Metal oxides such as VO2 undergo structural transitions to low-symmetry phases characterized by intricate crystalline order, accompanied by rich electronic behavior. We derive a minimal ionic Hamiltonian based on symmetry and local energetics which describes structural transitions involving all four observed phases, in the correct order. An exact analysis shows that complexity results from the symmetry-induced constraints of the parent phase which forces ionic displacements to form multiple interpenetrating groups using low-dimensional pathways and distant neighbors. Displacements within each group exhibit independent, quasi two-dimensional order, which is frustrated and fragile. This selective ordering mechanism is not restricted to VO2: it applies to other oxides which show similar complex order.Introduction.-Vanadium dioxide (VO 2 ) undergoes a transition from a high-symmetry rutile structure to a lower-symmetry monoclinic M1 phase, with intricate antiferroelectric (AFE) crystalline order [1][2][3][4]. The lowering of symmetry doubles the unit cell, changing the electronic band structure and converting a metal into a dimerized Mott insulator which shows unusual metalinsulator coexistence near T c . The insulating phase is structurally soft, as two other variants appear with doping or application of strain [5][6][7][8]. Although such materials are promising for applications (including lowdissipation logic [9-11] and many others), the lack of a microscopic theory has hindered progress. VO 2 is not alone: similar complex ordering also occurs in many other crystals [12][13][14][15], exhibiting diverse electronic and magnetic phases [16] whose properties would certainly reflect the intricacies of the underlying crystalline order. To understand exactly how this complexity appears and its effect on finite temperature properties, one needs a dynamical model to describe the structural phases.