Diatoms are single-celled organisms with a cell wall made of silica, called the frustule. Even though their elaborate patterns have fascinated scientists for years, little is known about the biological and physical mechanisms underlying their organization. In this work, we take a top-down approach and examine the micrometer-scale organization of diatoms from the Coscinodiscus family. We find two competing tendencies of organization, which appear to be controlled by distinct biological pathways. On one hand, micrometer-scale pores organize locally on a triangular lattice. On the other hand, lattice vectors tend to point globally toward a center of symmetry. This competition results in a frustrated triangular lattice, populated with geometrically necessary defects whose density increases near the center.