DNA origami tilings provide methods for creating complex molecular patterns and shapes using flat DNA origami structures as building blocks. Square tiles have been developed to construct micrometer-scale arrays and to generate patterns using stochastic or deterministic strategies. Here we show triangular tiles as a complementary approach for enriching the design space of DNA tilings and for extending the shape of the self-assembled arrays from 2D to 3D. We introduce a computational approach for maximizing binding specificity in a fully symmetric tile design, with which we construct a 20-tile structure resembling a rhombic triacontahedron. We demonstrate controlled transition between 3D and 2D structures using simple methods including tile concentration, magnesium, and fold symmetry in tile edge design. Using these approaches, we construct 2D arrays with unbounded and designed sizes. The programmability of the edge design and the flexibility of the structure make the triangular DNA origami tile an ideal building block for complex self-assembly and reconfiguration in artificial molecular machines and fabricated nanodevices.