Joel Madly - Triangular mesh with fractal behaviour

 

A triangular mesh showing fractal behaviour

This triangular mesh is the accidental result of a dissertation on adaptive finite element method algorithms. The fractal behaviour was first observed in a stress test of a proprietary refinement algorithm, similar to Kobbelt's root 3 subdivision. 

The core idea of the refinement came from splitting triangles at their centre of mass. From there I needed to find suitable additional rules to keep the refinement from degenerating. Thus, for each triangle

  • If it is an acute triangle (at most right angled): split it at the centre of mass
  • If it is an obtuse triangle: refine the neighbour opposite its long edge until it is also an obtuse triangle, where the common edge is the longest in each triangle. Then remove the common edge and connect the non-shared vertices of the two triangles. 
  • If it is an obtuse triangle having the long edge on the boundary of the mesh, split the long edge into thirds, generating 3 new triangles.

While the resulting images have no practical benefits, they are aesthetically pleasing, so I continued creating variants, trying to understand why they arise. Unfortunately, their secret still eludes me. The sequence of meshes that generated this image can be seen here https://imgur.com/gallery/6CtC9ur, as well as a pleasant surprise, if one zooms in at the very centre of the image.

Joel Madly is a postgraduate student at Wadham College.