Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

**Mathematical Bellows and Volume Formulas
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Robert Connelly, Cornell University**

Consider a triangulated polyhedral surface in three-space. Regard each edge of the surface as a rigid bar, and regard the bars as joined at ideal universal joints at the vertices of the surface. In 1978 examples were found where such a surface was a mathematically exact (flexible) mechanism. But all these examples had the property that the volume they enclosed during the flex was constant. In other words none of these examples could be used as a perfect mathematical bellows that would inhale and exhale. Recently, I. Sabitov showed that all such flexible surfaces must flex with constant volume. Indeed, there is a polynomial, with leading coefficient 1 and whose other coefficients are polynomials in the edge lengths of the surface, that is satisfied by the enclosed volume. Following the work of Anke Walz, we show how such a polynomial can be found for certain triangulated surfaces in higher dimensions.

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