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Calabi-Yau manifolds are important objects in algebraic geometry and intheoretical physics. A hypothesis of mirror symmetry is that Calabi-Yaumanifolds of dimension 3 come in pairs, with the Hodge numbers of one manifoldmirroring the Hodge numbers of the other.  In this thesis, Calabi-Yau manifolds are constructed by smoothingStanley-Reisner schemes of triangulations of 3-spheres. The triangulations ofthe 3-sphere with 7 or 8 vertices have been constructed and classified byGr\"unbaum and Sreedharan (1967). In this thesis we characterize the Calabi-Yaumanifolds obtained from the triangulations with 7 vertices. They are embeddedin six-dimensional projective space. Mirrors of such manifolds are constructedby resolution of singularities performed on a singular fiber in the deformationfamily of the Stanley-Reisner scheme. In this way, we reproduce the mirror byR{\o}dland (1999) of a degree 14 pfaffian Calabi-Yau threefold, and the mirrorcandidate by B\"ohm (2007) of a degree 13 pfaffian Calabi-Yau threefold. In thelatter case we verify that the Hodge numbers in fact constitute a mirror, usingtoric geometry.

Pfaffian Calabi-Yau threefolds, Stanley-Reisner schemes and mirror symmetry