Certain zonohedra surrounded by sets of parallel prisms. These align with and extend the planes of the faces of the zonohedra. The profile of these sticks, or blocks, is determined in part or completely by adjacent sticks. The portions of the sticks that extend beyond the contained zonohedron, in this case a cube, can be cut in various ways. In this example, the sticks have a cross-section of 1:2 proportion, and are cut to meet the faces of a Truncated Octahedron. The assembly is shown “exploded.” This construction is one of the type described in Three-axis coordinate compositions
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A cube, and more generally any zonohedron, can have its zones varied independently. Thus, a cube can have one dimension reduced. Here is an example of the same arrangement as the above, but with the cube reduced in height.
These sets of parallel sticks, or blocks, can be defined in more than one way. One can start with rods, either in tensegrity models, or as edges of polyhedra, as explained in Transformation of edges of polyhedra. Once rods are aligned as parallel beams with center lines that correspond to symmetry axes, the cross-section profile of the rods can be expanded to fill out the space defined by the adjacent rods in the array. Another approach is to start with a zonohedron, and extend its face planes to define the prismatic profile of the sticks. Either way, the result is like this example of a four axis intersection. Here, a Rhombic Dodecahedron is the zonohedron surrounded by 4 beams composed of 3 sticks each, with equilateral triangular cross-section.
These arrays of sticks, or blocks surrounding zonohedra correspond directly to Multi-axis composite beams. Here is a 4 beam intersection, which we call a Tetraxis, in exploded view, and in complete form.
Complete
Building these forms starting with sticks and string or elastic bands led to an understanding of them as mutable entities, where the symmetries and geometric features are realized by the physical interaction of tension and compression elements. Thus, we developed flexible, even foldable versions of these arrays. This variability of form can be expressed by models like this, where the surrounded zonohedron is an affine Rhombic Dodecahedron, and the 4 beams surrounding it have isosceles triangular profiles. Here is shown this variation of Tetraxis in exploded view, and as complete.
Complete
For a continuation of this series of models and images, see Part 2
Some of the forms shown here, or variations of them, have also been explored by others, and used extensively in puzzle designs, notably by Stewart Coffin