A node of valence k in an edge-to-edge tiling is a point that is the common vertex of k tiles. We show that an edge-to-edge tiling of plane by pentagons each of which has m nodes of valence 3 and 5 − m nodes of valence k has properties of (m, k) = (3, 4) or (m, k) = (4, 6) if it is normal. Then we discuss tilings by congruent convex pentagons using the properties.

Monodispersed intermetallic AuCu pentagonal nanorods with controlled size and composition have been developed by a seed-mediated growth route. The AuCu/C nanomaterial catalyzed the coupling reaction of sulfonamide with benzyl alcohol in good to excellent yields.

The paper introduces conformal tilings, wherein tiles have specified conformal shapes. The principal example involves conformally regular pentagons which tile the plane in a pattern generated by a subdivision rule. Combinatorial symmetries imply rigid conformal symmetries, which in turn illustrate a new type of tiling self-similarity. In parallel with the conformal tilings, the paper develops d...

the aim of this study was to find any structural differences between the digital pads of forelimbs and hind limbs as well as more careful investigation of the internal and external structures of the toe-pad. in this study, pad morphology and cytology in litoria caerulea is described using sem, tem and light microscopy. at the gross anatomical level, toe-pads in hind limbs were subdivided into m...

We consider tilings of the sphere by congruent pentagons. The basic example is dodecahedron, which has 12 pentagonal tiles. To simplify the problem, we assume the tiling is edge-to-edge and all vertices have degree ≥ 3.

Recently Zagier proved a remarkable q-series identity. We show that this identity can also be proved by modifying Franklin’s classical proof of Euler’s pentagonal number theorem. Mathematics Subject Classification (2000): 05A17 11P81

Field-induced slow magnetic relaxation was observed for air-stable mononuclear cobalt(II) compounds with pentagonal bipyramid geometry. These are the first examples of such behavior observed in the seven-coordinated mononuclear 3d metal compounds.