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Anyone able to explain this and how to kill it? New to Blender. Not sure of any terminology otherwise I'd try to search for my own answer. I even tried ⎈ CtrlF to create new faces hoping to eliminate this problem.

weird connections between vertices

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This is because you have a non-planar ngon, i.e. a non-flat face with more than four sides. This is particularly problematic with an ngon as blender has no indication of what the topology should be inside the face, resulting in ambiguous geometry. For display purposes there must be something there, so it takes a guess.

You should try to avoid such faces, as they will not behave well with subsurf or deformations (to name a few issues).

To fix this, you can either

  • Delete the face and fill in the hole with planar quads. Grid fill might be of help here.

  • Triangulate the face (⎈ CtrlT), splitting the large ngon into multiple planar triangles:

    enter image description here

    You could also try Mesh > Cleanup > Split non-planar faces.

    Note that this still won't subdivide or deform nicely.

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