Cubic
Lattice 
Tetrahedral Lattice 
Pyramidal Lattice 
Prismatic Lattice 




This lattice is equivalent to
that formed by cubes. 
These lattices are identical. Each is equivalent to
those formed by rhombic dodecahedra (polyrhons),
and truncated octahedra. 
This lattice is equivalent to
that formed by prismatic hexes 



Planar pieces are equilvalent
to solid polyominoes. Threedimensional pieces are
equivalent to polycubes. 
Both solid polyominoes and
solid polyhexes can be placed in these lattices. 
The pieces
Below are all the planar pieces of
orders 1  4. On the left are the pieces equivalent to the solid
polyhexes, with the solid polyomino equivalents on the right.
The linear pieces are common to both subsets. 

These pieces can be used to construct
several threedimensional figures. The colours above are used to
identify the pieces. The
level
5 pyramid, the
level 5 tetrahedron, the
level 6 tetrahedron, the
level 4 octahedron,
7
x 4 hipped roof,
two
pyramidal roofs, the
level 5 double tetrahedron, the
level 3 hexagonal prism,
hexagonal grapes, 
Nonplanar pieces 
Cubic Lattice
Pieces shown in isometric
projection, with the equivalent polycubes. 
Pyramidal Lattice
Polysphere pieces shown in plan
view. Equivalent truncated octahedra shown in isometric
projection, with those parts on the second level highlighted.
Thanks to Jan de Ruiter for his help in completing these
figures.

Prismatic Hexagonal Lattice
Pieces shown in isometric
projection, with the equivalent prismatic polyhexes.


Planar
pentaspheres


There are 33 planar pentaspheres, which
can exactly fill a
level 9 tetrahedron 