Pentaheptite modifications of the graphite sheet

Pentaheptites (three-coordinate tilings of the plane by pentagons and hepta gons only) are classified under the chemically motivated restriction that a ll pentagons occur in isolated pairs and all heptagons have three heptagona l neighbors. They span a continuum between the two lattices exemplified by the boron nets in ThMoB4 (cmm) and YCrB4 (pgg), in analogy with the crossov er-from cubic-close-packed to hexagonal-close-packed; packings, in 3D. Symm etries realizable for these pentaheptite layers are three strip groups (per iodic in one dimension), p1a1, p112; and pill, and five Fedorov groups (per iodic in two dimensions), cmm, pgg, pg, p2, and p1. All can be constructed by simultaneous rotation of the central bonds of pyrene tilings of the grap hite sheet. The unique lattice of cmm symmetry corresponds to the previousl y proposed pentaheptite carbon metal: Analogous pentagon-heptagon tilings o n other surfaces including the torus, Klein bottle, and cylinder, face-regu lar tilings of pentagons and b-gons, and a full characterization of tilings involving isolated pairs and/or triples of pentagons are presented. The Ke lvin paradigm of a continuum of structures arising-from propagation of two original motifs has many potential applications in 2D and 3D.