Six is the number of fibre orientations necessary and sufficient to remove all zero-valued shear moduli from fibre arrays that have cubic symmetry. Of particular interest is a weave, of just four families of flexible fibres, that has equal numbers of equal fibre lengths parallel to the six face diagonals of a cube. The relative positions of fibre lengths in this weave can be visualised as the edges of space-filling equilateral truncated octahedra.