Penrose Tilings as Jammed Solids

Abstract

Penrose tilings form lattices, exhibiting 5-fold symmetry and isotropic elasticity, with inhomogeneous coordination much like that of the force networks in jammed systems. Under periodic boundary conditions, their average coordination is exactly four. We study the elastic and vibrational properties of rational approximants to these lattices as a function of unit-cell size $N_S$ and find that they have of order $\sqrt{N_S}$ zero modes and states of self stress and yet all their elastic moduli vanish. In their generic form obtained by randomizing site positions, their elastic and vibrational properties are similar to those of particulate systems at jamming with a nonzero bulk modulus, vanishing shear modulus, and a flat density of states.

5 pages, 4 figures

References in corpus (6)

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• Mechanics of fiber networks under a bulk strain
• Configuration space partitioning in tilings of a bounded region of the plane
• Unearthing the anticrystal: criticality in the linear response of disordered solids