Johnson Solids J1 and J8

The number associated with my diurnal age today, 27126, introduced me to the notion of Johnson solids via its membership in OEIS A227221:

A227221

Volume of Johnson square pyramid placed upright on cube (rounded down) with edge lengths equal to \(n\).

The members of this sequence, up to 40000, are:

1, 9, 33, 79, 154, 266, 423, 632, 900, 1235, 1644, 2135, 2714, 3390, 4170, 5061, 6071, 7206, 8475, 9885, 11443, 13157, 15034, 17082, 19307, 21718, 24322, 27126, 30137, 33363, 36812

The formula for the area is given by \(  (1+\dfrac{ \sqrt{2}}{6}) \times n\) where \(n\) is the edge length. For 27216, \(n=28\). This shape is known as an elongated square pyramid and represent Johnson solid J8. A Johnson solid is a convex polyhedron with all edges equal and there 92 distinct types. The equilateral square pyramid sitting on one of the faces of the cube is Johnson solid J1. See Figures 1 and 2.

Figure 1: Johnson solid J1 (source)

Figure 2: Johnson solid J8 (source)

Figure 3 shows a octahedron, one of the five Platonic solids, that can be considered a square bipyramid, i.e. two Johnson square pyramids connected base-to-base.

Figure 3: octahedron (source)

Figure 4 shows a tetrakis hexahedron that can be constructed from a cube with Johnson square pyramids added to each face. It is a Catalan solid.

Figure 4: tetrakis hexahedron (source)