Recent results of string theory have shown that while the traditional cycles studied in Calabi-Yau 4-manifolds preserve half the spacetime supersymmetry, the more general class of Cayley cycles are novel in that they preserve only one quarter of it. Moreover, Cayley cycles play a crucial role in understanding mirror symmetry on Calabi-Yau 4-manifolds and Spin manifolds. Nonetheless, only very few nontrivial examples of Cayley cycles are known. In particular, it would be very useful to know interesting examples of Cayley cycles on the complex 4-torus. This thesis will develop key techniques for finding and constructing lattice periodic Cayley manifolds in Euclidean 8-space. These manifolds will project down to the complex 4-torus, yielding nontrivial Cayley cycles.