In 1997, Schaeffer described a bijection between Eulerian planar maps and some trees. In this work we generalize his work to a bijection between bicolorable maps on a surface of any fixed genus and some unicellular maps with the same genus. An important step of this construction is to exhibit a canonical orientation for maps, that allows to apply the same local opening algorithm as Schaeffer.As an important byproduct, we obtain the first bijective proof of a result of Bender and Canfield from 1991, when they proved that the generating series of maps in higher genus is a rational function of the generating series of planar maps.

Figure 1: Given a set of planar input contours (left), our system computes a stable, self-supporting wire sculpture. The physical prototype (right) can be easily fabricated using a 2D wire bending machine and assembled without the need of connectors between crossing wires. AbstractWe present a computational method for designing wire sculptures consisting of interlocking wires. Our method allows the computation of aesthetically pleasing structures that are structurally stable, efficiently fabricatable with a 2D wire bending machine, and assemblable without the need of additional connectors. Starting from a set of planar contours provided by the user, our method automatically tests for the feasibility of a design, determines a discrete ordering of wires at intersection points, and optimizes for the rest shape of the individual wires to maximize structural stability under frictional contact. In addition to their application to art, wire sculptures present an extremely efficient and fast alternative for low-fidelity rapid prototyping because manufacturing time and required material linearly scales with the physical size of objects. We demonstrate the effectiveness of our approach on a varied set of examples, all of which we fabricated.

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