![]() ![]() Then we denote a( w, i) the angle value in the i th vertex of triangle w. Let w be the word that names the triangle and i= be the top, bottom left, and bottom right vertices of any such triangle, respectively. We can denote angle positions on a curved face uniquely by utelizing the usual SG notation. We attatch the final code for the FEM as well as plots of normalized eigenvalues, which will be used to approximate the spectrum of the laplacian of the SG-octahedron. We used the information in the next sections to build a Matlab code which implements the finite element method over different levels m of the SG-octahedron and further subdivision at each level. The images above show discrete approximations at levels m=1 in blue and m=2 in green of the SG-octahedron, with curvature distributed across the colored SG-faces and folding occuring on the golden flat-faces. At each level m of discrete approximation the curvature determines the angles which determine the side lengths up to a constant. We enforce the flat faces (faces with 0-curvature) to be equilateral triangles (this includes half of the faces on the original octahedron, and the interior triangles on the SG-faces). (3 m+1) vertices at each of the 4 SGs, giving each vertex a weight of 2π/3 m+1. This measure is a limit of discrete measures with weights at the 2. We refer to the surface we explored as the SG-octahedron and it is constructed by taking a regular octahedron and putting a standard SG measure on half the faces with total mass π. We began this REU with an interest in building a surface which is a limit of convex polyhedra so that the curvature is distributed according to a fractal measure of a total mass 4π. By the Gauss-Bonnet theorem, the sum of curvature over the vertices equals 4π. Necessarily reflect the views of the National Science Foundation.Ī convex polyhedron is a surface that is flat (curvature = 0) everywhere except at the vertices, where it is a positive multiple of a delta mass. * Any opinions, findings, and conclusions or recommendationsĮxpressed in this material are those of the authors and do not Program at Cornell University, grant-1156350 Through the Research Experience for Undergraduates Research support by the National Science Foundation Iancu Dima: Popp: advisor: Professor Robert S. Surfaces with Fractal Curvature (In progress) ![]()
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