Bead-by-bead packing makes minimal connectivity
Each bead was placed at a stable point. (no friction)
Adding one bead adds 2 contacts
Number of contacts = 2N
Number of bead position variables (x, y) = 2N
Isostatic network:
contact constraints are just sufficient to determine bead positions.
2N contact forces are just sufficient to balance forces on particles
N beads
Isostatic networks are floppy: if we rebuild the pile with walls shifted by d,we create a deformation that costs no energy: a Òfree modeÓ where top shifts by e 
s
yy
s
xx
Internal stresses 
s
xx
s
yy
must balance to avoid motion of free mode
d
s
xx
e
s
yy
+
= 0
The medium must obey this stress balance condition.
Cf. Cates Bouchaud, Claudel, Wittmer, 1990Õs