A series of related models and images showing transformations of edges of polyhedra. These kinds of transformative re-orientations of edges of Platonic Solids and other polyhedra can be done in Tensegrity models, but can also be done with compression elements in contact with adjacent elements. The theme is, edge oriented struts can be rotated about a radial axis and moved in or out on that axis to make other configurations, such as triangles, and parallel groupings aligned with axial symmetries. This can be called an inversion from edge structure to body centered axis systems. In this model of a Tensegrity the blue struts represent tension elements, and the black struts represent compression elements, which are in 3 groups of 2 parallels oriented to x,y,and z axes.
Here, a 6-strut Tensegrity is shown as heartlines of square prisms that are in contact with each other at adjacent planes. This can be seen as expanding, or thickening the six struts until they meet each other. At this size, although it is the same set of struts converted to sticks or blocks, in the same relation to each other, it no longer meets the accepted definition of Tensegrity. It is apparent that the tension network can be replaced with adhesive, fasteers, or magnets, etc. We refer to such assemblies as this as examples of “Bypass Coupling,” where the elements deriving from compression members do not intersect, but pass by each other in contact without any alteration or deformation, and are connected by some tension elements. In this form, configurations like this can be used as modules of structural integrity, to build arrays without hub fixtures.
In this model, edges of a Tetrahedron are shown rotating about their midpoints through a sequence of skew positions to the parallel condition as shown above. In the sequence shown, the rotation continues through to the Tetrahedron in opposite orientation, the dual of the first one. In Desktop vZome, titled bookmarks show the full sequence of positions in order, top to bottom, including a second sequence bookmarked “Reflected to Orange frame,” the last bookmark. Struts are spaced apart, as they would be in a Tensegrity, but tension elements are not shown.
Another framework showing edges of a Tetrahedron rotating through a different sequence. Like the previous version, this has an ordered series of titled bookmarks in Desktop vZome.
3 axis transformation of a Tetrahedron shows edge struts maintaining contact with adjacents through the full sequence. Note that in this construction, rods are constant in length, and move in and out on x,y,z axes as they rotate, whereas in the previous models, the struts rotate about points on the axes, while their lengths change with the sequence.
This is a 12-strut Tensegrity model. Note that this is a chiral composition. The Six-strut models are achiral.
This shows two versions of 12-strut model, a left handed and a right handed one, as mirror copies. With six-strut models, a mirror copy is superimposeable on the original.
This shows an analogous transformation of edges, but rather than tetrahedra, it is the 12 edges of an Octahedron that rotate through a sequence that includes non-intersecting skew orientations and parallel sets, and also intersecting struts forming equilateral triangles, to alignment with edges of a Cube. As with the Tetrahedron transformation, this can be carried through an alternate, opposite handed sequence. This sequence is not shown, but can be explored in Desktop with titled bookmarks, in order from top down. Note that the yellow struts are in 4 sets of 3 in parallel, which can be extended, forming 4 composite beams with centerlines intersecting at a point, aligned with Body Centered Cubic (BCC) axes.
This is a representation of a 30-strut Tensegrity. Like the 12-strut models, it is chiral.
Analogously, this shows edges of an Icosahedron rotating through a sequence which encompasses several configurations. These include intersecting forms, such as 10 triangles, 5 tetrahedra, 6 pentagons, as well as 6 sets of parallels, 5 in each “beam,” and 10 sets of parallels, 3 in each “beam,” to aligning with edges of a Dodecahedron. Note that this is the sequence for one chirality, as shown, and there is an alternate sequence that passes through the same forms in the opposite chirality. As with the other transformation models, there are an ordered series of bookmarks that shows the alternate sequence, and also shows extensions of struts as stellations and extensions of parallels as composite beams.
These images show structures derived from these transformations as “Bypass coupling.” The first 2 are views of 12 strut composition of round, i.e.cylindrical compression elements in parallel mode.
This is a 5-Tetrahedron interlink, where the rods do not intersect at tetrahedron vertices, but pass by each other in contact. Tetrahedron “vertices” are at points surrounded symmetrically by the rods in groups of 3. Note that there are 20 small Tensegrity-derived tetrahedra in this composition. These rods, cylindrical in shape, pass by each other in contact at all these points with no deformation. This is quite extraordinary.
Some other configurations that are not directly derived from edges of Platonics, but from other polyhedra are:
Ten strut Tensegrity, 5 pairs of parallels. These are achiral, a mirrored copy is superimposeable on the original. Here, the compression elements are in green, the tension elements are in blue.
A vZome replication of a table I once made with brass rods and stainless wire.
Models representing achiral structures that could be made as Tensegrities, shown as “condensed” to bypass coupling, approximately, given strut shapes.
A nine pair composition with a single axis of rotational symmetry.
Likewise, 15 pair
