The quadratic polynomial f1/4f1/4 is the quadratic ploynomical f(z) = z2 + c, where c≈ -0.228155 + 1.11514 i is a zero of the cubic polynomial c3 + 2c2 + 2c + 2.
The notation f1/4 comes from external rays. In the complement of the filled Julia sets, the external ray with angle 1/4 (mod 1) lands at the critical value c. From this it follows that f(c) has angle 1/2 = 2 x 1/4, and f2(c) has angle 1 = 2 x 1/2. Since 2 x 1 ≡ 1 (mod 1), f2(c) is fixed and f has four postcritical points (∞, c, f(c), and f2(c). The Julia set for f1/4 is shown in the figure below, which was created from Mathematica.
The diagram for f1/4