An Explicit Isometric Reduction of the Unit Sphere into an Arbitrarily Small Ball

  • Evangelis Bartzos
  • Vincent Borrelli
  • Roland Denis
  • Francis Lazarus
  • Damien Rohmer
  • Boris Thibert
Article

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.

Keywords

Isometric embedding Convex integration Sphere reduction Boundary conditions 

Mathematics Subject Classification

Primary 35-04 Secondary 53C21 53C23 53C42 57R40 

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Copyright information

© SFoCM 2017

Authors and Affiliations

  • Evangelis Bartzos
    • 1
  • Vincent Borrelli
    • 2
  • Roland Denis
    • 3
  • Francis Lazarus
    • 4
  • Damien Rohmer
    • 5
  • Boris Thibert
    • 6
  1. 1.Lab of Geometric and Algebraic Algorithms, Department of Informatics and TelecommunicationsUniversity of AthensAthensGreece
  2. 2.Institut Camille JordanUniversité Lyon ILyonFrance
  3. 3.CNRSInstitut Camille JordanLyonFrance
  4. 4.GIPSA-LabCNRSGrenobleFrance
  5. 5.Inria/LJK (CNRS, CPE Lyon)Université Grenoble AlpesGrenobleFrance
  6. 6.Laboratoire Jean KuntzmannUniversité Grenoble-AlpesGrenobleFrance

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