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

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 C1 fractal equatorial belt. An intriguing question then arises about the transition between the smooth and the C1 fractal geometries. We show that this transition is similar to the one observed when connecting a Koch curve to a line segment.