To get [the D4 lattice], first take a bunch of equal-sized spheres in 4 dimensions. Stack them in a hypercubical pattern, so their centers lie at the points with integer coordinates. A bit surprisingly, there’s a lot of room left over – enough to fit in another copy of this whole pattern: a bunch of spheres whose centers lie at the points with half-integer coordinates!
If you stick in these extra spheres, you get the densest known packing of spheres in 4 dimensions. Their centers form the “D4 lattice”. It’s an easy exercise to check that each sphere touches 24 others. The centers of these 24 are the vertices of a marvelous shape called the “24-cell” – one of the six 4-dimensional Platonic solids. It looks like this:
Colour images by eusebia