= Transplantation theorem
{title2=$CM_s=N_sC$}
For two assemblies of congruent Euclidean tiles, let $M_s,N_s$ encode the gluing or boundary reflection at each labelled face. An invertible constant matrix $C$ satisfying $CM_s=N_sC$ carries tile restrictions of <Laplacian eigenfunctions> bijectively to those on the second assembly. Boundary values satisfy $M_su=u$ and outward <normal derivatives> satisfy $M_sv=-v$, so the intertwining identities preserve matching and <boundary conditions>. Use diagonal $-1$ for a <Dirichlet boundary condition> and $+1$ for a <Neumann boundary condition>.
Back to article page
