Month: April 2022

Inter-Geared Mobilities of the De-Nucleated Vector Equilibrium

Greg FrederickLeave a comment

See also: Vector Equilibrium and the (VE)

If we close-pack 12 spheres around a central nucleus and then remove the central sphere, the remaining spheres are free to rotate along twelve axes perpendicular to the radii connecting the sphere centers with their common center. The combined axes describe the regular octahedron.

Twelve close-packed spheres synchronously spinning on axes configured as the edges of a regular octahedron.
With the nuclear sphere removed, the surrounding twelve spheres rotate freely as synchronous gears

We can also replace the spheres by wheels, which is mesmerizing, though perhaps not very helpful to conceptualizing the geometry.

Twelve disks synchronously spinning on axes configured as the edges of a regular octahedron.
De-nucleated VE with spheres reduced to disks

The effect is related to, but not identical with the jitterbug. Both are related to the removal of the nucleus, but with the jitterbug, there are only four axes of spin, and these are identical with, rather than perpendicular to, the radii.

The eight equilateral triangles of the vector equilibrium (VE) rotating on their radial axes in the classic model of the jitterbug. With the top and bottom triangles fixed, the remaining six triangles will rotate around the equatorial axis.
The classic model of the jitterbug

This is perhaps made more clear if we replace the eight triangles of the classic model of the jitterbug with eight wheels rotating synchronously on the four radial axes of the cube.

Eight wheels centered on the corners of the cube and synchronously spinning along the cube's four radial axes. With the top and bottom wheels fixed, the remaining six wheels rotate around the equatorial axis.
The eight triangles of the classic model of the jitterbug replaced with wheels rotating on the four radial axes of the cube.

The eight wheels may be replaced with spheres.

Eight spheres in a symmetrical cubic arrangement synchronously spinning on radii connecting their their common center with their centers of volume. With the top and bottom spheres fixed, the remaining six spheres rotate around the equatorial axis.
The eight triangles of the classic model of the jitterbug replaced with spheres.