The squarographites: A lesson in the chemical topology of tessellations in 2- and 3-dimensions

This paper describes the 3-dimensional (3D) crystal structure of a novel, hypothetical 5-,6-connected tetragonal structural pattern with the binary stoichiometry AB(4). The novel pattern lies in symmetry space group P4/mmm, number 123. It possesses the Wells point symbol (4(8)6(4))(4(7)6(2))(4) and has the Wellsean Schlafli index (4(1/2), 5(1/5)). In plane projection, the topology of the network places it, on the one hand, between 4 the 2D tessellation (4, 4), in plane group p4mm, with the Wells point symbol 4(4), commonly known as the square net, and the 2D tessellation (6, 3 3), in plane group p6mm, with the Wells point symbol 6, which lies diagonally disposed to the square net, to the left and beneath (4, 4). On the other hand, the topology of the pattern places it, also, in the space between (4, 4) and the 3D network (4, 6), in the cubic symmetry space group Pm3m, with the Wells point symbol 4 6, also known as the primitive cubic net. It has therefore been given the name alpha-squarographite, and represents a fusing together into 3D of what are termed 2D alpha-squarographene sheets. Such sheets can be envisioned as a projection of the corresponding 3-dimensional (3D) alpha-squarographite lattice in (001), and they possess the Wells point symbol (6(4))(4.6(2))(4) and the Wellsean Schlafli symbol given by (5(1/2), 3(1/5)). In a geometric idealization, they represent a tiling of the Euclidean plane with slightly distorted hexagons and perfect squares, where the slightly distorted hexagons can be seen to possess oppositely oriented apical right angles and obtuse equatorial angles of 135 degrees, respectively. In the next section of the paper, the alpha-squarographene sheet, which occurs in the tetragonal symmetry plane group p4mm, is compared to its topological isomer beta-squarographene, in the orthorhombic symmetry plane group p2mm. This paper thus reports the first occurrence of topological isomerism among the 2D tessellations. Previously it was thought by Wells that such isomerism was restricted to the 3D networks. The topologically isomeric squarographene sheets are then compared to the ordinary square net (4, 4), or 4 4, and graphene 3 grid (6, 3), or 6, of which they are hybrids, that occur in the tetragonal and hexagonal symmetry plane groups p4mm and p6mm, as mentioned above, respectively. Finally, a contrast and comparison of the network topologies of the alpha- and beta-squarographene plane tessellations is presented with respect to the more familiar, related 2-dimensional (2D) Archimedean network topology known as the square-octagon plane tiling 4.8(2), given by the Schlaffli symbol (6(2/3), 3). (C) 2007 Published by Elsevier Masson SAS.