This is a series of models showing a method of construction of polyhedra. In this first part, the polyhedra are rhombic zonohedra. As shown in Scenes, these are Cube, Rhombic Dodecahedron, Rhombic Triacontahedron, and Enneacontahedron. Each are constructed in much the same way; Starting with a scaled down set of the vertices of the polyhedron, and copying this set to each vertex of a framewok of polyhedron as a “hub.” Then hollow struts are built around edge lines. These struts are composed of planes which are extensions of faces of the hubs. In the case of the Cube, this is easy to see. The planes of the mini-cubes at the corners extend to form tubular struts with square cross-section with the original edges as heartlines.
The Rhombic Dodeca (RD) is constructed in a similar way, with vertices of a scaled down “mini-RD” copied to each vertex of an RD frame, then planes are extended to make tubular struts with regular hexagonal cross-section surrounding heartlines of the RD frame. In this case, though, the process is a little more involved than with the Cube, owing to the fact that the planes as defined by the “hubs” intersect beyond the vertices of the “hubs.”
It is much the same with the Triacontahedron, mini-triacon “hubs” are copied to the vertices of the RT frame, and tubular struts with regular decagonal cross-section are composed around heartlines. As with the RD, points must be defined where planes intersect beyond the vertices of the “hubs.”
The tubular Enneacontahedron is formed the same way, but is a lot more involved, as there are many mpre intersections of the planes of what are tubular struts with irregular nonagonal cross-section.
In general, models constructed by this method can be designed to be 3D printable, in parts or as whole objects. They can also be used to good effect in defining parts to be fabricated, such as wood pieces to have mitre cuts made for joinery.
A 3D design created in vZome. Use your mouse or touch to interact.
Here, the method for building RDs is used to compose tubular Diamond Lattice samples.
The same method can be applied to less familiar forms with different symmetry, such as this Polar Rhombic Icosahedron. This is also a zonohedron with 5 directions of edges. This is not the canonical Rhombic Icosa, which could also be built with tubular struts, but is a “Normalized” version, with 10 square faces and 10 rhombic faces. This, like the Enneacon, has several sets of points beyond the “hubs” which must be defined.
A continuation of this series of models can be viewed Here