by Marek14 » Thu Sep 08, 2005 5:09 pm
Although cube has six faces, they are squares (therefore the 4), and there are three squares meeting at every vertex of the cube (therefore the 3).
One way to look at Schlafli symbol is to understand it in term of angles. {4,3} would mean (square with vertex angle 360/3). This is 120-degree square which exists in spherical geometry. You can fit six of them on the surface of sphere. What we call "cube" is the same thing but with circle arcs replaced by straight lines.
Let's look at the possibilities: we take the triangles first.
{3,3} is 120-degree triangle, and you can cover the whole sphere with just 4 of these - thus, tetrahedron. As you decrease the angle down to 90, you can cover it with eight triangles, leading to octahedron. Shrinking further, we get to 72 degrees, and {3,5} covering - or icosahedron. If we go even further, then {3,6} are common 60-degree Euclidean triangles. {3,6} is, therefore, covering of the plane.
If we reduce the angle even more, we get into hyperbolic realm. The edge of triangle grows as the angle shrinks. The limit is {3,oo} covering with infinite triangles that have 0-degree angles.
Analogically, we find that 120-degree squares cover sphere, 90-degree ones cover plane and anything more covers H-plane. {5,3} is spheric (dodecahedron), while {5,4} already passes over 108-degree mark of Euclidean pentagons and is well into hyperbolic realm. {6,3} is Euclidean, while {6,4} and anything {m,n} with m>6 is hyperbolic.
Now, we can apply it in four dimensions too, if we treat the first two numbers as a polyhedron, and the last one as its "dihedral angle" - the angle between neighbouring faces.
For example, the Euclidean regular tetrahedron has dihedral angle around 70 degrees - the exact value is not important here, just that it's more than 60 and less than 72. This means that {3,3,3}, {3,3,4}, and {3,3,5} are all spherical tetrahedra which tile the hypersphere, and they can be made into polytopes.
If we start to shrink the dihedral angle of tetrahedron, we find that it can't be shrunk down to zero - not even in hyperbolic space. The limit is 60 degrees here - corresponding to {3,3,6} which is made up of infinitely large tetrahedra (but still with finite volume).
The same way, Euclidean cube has dihedral angle 90 degrees - 120-degree cube is part of {4,3,3} - tesseract, while 72-degree cube {4,3,5} tiles H3 space and {4,3,6} is infinite. 120-degree dodecahedron is spherical dodecahedron which is the cell of 120-cell {5,3,3}. {5,3,4} and {5,3,5}, made up of 90-degree and 72-degree dodecahedra, are hyperbolic tilings, {5,3,6} is once again infinite.
{3,4,3} is icositetrachoron, or a spherical tiling made up of 120-degree octahedra. 90-degree ones from {3,4,4} are hopelessly infinite, though.
{3,5,3} is already hyperbolic, as the dihedral angle of Euclidean icosahedron is greater than 120 degrees. You won't even get to 90 here by the time you hit infinity.
Shrinking dihedral angles of infinite constructions like {6,3} is also interesting, but beyond the scope of this topic.
The only interesting dimension remaining is 5 - it has three spherical tilings (and therefore three regular polytopes) {3,3,3,3}, {3,3,3,4} and {4,3,3,3}, but it also has three tilings of E4 - {3,3,4,3}, {3,4,3,3}, and {4,3,3,4}, five tilings of H4 - {3,3,3,5}, {4,3,3,5}, {5,3,3,3}, {5,3,3,4}, and {5,3,3,5} and a single tiling of H4 with infinite tiles - {3,4,3,4}.