Archimedean Solids · 3D

Truncated Octahedron

4.6.6 · t{3,4}

Cut the six corners off an octahedron a third of the way in: eight hexagons and six squares. The one Archimedean solid that fills space by itself.

Vertices: 24
Edges: 36
Faces: 6 ■ + 8 ⬢
Fills space: yes

Open in SacredForm

Turn it in your hands, strike it to hear the chord of its edges, slice it with a plane.

Facts

Copies of it pack together with no gaps, as cubes do: no other Archimedean solid can.
Its 24 corners are the 24 orderings of 1, 2, 3, 4: the permutohedron, seen in the space where those points lie.
It is the cell each atom claims in a body-centred cubic crystal, such as iron.
Every corner joins one square and two hexagons (4.6.6).

History and meaning

In 1887 Lord Kelvin asked how to divide space into equal cells with the least wall, as a foam does, and answered with slightly curved truncated octahedra. His answer stood for over a century, until Weaire and Phelan found a better foam in 1993.