The first results of a new three-dimensional , finite temperature Skyrme-Hartree-Fock+BCS study of the properties of inhomogeneous nuclear matter at densities and temperatures leading to the transition to uniform nuclear matter are presented .
Calculations are carried out in a cubic box representing a unit cell of the locally periodic structure of the matter .
A constraint is placed on the two independent components of the quadrupole moment of the neutron density in order to investigate the dependence of the total energy-density of matter on the geometry of the nuclear structure in the unit cell .

This approach allows self-consistent modeling of effects such as ( i ) neutron drip , resulting in a neutron gas external to the nuclear structure , ( ii ) shell effects of bound and unbound nucleons , ( iii ) the variety of exotic nuclear shapes that emerge , collectively termed ‘ nuclear pasta ’ and ( iv ) the dissolution of these structures into uniform nuclear matter as density and/or temperature increase .
In part I of this work the calculation of the properties of inhomogeneous nuclear matter in the core collapse of massive stars is reported .
Emphasis is on exploring the effects of the numerical method on the results obtained ; notably , the influence of the finite cell size on the nuclear shapes and energy-density obtained .
Results for nuclear matter in beta-equilibrium in cold neutrons stars are subject of part II .
The calculation of the band structure of unbound neutrons in neutron star matter , yielding thermal conductivity , specific heat and entrainment parameters , will be outlined in part III .

Calculations are performed at baryon number densities of n _ { b } = 0.04 - 0.12 fm ^ { -3 } , a proton fraction of y _ { p } = 0.3 and temperatures in the range 0 - 7.5 MeV .
A wide variety of nuclear shapes are shown to emerge .
It is suggested that thermodynamical properties change smoothly in the pasta regime up to the transition to uniform matter ; at that transition , thermodynamic properties of the matter vary discontinuously , indicating a phase transition of first or second order .
The calculations are carried out using the SkM ^ { * } Skyrme parameterization ; a comparison with calculations using Sly4 at n _ { b } = 0.08 fm ^ { -3 } , T = 0 MeV is made .