Tetrahelical triangular tiling

The tetrahelical triangular tiling or facetted halved mucube is a regular skew apeirohedron that consists of triangular helices, with three at a vertex. It is the Petrie dual of the trihelical square tiling. It is also the second-order facetting of the halved mucube, so the edges and vertices of the tetrahelical triangular tiling are a subset of those found in the halved mucube. The tetrahelical triangular tiling is a chiral polyhedron; its helices are either all clockwise or all counterclockwise.

Vertex coordinates
The vertex coordinates of a tetrahelical triangular tiling of edge length 1 are given by all permutations of: where $$i,j,k$$ range over the integers.
 * $$(\sqrt{2} i, \sqrt{2} j, \sqrt{2} k)$$
 * $$(\sqrt{2} i, \sqrt{2} j+ \frac{\sqrt{2}}{2}, \sqrt{2} k+\frac{\sqrt{2}}{2})$$