Sets of parallel struts that surround lines that intersect symmetrically. The first is 3 pairs of struts that surround 3 axes of symmetry. These pairs can be found in several ways. Here they are shown as chords of an Icosahedron. The heart lines of these beams are shown extending to vertices of an Octahedron.
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This is 4 sets of 3 struts each, that can be seen as composite beams with 4 heart lines that intersect symmetrically. These sets are shown as chords of a Truncated Octahedron. This intersection of lines is sometimes called Body Centered Cubic, (BCC) as the lines are shown extending to the corners of a Cube.
This shows 6 beams composed of 5 struts each. These are shown as chords of a Rhombicosidodecahedron. The heart lines of these beams are shown extending to the vertices of an Icosahedron.
Another composition of a similar set of 6 beams is shown here, as chords of a Truncated Dodecahedron.
This shows ten beams composed of 3 struts each. Here, these sets of parallels are shown as chords of a Truncated Dodecahedron. The heart lines are shown extending to vertices of a Dodecahedron.
Showing how both 6 beam and 10 beam configurations intersect at vertices of a Truncated Dodecahedron. The 10 beam struts are shown extending to be identical with chords of a Truncated Icosahedron. sometimes called a Buckyball.
While there is more than one way to compose the intersection of 6 beam and 10 beam configurations as chords of polyhedra, it is also possible to construct a non-intersecting coincidence of the 2. Here is shown such a composition with the spacing of the red struts adjusted to pass through the yellow struts, which are represented here as in close contact. Empirically, and by subsequent study, there is a ratio of the thickness of round rods in the 10 beam system to those in the 6 beam system at which this can happen without distortion of either system. Actually, it is a range of ratios, from approximately 5/6 to 7/8.
Another type of composite beam is this. 5 pairs of struts are shown as chords of a Dodecahedron. These green struts can be configured as 5 Tetrahedron interlinks. Here, they are selected in parallel pairs, with heart lines that extend to vertices of a Pentagonal Antiprism.
It’s noteworthy that while this intersection of 5 lines derives from a Dodecahedron, it only has one axis of symmetry. The lines themselves are not axes of symmetry in the same sense, but are intersecting axes that can accomodate rotation of the whole about each axis that is balanced, as it is in crankshafts. Because there is only polar symmetry, the lines can be adjusted to any angle vis a vis the polar axis. Here is shown 5 pairs of beams that have heart lines that intersect at a different suite of angles, and extend to vertices of a “taller” Pentagonal Antiprism. In this arrangement, the 5 lines intersect at 90° between non-adjacents and arcsin 1/√𝜑 = 51 827…° between adjacents. This can be called a normalized 5-directional system, which has very interesting features, and is the basis of the √𝜑 field in vZome.
Many related composite beam arrangements can be built with polar symmetry, such as 6 directional, 7 directional, etc. There are also multiples of these primary composite beams, one example shown Here
These arrays of parallel struts as beams can be defined as chords of polyhedra, but can also be arrived at by Transformations of edges of polyhedra as discussed. This can be accomplished in Tensegrity form, using simple materials, like sticks and string. Once defined, the parallel groups can be drawn together into contact, and the tension members can be replaced. This represents a 4-axis tensegrity composition, with the struts extended as beams.
Using round rods with the beams drawn together, this object has great structural integrity, the physicality of it works to have the beams displace each other toward a self-defining symmetrical equilibrium. This is true in varying degrees of all these composite beams, though some require being built out more, i.e. multiples of primary beams, for stability and definition.
These configurations have been made as artwork for many years, as shown on this WebPage
Many of the same structures have been independently discovered and explored by others, notably Akio Hizume
More about these arrays can be seen Here