If a surface is a 2D manifold then it satisfies Euler's polyhedron formula. Does this mean that all polyhedra are 2D manifolds? Prof Ren seems to add the last bullet as an interesting afterthought, but I wonder if the polyhedron formula can actually be useful in constructing meshes.

hfan9

Wikipedia says that polyhedron are solids in 3D. Since manifolds are 2D, I don't think you can equate them. However, the last bullet notes that the formula holds when the surface (the 2D manifold) is topologically equivalent to a sphere, which I take to basically mean that the surface is closed (no boundaries, and it doesn't look like a ring). Then this manifold would be the surface area of a polyhedron, which is why the formula holds. The polyhedron itself is the 3D solid, but if you only looked at its 2D surface, you would get the manifold.

If a surface is a 2D manifold then it satisfies Euler's polyhedron formula. Does this mean that all polyhedra are 2D manifolds? Prof Ren seems to add the last bullet as an interesting afterthought, but I wonder if the polyhedron formula can actually be useful in constructing meshes.

Wikipedia says that polyhedron are solids in 3D. Since manifolds are 2D, I don't think you can equate them. However, the last bullet notes that the formula holds when the surface (the 2D manifold) is topologically equivalent to a sphere, which I take to basically mean that the surface is closed (no boundaries, and it doesn't look like a ring). Then this manifold would be the surface area of a polyhedron, which is why the formula holds. The polyhedron itself is the 3D solid, but if you only looked at its 2D surface, you would get the manifold.