The elementary neighbourhood on such grids is defined as a cuboctahedron
(each pixel having twelve nearest neighbours) which is a shape closer to the sphere than the cube.
Joseph DeVincentis suggested the cuboctahedron
as a likely shape to use for solid forms--it offers a nice packing of reasonably long words.
At this point, they switched to the PASE module and read about the cuboctahedron
By using a sequence of elementary moves (moving a vertex along an edge), Morin transforms the cuboctahedron into a curiously shaped figure, which he calls the "central model," with only 12 faces but the same number of vertices as before.
If you start with the 12 vertices of a cuboctahedron, you see many phenomena," Morin adds.
In 1943, the Life magazine published a map projection on a cuboctahedron
, This projection was created by R.
However, his patented map projection (Fuller 1946) is based on the cuboctahedron
, which has eight triangular faces and six square faces.