Comparing 2D and 3D Systems - IAS

theorem, the number of grains, faces, edges, and vertices are related as follows:χ = G − F + E − V, (5.1)where G is the number of grains in a system, F the number of faces, E the number of edges, andV the number of vertices or quadruple points in the system; χ is known as the Euler characteristicof the underlying space, and in our case χ(T 3 ) = 0, where T 3 is the cube with periodic boundaryconditions.Since each vertex is incident with exactly four edges and each edge is incident withexactly two vertices, we have E =2V . Because every edge is adjacent to three faces, the averagenumber of edges per face is 3E/F ; because every face is adjacent to two grains, the average numberof faces per grain is 2F/G. If we use E =3E/F to denote the average number of edges per faceand F =2F/G to denote the average number of faces per grain, then the relationship betweenthese two averages is precisely:E =6− 12/F . (5.2)Therefore, only one of these two quantities is independent.As a short aside, we point out thatdespite the simple nature of this relationship, a number of papers have reported data that are clearlyinconsistent with it. For example, [96] reports that F ≈13.6 and E ≈5.65, two values thatcannot be simultaneously correct. This might point to some the difficulty in consistently identifyingedges and faces when using volume-based methods such as Potts and cellular automaton models. Ina second example, a front-tracking model [109] reports that F ≈13.8 and E ≈5.01. There seemsto be no easy to reconcile this data with itself. Both examples make use of periodic boundaries intheir simulations.In any case, it is not known what this number should be in the three-dimensional steady statestructure, although a few conjectures have been made.Several publications predict that F ≈13.397, for which E ≈5.104 [66, 122, 123, 137, 116] and one predicts F ≈13.564, for whichE ≈5.115 [124]. Our simulations indicate that F ≈13.766 and E ≈5.128.Likewise, there is no known analytic function that gives the probability distributions in Figure5.2, despite some recent attempts [138] for the 3D case. Nevertheless, the similarity in the shapesof the distributions for edges per grain in 2D and for faces per grain in 3D is quite striking, asis the difference in the distributions of edges per grain in the 2D and 3DX systems. These datamake clear the difficulty of directly comparing two-dimensional simulations with cross sections ofthree-dimensional experimental samples. The field of stereology studies the sort of data that can be137