Tilings and Nets

As alluded to in the preceding section, inflating the 3D-net and replacing it with connected polyhedra in principle makes it possible to use the concept of genus. However, Bonneau et al. have suggested a conceptually much easier definition as we will shortly see f 11]. 

The smallest piece of the net needed to form the whole 3D-net is the unit that is repeated only by translation throughout the structure. It can be seen as the unit cell contents of the net. For a six-connected 3D net ihe number of vertices in such a unit, called Zi, may be as small as one, but for a four-connected 3D net it is at least 2 since we need a minimum of six loose ends to build from. 

The number of new edges that has to be drawn is the cyclomatic number and equal to the genus, g, of the net. If the total number of edges and vertices are e and V then  

Wells classified the nets with the same (n,p) based on the Zy variable, the number of vertices in the minimal repeating unit, yielding list of nets with descending symmetry. Wc now sec that we can instead use a common property, the genus of the quotient graph, to classify nets with different connectivities. The connectivity does not need to be the same for all vertices either, so the 5 and 3 connected and 3 and 4 connected net can readily be included as well. 

For the three-connected = 3) nels the genus and the Z, variable are related since each edge shares two vertices) by the reformulation of equation 10.14  

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