# Chladni figures Barycentric refinement

Here are the Chladni figures of the 7 eigenfunctions of the Barycentric refinement G_{1}and the 25 eigenfunctions of the Barycentric refinement G

_{2}of the triangle K

_{3}. As defined in the Sard paper, the surfaces are in the Barycentric refinement. We are interested in these figures in order to understand the spectral gaps, the discontinuities of the spectral functions F. Like the gap occurring at x=1/2. See the preprint "The graph spectrum of barycentric refinements" for the math and mini blog. For the graph shown below, the eigenvalues are (rounded to one decimal digit) (0,0.82,0.82,1.8,1.8,2.7,2.8,3.4,3.4,3.4,3.6,3.6,5.6,5.9,5.9,6.2,6.2,6.3,6.3,6.8,6.8,7.5,7.5,7.6,13). In this case still, all of them can be expressed by radicals even so the expressions are complicated. Same pictures for G

_{4}, a polyhedron or a square.