Baird, Paul Constant mean curvature polytopes and hypersurfaces via projections. (English) Zbl 1284.52015 Differ. Geom. Appl. 33, Suppl., 199-212 (2014). Summary: Given a regular polytope in Euclidean space and an orthogonal projection to the complex plane, the function which assigns to each vertex its projected value satisfies a quadratic difference equation. The form of the equation is the same whatever the polytope, except for a real parameter \(\rho\) which varies from polytope to polytope. It is independent of the projection used and the size of the polytope. When we consider an orthogonal projection of a smooth hypersurface in Euclidean space, remarkably we find the same phenomena, namely that a smooth version of the equation is satisfied independently of the projection, where the parameter \(\rho\) depends only on the mean curvature. We therefore make an unconventional definition of a constant mean-curvature polytope as one which satisfies this same equation with \(\rho\) constant, independently of the orthogonal projection. We discuss some examples of constant mean curvature polytopes. Cited in 2 Documents MSC: 52B10 Three-dimensional polytopes 52B11 \(n\)-dimensional polytopes 39A14 Partial difference equations 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) Keywords:constant mean curvature hypersurface; constant mean curvature polytope; regular polytope; orthogonal projection PDF BibTeX XML Cite \textit{P. Baird}, Differ. Geom. Appl. 33, 199--212 (2014; Zbl 1284.52015) Full Text: DOI OpenURL References: [1] Baird, P.: A class of quadratic difference equations on a finite graph [2] Baird, P.; Wood, J. C.: Harmonic morphisms between Riemannian manifolds, London mathematical society monographs (New series) 29 (2003) · Zbl 1055.53049 [3] Connelly, R.; Jordan, T.; Whiteley, W. J.: Generic global rigidity of body-bar frameworks, EGRES [4] Eastwood, M. G.; Penrose, R.: Drawing with complex numbers, Math. intell. 22, 8-13 (2000) · Zbl 1052.51505 [5] Gauss, C. F.: Werke, zweiter band, (1876) [6] Hsu, L.; Kusner, R.; Sullivan, J.: Minimizing the squared mean curvature integral for surfaces in space forms, Exp. math. 1, No. 3, 191-207 (1992) · Zbl 0778.53001 [7] Jaco, W.; Rubinstein, J. H.: PL minimal surfaces in 3-manifolds, J. differ. Geom. 27, 493-524 (1988) · Zbl 0652.57005 [8] Karcher, H.: The triply periodic minimal surfaces of alan schoen and their constant mean curvature companions, Manuscr. math. 64, 291-357 (1989) · Zbl 0687.53010 [9] Monge, G.: Géométrie descriptive, (1811) [10] Polthier, K.: Polyhedral surfaces of constant mean curvature · Zbl 0980.53012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.