This paper considers the problem of placing mosaic tiles on a surface to produce a surface mosaic. We assume that the user specifies a mesh model, the size of the tiles and the amount of grout, and optionally, a few control vectors at key locations on the surface indicating the preferred tile orientation at these points. From these inputs, we place equal-sized rectangular tiles over the mesh such as to almost cover it, with controlled orientation. The alignment of the tiles follows a vector field which is interpolated over the surface from the control vectors, and also forced into alignment with any sharp creases, open boundaries, and boundaries between regions of different colors. Our method efficiently solves the problem by posing it as one of globally optimizing a spring-like energy in the Manhattan metric, using overlapping local parameterizations.We demonstrate the effectiveness of our algorithm with various examples.

In the first paper of this series, “Electrodynamics and the Gauss Linking Integral on the 3-sphere and in Hyperbolic 3-space,” we developed a steady-state version of classical electrodynamics in these two spaces, including explicit formulas for the vector-valued Green’s operator, explicit formulas of Biot-Savart type for the magnetic field, and a corresponding Amp`ere’s Law contained in Maxwell’s equations, and then used these to obtain explicit inte- gral formulas for the linking number of two disjoint closed curves. In this second paper, we obtain integral formulas for twisting, writhing, and helicity, and prove that link = twist + writhe on the 3-sphere and in hyperbolic 3-space. We then use these results to derive upper bounds for the helicity of vector fields and lower bounds for the first eigenvalue of the curl operator on subdomains of these two spaces. An announcement of these results, and a hint of their proofs, can be found in the Math ArXiv, math.GT/0406276, while an expanded version of the first paper, with full proofs, can be found at math.GT/0510388.

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L’objet de cet article est de répondre à une question posée par Gelfond en 1968 en montrant que la somme des chiffres des carrés écrits en base$q$⩾ 2 est équirépartie dans les progressions arithmétiques.

We define a partition of Z into intervals {$I$_{j}} and prove the Littlewood-Paley inequality ‖$f$‖_{$p$}≦$C$_{p}‖$Sf$‖_{$p$}, 2≦$p$<∞. Here$f$is a function on [o, 2π) and $$Sf = (\sum |\Delta _j |^2 )^{1/2} , \hat \Delta j = \hat f\chi _{Ij} $$ . This is a new example of a partition having the Littlewood-Paley property since the {$I$_{j}} are not of the type obtained by iterating lacunary partitions finitely many times.