Filling three-dimensional space with tetrahedra: A geometric and crystallographic problem

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Abstract
Exact, nonperiodic, close-packed structures with randomness that can fill three-dimensional space are found. We find many solutions for distances between atoms that satisfy the necessary conditions of filling three-dimensional space with tetrahedra formed with two kinds of atoms. Only three solutions that also satisfy the sufficient condition of filling three-dimensional space are discussed. They all involve periodic as well as nonperiodic structures resulting from the random stacking of layers. One solution corresponds to the NaCl structure. Another solution exhibits both tetragonal and hexagonal symmetry, which violates crystallography. A third solution has a unit cell whose surface exhibits distorted pentagonal symmetry and whose elementary unit is a 44-face polyhedron. Suggestions for a possible growth model in three dimensions with tetrahedra are discussed.