I learned the Schläfi symbol notation from Coxeter’s “Regular Polytopes”. I will proceed by example:

{3} denotes an equilateral triangle and {4} denotes a square.
{3, 3} denotes a tetrahedron bounded by triangles meeting at vertices by threes.
{4, 3} denotes a cube bounded by squares meeting by threes.
{3, 4} denotes an octahedron bounded by triangles meeting by fours.
{3, 5} denotes an icosahedron bounded by triangles meeting by five’s.
{5, 3} denotes a dodecahedron bounded by pentagons meeting by threes.

Note that reversing the numbers yields the dual figure; that pattern persists in more general cases below. Each of the above can be viewed as tiling the sphere. The general pattern so far is that curly braces with n elements denotes an n+1 dimensional polytope whose n dimensional faces meet at n−2 dimensional ‘edges’. (They also meet at n−1 dimensional places, but always by twos.)

{3, 6}, {4, 4}, and {6, 3} each tile the flat plane.
{5, 4} tiles the hyperbolic plane and my PostScript program draws that tiling on the Poincaré disk, and a movie of same. See this for more.

{4, 3, 4} tiles flat 3 space. {4, 3} denotes a cube and {4, 3, 4} denotes packed cubes, four about shared edges. In these constructions an n dimensional polytope meets neighboring polytopes across an n−1 dimensional polytope which is a face of each of the first two. Several congruent polytopes may ‘surround’ and meet at an n−2 dimensional polytope and the count of these provides the values in the Schläfi symbols. Such meetings are the bones of curvature; the space is flat elsewhere.

Aristotle thought that {3, 3, 5} tiled space, but he was wrong! Here is an applet to show {5, 3, 3} projected onto the screen. You can rotate it in 4-D.
The images in the Scientific American are {4, 3, 5} with five cubes about each edge. A movie has been made wandering about this negatively curved space.

There are two closely related spaces associated with each of these designations:

The same two ways of considering tilling applies to negatively curved spaces. For this note we shall henceforth imagine continuous curvature, in fact constant curvature. I think that there is a connection to homotopy theory.

For uniform curvatures the space is Euclidean in the small. If, in a 3-D tiling, we consider a small sphere about a vertex, the sphere is itself tiled. Being embedded in a Euclidean space, the sphere must itself have a positive curvature. This rules out {4, 3, 6} as a 3-D tiling for the said sphere would then be tiled by {3, 6} which does not have positive curvature. There are tedious elementary formulae indicating whether the tiled space is positively or negatively curved.

The tiling {4, 3, 5} tiles the vertex spheres as {3, 5}. The vertex of that tiling is thus like an icosahedron and there are 12 edges meeting that vertex.

See this for more about the geometry of curved spaces.