John Baez:

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&amp;#8217;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 &amp;#8220;D4 lattice&amp;#8221;. It&amp;#8217;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 &amp;#8220;24-cell&amp;#8221; - one of the six 4-dimensional Platonic solids. It looks like this:







&amp;#8230;
Colour images by eusebia