Lattice displays a rotating 3-dimensional slice through a collection of hyperspheres packed in various lattice arrangements. In each case, the collection consists of one representative hypersphere and all its nearest neighbours in the lattice, but only a certain number of those neighbours will intersect the 3-dimensional slice and be included in the view at any given moment. The lattice spacing is chosen so that each hypersphere makes contact with its nearest neighbours, though the 3-dimensional slice will generally not include any points of contact, so the spheres that are shown will not touch. Clicking on the applet displays different lattices.

In the **D4 lattice** in 4 dimensions, each hypersphere makes
contact with 24 nearest neighbours, which lie at the centres of the
24 two-dimensional faces of a
hypercube (centred on the representative hypersphere itself).
The 24 neighbours lie on 12 rays which split up into three tetrads of orthogonal
rays; the hyperspheres are coloured by the tetrad they belong to.

In the **E8 lattice** in 8 dimensions, each hypersphere makes contact with
240 nearest neighbours; 112 of these lie at the centres of the 112 6-cubes of
an 8-cube centred on the representative hypersphere, while the other 128 lie on
half the 256 vertices of another 8-cube which is half the size of the first.
The 240 neighbours lie on 120 rays which split up into 15 octads of orthogonal
rays, which again determine the colouring scheme.

In the **Leech lattice** in 24 dimensions, each hypersphere makes contact
with 196,560 nearest neighbours! This lattice is too complicated to describe here
briefly, but it has been shown to give the densest possible lattice packing
in 24 dimensions.
(However, the higher the dimension, the less dense the packing, and in this case the
applet has to cheat a bit and pick three-dimensional slices that are *not* oriented
randomly in the full 24 dimensions, or it would mostly be displaying no neighbours
at all.)
More on the Leech lattice can be found in:

“Kissing Numbers, Sphere Packings, and Some Unexpected Proofs” by Florian Pfender and Günter M. Ziegler,Notices of the AMSVol. 51 No. 8, September 2004, pp 873–883, available online (PDF file).