Examples of singularity models for $\mathbb{Z}\mathrm{/}2$ harmonic 1-forms and spinors in dimension 3

We use the symmetries of the tetrahedron, octahedron and icosahedron to construct local models for a $\mathbb{Z}\mathrm{/}2$ harmonic 1-form or spinor in 3-dimensions near a singular point in its zero loci. The local models are $\mathbb{Z}\mathrm{/}2$ 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.