2025-10-07

We want to create a non-trivial codimension 2 example in 4-D. Instead of starting with a mesh in 4-D and finding a 2-complex as input, we can do the reverse. That is, start with the 2-complex given as a mesh in 3-D, and embed the mesh into a 4-D cube.

Embedding a 3-D mesh in 4-D is quite straight forward. We can either set the 4th dimension $w$ to a constant value, or use some function of $x,y,z$.

The problem is that we need the triangles of the embedded 3-D mesh to correspond to triangles in the 4-D cube complex. The following is my strategy, which may or may not work.

• Start with a 3-D mesh (for example a torus)
• Embed the 3-D mesh into ${\mathbb{R}}^4$ by a linear combination $w=\alpha x+\beta y+\gamma z$ such that $x,y,z,w\in [0,1]$
• Create a uniform grid of points in $[0,1{]}^4$
  • Exclude all points within $ϵ$ units away from a vertex of the torus
• Take the $n$-d Delaunay complex using gudhi of scipy

Alternative strategy:

• Start with a 3-D mesh (for example a torus)
• Embed the 3-D mesh into ${\mathbb{R}}^4$ by a linear combination $w=\alpha x+\beta y+\gamma z$ such that $x,y,z,w\in [0,1]$
• For each triangle $[a,b,c]$, construct a tetrahedra $[a,b,c,d]$ by adding another vertex $d$ along the triangle’s normal vector
  • Requires transforming the normal vector during embedding
  • Perhaps also add in opposite normal direction to fill in the void of torus
• Fill the remaining of the space with grid of points in $[0,1{]}^4$ as before
• Take the $n$-d Delaunay complex

Strategy doesn’t appear to work.

New strategy:

• Start with the same method as before
• For each triangle in the mesh
  • Map the vertices to the closest vertex in the tetrahedralization
  • Take the new triangle formed

Concern is if the new triangle is not a valid triangle in the complex. Perhaps I could take a collection of triangles which make up the triangle?