Geometric sculptures carved from wood are presented to demonstrate the application of a creative discovery process utilising rhombic polyhedra. A surface tiling is applied to the faces of the rhombic dodecahedron and rhombic triacontahedron to generate three interwoven cage structures. The tiling is composed of tiles that produce a triaxial weaving pattern. These structures have been carved from a single piece of beech timber to form three completely separated but interwoven cages. These works were produced primarily with hand tools, and represent two members of an extended family of potential structures. Further family members could employ zonohedra, but would push the woodcarver to the practical limits in the mechanical properties of their medium, a topic explored within the context of the two geometric sculptures presented.