  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.
