Abstract The spin-dependent shape transition temperature T c2( I) is calculated for the first time. At this temperature there is a transition from a prolate noncollective equilibrium phase ( T < T c2) to an oblate noncollective equilibrium phase ( T > T c2). This shape transition line has a tricritical point at I c = 40 for 188Os and at I c = 50 for 196Pt. The transition is second-order for spins below I c, and first-order for spins above I c. Statistical shape fluctuations do not obscure the shape transition at T c2. At the temperature T c1( I), where T c1 < T c2, 188Os and 196Pt have a transition from an oblate (or triaxial) collective equilibrium phase ( T < T c1) to the prolate noncollective equilibrium phase ( T > T c1). For 196Pt T c1( I) and T c2( I) meet at I = 53, where the prolate noncollective phase ends.