Fang Fang, Richard Clawson, Klee Irwin (2017)

This paper presents various ways of encoding geometric frustration in tetrahedral packings, by introducing small gaps to form quasicrystalline order, by curving to the 4th dimension with discrete curvatures, by distortion of tetrahedral edges and by twisting the edge-sharing and/or vertex-sharing local tetrahedral clusters. The key to these methods is to encode the deficit of the tetrahedral dihedral angle in closing a circle which is the cause of the geometric frustration. A surprising connection between the discrete curvature method and the twisting method is that both the transformation angle and the joint angle are the same in the one case as in the other. This connection leads to a way of encoding discrete curvature with twisting, which may help to model spacetime based on a quasicrystalline network that serves as a discrete version of a pseudo-Riemannian space.