  By fine-tuning generic Cauchy data, critical phenomena have recently been
discovered in the black hole/no black hole "phase transition" of various
gravitating systems. For the spherisymmetric real scalar field system, we find
the "critical" spacetime separating the two phases by demanding discrete
scale-invariance, analyticity, and an additional reflection-type symmetry. The
resulting nonlinear hyperbolic boundary value problem, with the rescaling
factor Delta as the eigenvalue, is solved numerically by relaxation. We find
Delta = 3.4439 +/- 0.0004.
