Solid-solid phase transitions are the most ubiquitous in nature, and many technologies rely on them. However, studying them in detail is difficult because of the extreme conditions (high pressure/temperature) under which many such transitions occur and the high-resolution equipment needed to capture the intermediate states of the transformations. These difficulties mean that basic questions remain unanswered, such as whether so-called diffusionless solid-solid transitions, which have only local particle rearrangement, require thermal activation. Here, we introduce a family of minimal model systems that exhibits solid-solid phase transitions that are driven by changes in the shape of colloidal particles. By using particle shape as the control variable, we entropically reshape the coordination polyhedra of the particles in the system, a change that occurs indirectly in atomic solid-solid phase transitions via changes in temperature, pressure, or density. We carry out a detailed investigation of the thermodynamics of a series of isochoric, diffusionless solid-solid phase transitions within a single shape family and find both transitions that require thermal activation or are "discontinuous" and transitions that occur without thermal activation or are "continuous." In the discontinuous case, we find that sufficiently large shape changes can drive reconfiguration on timescales comparable with those for self-assembly and without an intermediate fluid phase, and in the continuous case, solid-solid reconfiguration happens on shorter timescales than self-assembly, providing guidance for developing means of generating reconfigurable colloidal materials.colloids | self-assembly | phase transitions | nanoparticles D espite wide-ranging implications for metallurgy (1), ceramics (2), earth sciences (3, 4), reconfigurable materials (5, 6), and colloidal matter (7), fundamental questions remain about basic physical mechanisms of solid-solid phase transitions. One major class of solid-solid transitions is diffusionless transformations. Although in diffusionless transformations, particles undergo only local rearrangement, the thermodynamic nature of diffusionless transitions is unclear (8). This gap in our understanding arises from technical details that limit what we can learn about solid-solid transitions from standard laboratory techniques, such as X-ray diffraction or EM (9). The use of a broader array of experimental, theoretical, and computational techniques could provide better understanding of solidsolid transitions if an amenable class of models could be developed (10). To develop minimal models, it is important to note that solid-solid transitions are accompanied by a change in shape of the coordination polyhedra in the structure (11). Coordination polyhedra reflect the bonding of atoms in a crystal, which suggests that minimal models of solid-solid transitions could be provided by systems in which the shape of coordination polyhedra is directly manipulated. Direct manipulation of coordination polyhedra may be achieved ...