cp's OEIS Frontend

These are "free polyforms" because they are counted up to rotation and reflection.

The triangular pyramidille is dual to the cantitruncated cubic honeycomb.

The polyhedral cells are each 1/24 of a cube and are similar to the convex hull of (0,0,0), (2,0,0), (1,1,0), and (1,1,1).

If S is an arrangement of non-overlapping balls of radius 1, the contact number of S is the number of pairs of balls that just touch each other.

a(13) >= 36 (take one ball and its 12 neighbors), so this is different from A008486.

If b(n) denotes the maximal contact number of any arrangement of n balls then it is conjectured that a(n) = b(n) for n <= 9. It is also known that b(10)>=25, b(11)>=29, b(12)>=33 and of course b(13) >= a(13) >= 36. [Bezdek 2012]

Note that Figure 1e of Bezdek's arxiv:1601.00145 shows at n=5 a sphere packing with 9 contacts on the hexagonal close package (!), not on the cubic close package (which equals the f.c.c.). [In Figure 1e there is one sphere that touches from above a set of 3 spheres in a middle layer right above the bottom sphere; so this needs the ABABA... layer structures of the h.c.p, and cannot be done with the ABCABC... layer structure of the f.c.c.] So Figure 1e is not demonstrating a(5)=9. The correct value for the f.c.c is apparently a(5)=8 (where two structures with 8 contacts exist.) - R. J. Mathar, Mar 13 2018

These are "free polyforms" because they are counted up to rotation and reflection.

A bisymmetric hendecahedron is an 11-sided polyhedron that is similar to the convex hull of (-2,1,-1), (-2,1,1), (-1,-1,0), (0,-1,-1), (0,-1,1), (0,0,-2), (0,0,2), (0,2,0), (1,-1,0), (2,1,-1), and (2,1,1).