33 research outputs found
Inextensible domains
We develop a theory of planar, origin-symmetric, convex domains that are
inextensible with respect to lattice covering, that is, domains such that
augmenting them in any way allows fewer domains to cover the same area. We show
that origin-symmetric inextensible domains are exactly the origin-symmetric
convex domains with a circle of outer billiard triangles. We address a
conjecture by Genin and Tabachnikov about convex domains, not necessarily
symmetric, with a circle of outer billiard triangles, and show that it follows
immediately from a result of Sas.Comment: Final submitted manuscript. Geometriae Dedicata, 201
Marginal stability in jammed packings: quasicontacts and weak contacts
Maximally random jammed (MRJ) sphere packing is a prototypical example of a
system naturally poised at the margin between underconstraint and
overconstraint. This marginal stability has traditionally been understood in
terms of isostaticity, the equality of the number of mechanical contacts and
the number of degrees of freedom. Quasicontacts, pairs of spheres on the verge
of coming in contact, are irrelevant for static stability, but they come into
play when considering dynamic stability, as does the distribution of contact
forces. We show that the effects of marginal dynamic stability, as manifested
in the distributions of quasicontacts and weak contacts, are consequential and
nontrivial. We study these ideas first in the context of MRJ packing of
d-dimensional spheres, where we show that the abundance of quasicontacts grows
at a faster rate than that of contacts. We reexamine a calculation of Jin et
al. (Phys. Rev. E 82, 051126, 2010), where quasicontacts were originally
neglected, and we explore the effect of their inclusion in the calculation.
This analysis yields an estimate of the asymptotic behavior of the packing
density in high dimensions. We argue that this estimate should be reinterpreted
as a lower bound. The latter part of the paper is devoted to Bravais lattice
packings that possess the minimum number of contacts to maintain mechanical
stability. We show that quasicontacts play an even more important role in these
packings. We also show that jammed lattices are a useful setting for studying
the Edwards ensemble, which weights each mechanically stable configuration
equally and does not account for dynamics. This ansatz fails to predict the
power-law distribution of near-zero contact forces, .Comment: final submitted versio