
- Gaussian Curvature: Intrinsic curvature of the surface
Platonic Solids:
These five regular polyhedra exhibit discrete holonomy due to their angular defects at vertices. The holonomy relates to the symmetry group and the total curvature enclosed by the loop.