Discrete minimal surfaces

Abstract

In this thesis, I propose a new numerical procedure to obtain discrete minimal surfaces with fixed or partially free boundaries. Using this procedure, I recover most of the minimal surfaces obtained by other mathematicians such as Hildebrandt, Karcher as presented in the monograph "Minimal Surfaces" by Dierkes et al (1992) [1]. The same procedure also gives rise to new graphics of partially free boundary minimal surfaces as depicted in the popular scientific account "The parsimonious universe : shape and form in the natural world" by Hildebrandt and Tromba (1996) [3]. The origin of the minimization algorithm comes from the paper of Pinkall and Polthier [4] published in the journal Experimental Mathematics in 1993. My contributions consists of : (i) improving the algorithm of Pinkall and Polthier so that one point at a time needs be minimized, as a consequence of which the computer code is greatly simplified. (ii) writing down the codes in the language of Mathematica and implementing it, (iii) providing the convergence of my algorithm for the fixed boundary case, (iv) using this algorithm to produce graphics of most of the famous minimal surfaces in the book on minimal surfaces by Dierkes et al [1] as well as some additional new minimal surfaces by pasting and refinement techniques starting from some simple fundamental pieces, and (v) writing down the Mathematica codes to handle the partially free boundary problem case.