Penrose Tilings as Jammed Solids

TL;DR

This paper investigates the elastic and vibrational properties of Penrose tilings and their approximants, revealing similarities to jammed particulate systems, including zero modes, vanishing elastic moduli, and flat density of states.

Contribution

It introduces a detailed analysis of Penrose tilings' elastic behavior, connecting their properties to jammed systems and exploring effects of approximant size and randomness.

Findings

Approximants have order √N_S zero modes and states of self stress.

All elastic moduli of approximants vanish.

Randomized Penrose tilings resemble jammed particulate systems with flat density of states.

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.