  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 rescaling of $\mathbb{R}^3$ with their zero locus
consisting of four or more rays from the origin. The rays point from the origin
to the vertices of a centered tetrahedron in one example; and they point from
those of a centered octahedron and a centered icosahedron in two others.
