the discrete Hodge star
In smooth geometry the Hodge star is the operator that knows about lengths and angles: it turns a k-form into the complementary (n-k)-form by using the metric, and it is what lets you take divergences, define the Laplacian, and measure the size of forms. The discrete Hodge star is its mesh counterpart, and it is the single place in discrete exterior calculus where geometry — not just connectivity — enters. Everything else (the exterior derivative as coboundary) is pure combinatorics; the star injects edge lengths and areas.
Mechanically, the discrete Hodge star maps a primal k-form, stored on the k-simplices of the mesh, to a dual (n-k)-form, stored on the cells of the dual mesh (the dual of an edge is the segment joining circumcenters of its adjacent triangles, and so on). The simplest version is the diagonal circumcentric (Hodge-DEC) star: the value on a primal k-cell is rescaled by the ratio of the dual cell's volume to the primal cell's volume. For a surface, the 1-form star carries exactly the cotangent weights — *_1 on edge (i,j) is (cot(alpha_ij) + cot(beta_ij)) / 2 — which is why the cotangent Laplacian factors as delta = -*_0^{-1} d^T *_1 d. A more accurate but non-diagonal option is the Galerkin/Whitney star, an honest mass matrix from interpolating Whitney forms, which sacrifices diagonality for better convergence.
The discrete Hodge star matters because it is what completes the discrete de Rham picture into a usable computational tool: composing d, the star, and d again builds discrete Laplacians, discrete codifferentials, and the discrete Hodge decomposition, underpinning everything from electromagnetic solvers to vector-field design. The honest pitfall is sharp: the diagonal circumcentric star can fail to be positive definite when a triangle's circumcenter lies outside it (an obtuse triangle), producing negative diagonal entries; this is exactly the obtuse-triangle problem behind non-positive cotangent weights, and it is why Delaunay or intrinsically-Delaunay meshes are preferred. Choosing the star is a genuine modeling decision, not a forced one.
For a triangulated surface, assembling d (vertex->edge incidence) with the diagonal 1-form Hodge star whose edge weight is (cot(alpha) + cot(beta))/2 reproduces exactly the cotangent Laplacian L = d^T * d. If a triangle is very obtuse, cot of its large angle is negative, the star weight can go negative, and L loses the maximum principle — the classic warning sign that the mesh should be made Delaunay.
The 1-form Hodge star carries the cotangent weights; obtuse triangles can make those weights negative.
The diagonal circumcentric star is only positive definite for well-centered (each circumcenter inside its simplex) or Delaunay meshes; on obtuse triangles its weights go negative and the resulting operators lose positivity. The Galerkin/Whitney star avoids this at the cost of being non-diagonal and more expensive.