We use the symmetries of the tetrahedron, octahedron and icosahedron to construct local models for a \(\mathbb{Z}/2\) harmonic 1-form or spinor in 3-dimensions near a singular point in its zero loci. The local models are \(\mathbb{Z}/2\) harmonic 1-forms or spinors on \(\mathbb{R}^3\) that are homogeneous with respect to the rescaling of \(\mathbb{R}^3\) with their zero loci consisting of 4 or more rays from the origin. The rays point from the origin to the vertices of a centered tetrahedron in one example, and to those of a centered octahedron and a centered icosahedron in two others.