Vector Equilibrium
Pack twelve equal spheres as tightly as they will go around a thirteenth, join their centres, and you get the vector equilibrium: a shape whose every edge is exactly as long as the line from its centre to a corner.
Vector Equilibrium is also the title of a SUMUS track: listen to it here.
Twelve around one Jitterbug Great circles Vector matrix
Each figure is drawn live in 3D in your browser. Drag one to turn it; the pause button stops it. The figures only move while they are on screen, and with reduced motion switched on they hold still.
Twelve around one
Exactly twelve spheres fit around one of the same size, each touching the nucleus and four of its neighbours. The lines between their centres form a cuboctahedron: 12 corners, 24 edges, 8 triangles and 6 squares. Buckminster Fuller called it the vector equilibrium, because its radial lines, pushing out, are just as long as its edges, holding the outside together, so the two are in balance.
Keep packing and every new layer keeps the same shape. Shell number F holds 10F² + 2 spheres: 12, 42, 92, 162… With the nucleus that makes 13, 55 and 147 spheres after three shells.
The jitterbug
Build the vector equilibrium from its eight triangles only, hinged at the corners, and leave the six squares open. Twist it and it cannot stay still: the triangles turn, the squares fold into pairs of triangles and the figure passes through the icosahedron, then closes into the octahedron. Fuller named this motion the jitterbug.
Fold once more and the eight triangles pair up into a tetrahedron, each face two triangles thick. Measured in tetrahedra of the same edge, as Fuller did, the volume goes from 20 for the vector equilibrium to about 18.51 for the icosahedron, 4 for the octahedron and 1 for the tetrahedron.
Four great circles, seven axes
Four planes through the centre each cut the vector equilibrium in a regular hexagon through six of its twelve corners. On the surrounding sphere they become four great circles, and every edge of the figure lies on one of them.
Spin it around a line through two opposite triangles and you get one of four axes; through two opposite squares, one of three. Together those are seven axes: the four with the gold dashes and the three in green.
The isotropic vector matrix
Repeat the packing in every direction and join each sphere's centre to its twelve neighbours. Every line has the same length, every point looks the same, and space fills completely with tetrahedra and octahedra, nothing left over. Fuller called it the isotropic vector matrix: isotropic because it is the same in all directions.
Every point of it is the centre of a vector equilibrium. The figure shows 55 of its points, a second-frequency vector equilibrium, with the first one in gold at its heart.
Sources
- Greg FrederickVector Equilibrium and the “VE”, The Geometry of Thinking, 2023. A detailed article with the author's own models and animations, the inspiration for this page.
- R. Buckminster FullerSynergetics (1975) and Synergetics 2 (1979), where the vector equilibrium, the jitterbug and the isotropic vector matrix are developed.
- These figuresDrawn by SUMUS in the browser with a plain 2D canvas and a small 3D projection, no libraries and no borrowed images.