Cubic honeycomb

Simple cubic lattices can be distorted into lower symmetries, represented by lower crystal systems:

There is a large number of uniform colorings, derived from different symmetries. These include:

The rectified cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

The truncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

The bitruncated cubic honeycomb shown here in relation to a cubic honeycomb

The cantellated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

It has irregular triangle bipyramid cells which can be seen as 1/12 of a cube, made from the cube center, 2 face centers, and 2 vertices.

The cantitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

A cell can be as 1/24 of a translational cube with vertices positioned: taking two corner, ne face center, and the cube center. The edge colors and labels specify how many cells exist around the edge.

Despite being non-uniform, there is a near-miss version with two edge lengths shown below, one of which is around 4.3% greater than the other. The snub cubes in this case are uniform, but the rest of the cells are not.

The runcitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

Cells are irregular pyramids and can be seen as 1/24 of a cube, using one corner, one mid-edge point, two face centers, and the cube center.

The omnitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

24 cells fit around a vertex, making a chiral octahedral symmetry that can be stacked in all 3-dimensions:

Individual cells have 2-fold rotational symmetry. In 2D orthogonal projection, this looks like a mirror symmetry.

It is constructed from a truncated square tiling extruded into prisms.