Trihelical square tiling

The trihelical square tiling or petrial facetted halved mucube is a regular skew apeirohedron that consists of square helices, with three meeting at each vertex. All adjacent square helices in the trihelical square tiling that share a vertex are perpendicular to each other. The trihelical square tiling is a chiral polyhedron; its helices are either all clockwise or all counterclockwise.

The trihelical square tiling is the Petrie dual of the tetrahelical triangular tiling. It also is the second-order facetting of the petrial halved mucube, so the edges and vertices of the trihelical square tiling are a subset of those found in the halved mucube.

Vertex coordinates
The vertex coordinates of a trihelical square tiling of edge length 1 are given by all permutations of:


 * $$(\sqrt{2} i, \sqrt{2} j, \sqrt{2} k)$$
 * $$(\sqrt{2} i, \sqrt{2} j+ \frac{\sqrt{2}}{2}, \sqrt{2} k+\frac{\sqrt{2}}{2})$$

where $$i,j,k$$ range over the integers.