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Numerical modelling by finite element methods provides two significant insights into the formation of the giant amethyst geodes of the Paraná volcanic province: the conditions needed to open the cavities and the conditions that control their size and shape. Giant amethyst geodes were formed in the Cretaceous (135 Ma) in altered volcanic rocks by water vapour pressure (Δp) at about 0.5 MPa under an altered basalt cover of 5-20 m. Only rocks with Young's modulus values (E) in the range 1-2 GPa can sustain ballooning, which is the growth of a cavity in a ductile medium by the pressure of water and its vapour. The size of the proto-geode is dependent on the water vapour pressure, which is directly related to thickness of the overlying basalt. Varying the yield points causes the formation of either prolate or oblate cavities. A low transition point (smaller than 0.18 MPa) generates a prolate-shaped cavity, whereas a high transition point (larger than 0.18 MPa) generates oblate proto-geodes. Proto-geodes are smaller when Young's modulus is higher (rock is less altered) or when water vapour pressure is lower (because of thinner overburden of basalt). The calculations are an indication that the processes operative in the altered basalts led to the opening of giant cavities by ballooning. [ABSTRACT FROM AUTHOR]