The Topology and Geometry of Surface Reticulations in E³

We call the projection of a 2D hyperbolic net onto a TPMS a surface reticulation. The vertices and edges of the reticulation define a geometric embedding of a 3–periodic network. The underlying topology of the 3D network is defined by forming its quotient graph (take a translational unit cell of the reticulation and join edges that are translational copies of one another).

For a fixed periodic minimal surface, each distinct geometric realisation of a 2D hyperbolic tiling (determined as described in the hyperbolic tiling page) maps to a distinct surface reticulation. It is possible, however, that two different reticulations of the same surface may give a network with the same underlying topology. In some cases it is possible that one reticulation may be a self–knotted or self–catenated version of the other.

There are also examples where two different tilings mapped onto two different surfaces generate nets with the same topology, but (naturally) different geometry.

As an illustration of both the above situations, consider the following four surface tilings.

Simple cubic lattice
on the P surface
pcu on P surface pcu on P surface
Simple cubic lattice
on the D surface
pcu on D surface pcu on D surface

The above figures show four different surface tilings whose edge skeletons each have the topology of the simple, or primitive, cubic lattice ( pcu in the notation O'Keeffe uses within the RCSR database).

In general, there is no effective algorithm for determining when two quotient graphs represent the same periodic net. However, for a large class of euclidean nets, (those without vertex collisions Olaf Delgado–Friedrichs and Michael O'Keeffe have defined a canonical form for the quotient graph and this has led to efficient algorithms for comparing the topology of 3–periodic nets.

These algorithms are grouped into a package called SyStRe. The technique works by embedding the net according to an equilibrium placement of its vertices (each vertex is at the centre of mass of its edge–bonded neighbours). Olaf Delgado–Friedrichs also proves that this embedding of the net has the highest possible symmetry.

We use the systre cannonical form where possible to identify surface reticulations with the same underlying topology, and give the equilibrium placement. Not all our networks are amenable to analysis using systre, so we have not identified identical topologies in these cases.

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