DEFORMATION OF THE RIEMANN SURFACE OF AN ALGEBRAIC FUNCTION
Catalog-number 229 (year 1983). (Theory of algebraic functions).
Underlying this twisted deformation space, where long tubes interwine to weave a tortuous egg-like shape, is a certain three-dimensional model. The model shows a deformation of a Riemann surface of a special algebraic function, set in four-dimensional Euclidean space. This surface is also considered to be homeomorphic to a two-dimensional sphere with one hundle as well as a two-dimensional torus. In terms of the theory of algebraic functions, we can construct this kind of Riemann surface by taking two copies of a two-dimensional sphere, making two cuts on each and then gluing the corresponding cuts together. What we obtain is a torus, or donut-like object, represented as two spheres joined together by two tube-like cylinders (which are shown in this image). A curious feature of this form is that if we deform the underlying function, a polynomial, such that its roots coalesce, then so appear, singular points arise, and the surface loses its smoothness. In this image, two roots seek to coalesce into one, and, as a result, the upper sphere gets smaller while the lower one grows larger, a process somewhat visible through the object’s cut-away sections.