A finite subdivision rule for the 4-dimensional torus. These pictures need their orientation to be specified better.
A finite subdivision rule for the Borromean rings, with some subdivisions of the blue tile type.
A diagram displaying the known information about subdivision rules associated to 3-manifolds of various geometries.
1. Sol geometry cannot have a finite subdivision rule, because it is not almost convex with respect to any generating set. Thus, instead of showing a subdivision rule, we show a fundamental domain for a typical Sol manifold, to encourage the reader to try to construct a subdivision rule themselves and see where it goes wrong.
2. Nil geometry may or may not have a finite subdivision rules. As with Sol geometry, we show a fundamental domain for the quotient of Nil space by the integer Heisenberg group, to encourage people to experiment.
3. All manifolds in spherical geometry have eventually empty boundary, and the only possible subdivision on empty boundary is the empty subdivision.
4. Spherical cross Euclidean geometry manifolds have two boundary spheres, and must have eventually constant subdivision on each.
1. Euclidean geometry manifolds have polynomial growth, so no subdivision of any tile can contain more than one copy of itself, and the majority of tiles must eventually stop subdividing.
2. and 3. Hyperbolic cross Euclidean and SL2(R) geometries. These diagrams are the same on purpose, because these geometries are indistinguishable with respect to subdivision rules in a certain sense. They contain a circle corresponding to the circle at the boundary of hyperbolic 2-space, and the rest behaves like Euclidean space.
4. Hyperbolic 3-manifolds have conformal subdivision rules with combinatorial mesh approaching 0.
A finite subdivision rule for the 2-sphere cross a circle. The fundamental domain is a thickened sphere, and the universal cover is created by nesting these spheres inside of each other. The boundary is two spheres with no cell-structure, and subdivision does not change this, so the subdivision rule is the identity.
A finite subdivision rule associated to the 3-torus, a Euclidean 3-manifold. The universal cover is creating by stacking cubes. The subdivision complex depicted consists of the top and sides of a cube.
A finite subdivision rule associated to a hyperbolic surface crossed with a circle. Tiles are dodecahedral prisms with angles 2pi/3 on the vertical sides are pi/2 on the horizontal sides. The initial subdivision complex is the top and sides of a dodecahedron. Notice how the circle at infinity of the surface appears in the subdivision rule.
A finite subdivision rule associated to the right-angled dodecahedral 3-manifold. This is a conformal subdivision rule with combinatorial mesh approaching 0. Cannon, Floyd and Parry found a subdivision rule for this manifold earlier, as shown in http://www.math.vt.edu/people/floyd/subdivisionrules/gallery/gallery.html (where it is listed as "Dodecahedral subdivision rule").
A finite subdivision rule associated to the complement of a tubular neighborhood of the Hopf link. Blue faces correspond to the toroidal boundary of the tubular neighborhood. Notice how the number of tiles grows very slowly, and the white tiles (which represent elements of the fundamental group) grow only in one direction.
A finite subdivision rule with the same set-up as above, except now we are taking the complement of a tubular neighborhood of the trefoil knot. Again, blue tiles correspond to boundary and white tiles correspond to elements of the fundamental group. Here, the fibering of this knot can be seen: there is a hyperbolic direction (the vertical direction) where tiles subdivide to form a Cantor set, and a Euclidean direction (the horizontal direction) where tiles divided linearly.
This example is just like the previous two examples, except now we are looking at the complement of a neighborhood of the Borromean rings, a hyperbolic link complement. Notice how we have exponential growth in every direction.
1. Bounded growth of valence, as shown in the hexagonal replacement subdivision rule on the left,
2. Linear growth of valence, as shown in the subdivision rule in the middle associated to the Borromean rings' complement, and
3. Exponential growth of valence, as shown in barycentric subdivision on the right.
The `circles' about vertices (i.e. stars of vertices) in picture 2 will shrink away logarithmically, while those in picture 3 will not shrink away. Stars of vertices for picture 1 shrink quickly.
Examples of subdivision rules leading to nested rings, where each subdivides more tangentially than radially, an equal amount, or more radially.
The space at infinity for the subdivision rule associated to the 3-torus shown near the top of this page.
This was an experiment with blowing up vertices. If you take Cannon, Floyd and Parry's classic dodecahedral subdivision rule at http://www.math.vt.edu/people/floyd/subdivisionrules/gallery/gallery.html and replace each vertex in the middle with a square and those at the edges with triangles (so they meet up with squares) and then run subdivision like usual (with squares not subdividing), will those added square blow up, or shrink away? These pictures (and explicit calculations) show that they shrink away.