Archimedean Solids · 3D

Vector Equilibrium

Cuboctahedron · 3.4.3.4

Eight triangles and six squares, twelve corners all at the same distance from the centre as from each other. Fuller called it the vector equilibrium: the one shape in which every force is in balance.

Vertices: 12
Edges: 24
Faces: 8 ▲ + 6 ■
Edge = radius: yes

Open in SacredForm

Turn it in your hands, strike it to hear the chord of its edges, slice it with a plane or lift it into four dimensions.

Facts

Its edge length equals its radius: the twelve radial lines are as long as the edges.
Its twelve corners are the centres of twelve spheres packed around one.
The four hexagons through its centre (the radial planes) form its equators.
Its dual is the rhombic dodecahedron, the cell that fills space in that packing.
Cut the 24-cell through its middle and this is the solid you find.

History and meaning

Archimedes described it among his thirteen semi-regular solids. Buckminster Fuller made it the zero point of his synergetic geometry: twist it and it collapses through the icosahedron into the octahedron (the Jitterbug).

Across the dimensions

Equator of 24-CellCut the 24-cell through its middle and the vector equilibrium appears: it is the 24-cell’s equator.
Dual Rhombic Dodecahedron