Algorithm 1 Decide existence of strictly auxetic deformations.

Precondition: ℱ is a regular periodic framework.

function Auxetic(ℱ)

Step 1. - Set up the linear system for periodic infinitesimal deformations. - Solve it in terms of 3 independent variables chosen from the 6 giving the infinitesimal deformations of the Gram matrix. - If this is not possible, STOP: the framework is not regular.

EndStep 1.

Step 2. - M ← Substitute the resulting linear forms in the 3 × 3 matrix of infinitesimal deformations of the Gram matrix. - Compute the determinant Det(M). The result is a cubic form C(X, Y, Z) in 3 variables, called X, Y and Z.

EndStep 2.

Step 3. - Compute the Aronhold invariants S and T for C(X,Y,Z). - Using S and T, compute the discriminant D of C(X,Y,Z). - If D = 0, cubic is singular. return “not regular”. - If D < 0, return NO: the framework does not have infinitesimal auxetic deformations.

EndStep 3.

Step 4. If D > 0: - compute the 3 × 3 linear matrix L for the transformation of the cubic C(X, Y, Z) to the Hesse normal form H(x, y, z). - Compute the pre-image of the point (1, 1, 1). This is a vector of specific values for (X, Y, Z). - The corresponding constant symmetric matrix M is then tested for being definite (either positive or negative definite) and the corresponding output is produced: - return YES, if definite or return NO, if indefinite.

EndStep 4.

end function