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

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.