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Our strategy is to pack cycles into sequences of linked octahedra.
We still have a lot of choice as to which octahedra are modified this way.
For the octahedra this is at most 5.
We may also have missed at most 34 x 4 squares and triangles when we changed back some octahedra.
We shall now modify the octahedra in the trail so as to include all the additional edges.
We shall then pack these trails of octahedra with cycles of arbitrary lengths.
Octahedra were chosen because they are large enough to allow a wide range of packings by cycles but sufficiently small to analyse these packings.
Unfortunately, we now need to pack cycles into these modified octahedra, and for very short cycles we shall need to introduce several alternative modifications to the octahedra.
It is composed of truncated cubes and octahedra in a ratio of 1:1.
Each vertex is surrounded by four cells: two truncated octahedra, and two hexagonal prisms, arranged in two chiral irregular tetrahedral vertex figures.
Ulexite is a structurally complex mineral, with a basic structure containing chains of sodium, water and hydroxide octahedra.
The cells are 5 cuboctahedra, 5 octahedra, and 10 triangular prisms.
As a result, the triangular prisms expand into hexagonal prisms, and the truncated tetrahedra expand into truncated octahedra.
At each linked edge two oxygen ions are shared between two octahedra, and each octahedron must have two shared edges to form a chain.
It is a layered silicate mineral, with one tetrahedral sheet linked through oxygen atoms to one octahedral sheet of alumina octahedra.
