This website provides access to data for the work done during and after the Summer Program for Undergraduate Research 2014 at Cornell University in the Analysis on Fractals Group.
We construct a surface that is obtained from the octahedron by pushing out 4 of the faces so that the curvature is supported in a copy of the Sierpinski gasket in each of them, and is essentially the self similar measure on SG. We then compute the bottom of the spectrum of the associated Laplacian using the ?nite element method on polyhedral approximations of our surface, and speculate on the behavior of the entire spectrum.
The current version of our writedown is here: Paper
An earlier version of this website can be found here: Website
Spectra
We computed the spectrum of the polyhedron level 1-7 without subdivisions. The multiplicities are given in brackets after the respective eigenvalue. The extrapolation refers to the multiplicities of level 7.
To understand the eigenfunctions, we subdivided each edge 10 times and produced a quality mesh. To put the 4 SG faces and 4 ?at faces together, we will need make identi?cations inside the SG-faces to stich up the small slits and also identify the sides of SG-faces and ?at faces. For level 3, this is shown here, where two vertices are identi?ed when they are connected by a blue line. This is done in a way that the 2*2 left faces are rotated by 90° to the right and put ”on top” of the 2*2 faces on the right.
We compute the first 500 eigenfunctions of level 1 and 2.
Iancu Dima, Ithaca College, Department of Mathematics, idima@ithaca.com
Rachel Popp, The Graduate Center, CUNY, rachel.popp@gmail.com
Robert Strichartz, Cornell University, Department of Mathematics, str@math.cornell.edu
Samuel Wiese, University of Leipzig, Department of Mathematics, sw31hiqa@studserv.uni-leipzig.de
