Vector Equilibrium and its Transformation Pathways

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Vector Equilibrium and its Transformation Pathways

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The pathways illustrated on the map are the main connections between predominantly symmetric figures which can he derived by subjecting the vector equilbrium (also named a "jitterbug") to various processes: folding, compressing, expanding, rotating, etc

"Jitterbug" name coined by the American mathematician Buckminster Fuller for the geometrical transform- ations of a flexible jointed cuboctahedron. (R. Buckminster Fuller. Synergetics: Explorations in the Geometry of Thinking MacMillan, 1975)
Omnidirectional closest packing of spheres defining nodes of the Cuboctahedron Vector equilibrium The cuboctahedron is the polyhedron obtained by bisecting the 12 edges and truncating the eight corners of the cube. It can also be developed, however, from the omnidirectional closest packing of spheres around one nuclear sphere. The centres of 12 such spheres define the 12 nodes of the cuboctahedron. As all spheres are the same size it can be seen that the length of the cuboctahedron's edges equal the distance from its centre to its 12 nodes. Thus the form can be considered to be a system of equal vectors which are in equilibrium a VECTOR EQUILIBRIUM - where the outward radial thrust of the vectors from the centre is balanced by the circumfer- entially restraining chordal vectors. The explosive forces perfectly balance the implosive forces. Because energetic forces are in such an unstable equilibrium, as Buckminster Fuller says,.... "the vector equilibrium is a condition in which nature never allows herself to tarry. The vector equilibrium itself is never found exactly symmetrical in nature's crystallography. Ever pulsive and impulsive, nature never pauses her cycling at equilibrium : she refuses to get caught irrecover- ably at the zero phase of energy. She always closes her transformative cycles at the maximum positive or negative asymmetry stages." The JITTERBUG demonstrates complex geometric relationships in a very simple and entertaining educational device for children and adults. Additional inform- ation is enclosed. Made in Australia. The geometric forms we observe in nature are often those of the JITTERBUG'S transformational patterns such as the icosahedron, the octahedron and the tetrahedron where the enegetic vectors temporarily "fix". The JITTERBUG, then, is a vector equilibrium with the internal radially thrusting vectors removed so that, in manipulating it, the pull of your hands replaces the push of nature, and one can feel nature's energy circuits. HOW TO JITTERBUG With two hands holding opposite triangles and without twisting its axis, the JITTERBUG is symmetrically contracted, firstly to an ICOSAHEDRON and then to an OCTAHEDRON. At this stage, all the struts are doubled together in tight parallel (simulating chemical double bonding of atoms). Now, by rotating the top triangle of the composite octahedron by 60 degrees, and then lowering it, the whole structural - system will transform into a two frequency triangle whose outer triangles can now be folded inwards to form a six edge pattern of the tetrahedron which organic chemists would describe as a quadrivalent tetrahedral structure. Finally, the model of the tetrahedron can be turned inside out, opened up, and the previous contractive sequence of events is reversed. With practice, a rythmic contraction - expansion move- ment can be acquired with alternative clockwise and anti-clockwise rotations. Happy jitterbugging! TETRA TOYS 1 Queens Ave: McMahons Point N.S.W. 2060 Tel: (02)436-1399

The figures emerge whenever a new pattern of triangulation occurs, namely whenever any of the processes brings vertices into triangulation distance, or takes them out of it. Example: in compressing to opposite triangles from the vector equilibrium position, a point is reached when the 6 squares are converted into 12 possible triangles with the (imaginary) addition of six extra edges across the original diagonals

The map is not complete even for the portion shown. The 3-dimensional asymmetric portion is quite complicated and difficult to disentangle. It is interesting that most of those structures should emerge via the hexagonal wheel

Map of transformations of the vector equilibrium by manipulation of the jitterbug (partially relabelled in 2021)

Clarification (2021): The so-called jitterbug dynamic relates solely to the transformations between cuboctahedron, icosahedron and octahedron (in the upper left region of the image above). There are many accessible videos of this dynamic (Buckminster Fuller's Jitterbug, YouTube, 6 May 2007; Buckminster Fuller Explains Vector Equilibrium, YouTube, 29 November 2014). One physical model by which this dynamic can be observed through its manipulation is named "jitterbug". However this model also allows related forms in 3D and 2D to be explored through its manipulation, most obviously by variously overlapping its triangular features. It is these additional forms which feature in other portions of the map above and in that below. A complete mapping could most usefully be articulated by algorithmic manipulation of the structure in virtual reality.

Map of "secondary" transformations of the vector equilibrium by manipulation of the jitterbug (portion of original manual map, not amended in 2021)

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