Planar Disk Filters and Real Chromatic Roots at Beraha Numbers

We construct simple plane triangulations whose chromatic polynomials have real roots accumulating at every noninteger Beraha number $B_\ell=4\cos^2(π/\ell)$; the integer values $0,1,2,3$ are roots of $K_4$. The construction rests on the observation that a triangulated disk with four boundary vertice