Abstract
Spheres are known to be rigid geometric objects: they cannot be deformed isometrically, i.e., while preserving the length of curves, in a twice differentiable way. An unexpected result by Nash (Ann Math 60:383–396, 1954) and Kuiper (Indag Math 17:545–555, 1955) shows that this is no longer the case if one requires the deformations to be only continuously differentiable. A remarkable consequence of their result makes possible the isometric reduction of a unit sphere inside an arbitrarily small ball. In particular, if one views the Earth as a round sphere, the theory allows to reduce its diameter to that of a terrestrial globe while preserving geodesic distances. Here, we describe the first explicit construction and visualization of such a reduced sphere. The construction amounts to solve a nonlinear PDE with boundary conditions. The resulting surface consists of two unit spherical caps joined by a \(C^1\) fractal equatorial belt. An intriguing question then arises about the transition between the smooth and the \(C^1\) fractal geometries. We show that this transition is similar to the one observed when connecting a Koch curve to a line segment.
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Notes
Very recently, a formal construction of a deformed isometric sphere was obtained by considering isometric extensions [13, Cor. 1.3]. However, one equator is left unchanged in this approach, which prevents the sphere to be globally reduced.
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The authors thank the anonymous referees for their scrutinous and insightful comments.
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Communicated by Philippe G. Ciarlet.
Dedicated to the memory of David Spring.
This work is part of the Hevea project and was partly supported by the LabEx Persyval-Lab ANR-11-LABX-0025-01. The first author was in internship at the Institut Camille Jordan. The third author was a postdoc financed by the Matstic grant First from University Joseph Fourier and by Laboratoire Jean Kuntzmann. We are also thankful to the Grenoble University High Performance Computing Centre project (Ciment) for providing access to its computing platform.
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Bartzos, E., Borrelli, V., Denis, R. et al. An Explicit Isometric Reduction of the Unit Sphere into an Arbitrarily Small Ball. Found Comput Math 18, 1015–1042 (2018). https://doi.org/10.1007/s10208-017-9360-1
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DOI: https://doi.org/10.1007/s10208-017-9360-1