The simplest way to arrange spheres in space is to use the cubical lattice. This is the obvious generalization of the checkerboard, and it lends itself naturally to a coloring with two colors such that neighboring spheres are differently colored. While this is not the densest sphere packing, it will be pretty dark inside.
Leaving out the spheres of one color, painting the rest with most of RGB color space creates the following arrangement of spheres, and makes enough room for light to get through.
Now imagine yourself inside of it, and all spheres being reflective in addition to being colored. The formerly simplistic object becomes a dazzling fractal-like maze.
The original bicolored sphere packing is related to a packing of space by octahedra (one for each orange sphere).
Two octahedra share then at most an edge, and the gaps can be filled with regular tetrahedra of the same edge length.
Minkowski discovered that octahedra can be packed much more densely. The gaps can still be filled with regular tetrahedra, but their edge length is only one third of the edge length of the octahedra.