Exact construction of periodic minimal surfaces : the I-WP surface and its isometries

Abstract

We have analysed in detail the geometry of the I-WP infinite periodic minimal surface, discovered by Alan Schoen in the 1960's. An exact parametrisation has been found, using the local Weierstrass equations for the surface, involving modified hyperelliptic integrals. We have used rapidly converging integration techniques to calculate the surface to volume ratio of this surface, and found that it differs from the value conjectured by Anderson. Further, the family of isometric minimal surfaces related to the I-WP surface has been found to exhibit a novel sequence of structures; viz. repeated formation of the I-WP surface itself On analyse en detail la geometrie de la surface minimale infinie et periodique I-WP, decouverte par Alan Schoen dans les annees 60. En utilisant les equations de Weierstrass locales pour la surface, on obtient une parametrisation exacte a l'aide d'integrales hyperelliptiques modifiees. A l'aide de methodes d'integrations numeriques rapides, le rapport surface/volume de cette surface est calcule; il differe de la valeur conjecturee par Anderson. De plus, on trouve que la famille de surfaces minimales isometriques reliees a la surface I-WP possede une nouvelle suite de structures qui reproduisent la surface I-WP elle-meme