where sign ( x ) gives the sign of argument x . Let the centroids of the three fixed neighbors be ( a$_{i}$, b$_{i}$ ) ( i = 1 , 2 , 3), and the orientations be θ$_{i}$ . Their boundaries are then given by Similarly, if the centroid of the central particle is at ( a$_{o}$, b$_{o}$ ) and its orientation is characterized by θ$_{o}$ , its boundary is specified by Since the positions and orientations of the three neighbors are fixed, the four-particle system has four degrees of freedom, namely the position ( a$_{o}$, b$_{o}$ ) and orientation θ$_{o}$ of the central particle, as well as the radius R of all particles, as discussed in Sec. III.C. In the jammed configuration, the central particle contacts all its three neighbors. From Eqs. (A-1) and (A-4), the contact point ( x ( i ) 1 $_{c}$, x ( i ) 2 $_{c}$) between neighbor particle i and the central particle can be expressed in terms of ( a$_{o}$, b$_{o}$, θ$_{o}$, R ), which must also lie on the boundary of the neighboring particle i , i.e., for i = 1 , 2 , 3. This leads to three equations in the variables a$_{o}$, b$_{o}$, θ$_{o}$ , and R . In addition, to achieve jamming the three normals at contacts must meet at a common point, which guarantees torque balance. The normals at contacts are along the lines The aforementioned torque balance condition requires that the three lines given by Eq.(A-6) must intersect at a common point. This leads to another equation in the variables a$_{o}$, b$_{o}$, θ$_{o}$ , and R . Thus, there are four independent equations for the four degrees of freedom and ( a$_{o}$, b$_{o}$, θ$_{o}$, R ) can be completely determined for a locally jammed four-particle configuration. 17 ( x$_{2}$ − x ( i ) 2 $_{c}$) = − dx ( i ) 1 /dθ dx ( i ) 2 /dθ$^{∣}$ ∣ ∣ ( x ( i ) 1 c ,x ( i ) 2 $_{c}$) ( x$_{1}$ − x ( i ) 1 $_{c}$) . (A-6) ∣ ∣ ∣ x ( i ) 1 $_{c}$( a$_{o}$, b$_{o}$, θ$_{o}$, R ) − a$_{i}$ R ∣ ∣ ∣ 2 p + ∣ ∣ ∣ x ( i ) 2 $_{c}$( a$_{o}$, b$_{o}$, θ$_{o}$, R ) − b$_{i}$ R ∣ ∣ ∣ 2 p = 1 , (A-5) x o $_{1}$( θ ) = a$_{o}$ + cos θ$_{o}$ | cos θ | 1 p · R · sign (cos θ ) + sin θ$_{o}$ | sin θ | 1 p · R · sign (sin θ ) , x o $_{2}$( θ ) = b$_{o}$ − sin θ$_{o}$ | cos θ | 1 p · R · sign (cos θ ) + cos θ$_{o}$ | sin θ | 1 p · R · sign (sin θ ) . (A-4) x ( i ) 1 ( θ ) = a$_{i}$ + cos θ$_{i}$ | cos θ | 1 p · R · sign (cos θ ) + sin θ$_{i}$ | sin θ | 1 p · R · sign (sin θ ) , x ( i ) 2 ( θ ) = b$_{i}$ − sin θ$_{i}$ | cos θ | 1 p · R · sign (cos θ ) + cos θ$_{i}$ | sin θ | 1 p · R · sign (sin θ ) . (A-3)