This project will deepen the understanding of differential geometric concepts in relation to 3D surfaces and the ability to investigate these properties numerically using computational geometric methods. Bicontinuous minimal surfaces are smooth infinite surfaces that partition space into two intertwined labyrinthine domains, and a naturally occurring nanostructure across all kingdoms of life. One such family of minimal and constant-mean-curvature (CMC) surfaces is the lonsdaleit structure, also known as hexagonal diamond. A commonly used model for bicontinuous and CMC surfaces is the so-called nodal surface representation, an approximation to the exact geometry of these surfaces. I propose a study to investigate the exactness of the nodal surface representation for these cases, using and comparing numerical and analytical methods. I will generate minimal and CMC surface versions of the lonsdaleit (hexagonal diamond) structure and create mesh approximations using appropriate software packages. I will then measure the area and curvature properties of each model and compare the accuracy of the nodal approximations in order to determine the range for which nodal approximations provide good approximates to the CMC surfaces.
Monica Seeber is an aspiring polymath and has a Bachelor of Arts (with honours) majoring in philosophy from The University of Western Australia and a Master of Arts in professional writing from Deakin University. She has almost completed a Bachelor of Science at Murdoch University, majoring in mathematics and statistics with a secondary major in philosophy. When people are surprised by the combination of mathematics and philosophy, she cheerfully explains in excruciating detail the intimate connection between the two disciplines. Monica enjoys all fields of mathematics, including statistics, but she is especially fond of calculus and propositional logic. Her project “Accuracy and Limits of Nodal Surface Approximations of the Lonsdaleit Minimal Surface” combines her love of calculus, passion for research and her insatiable curiosity about the bizarre things occurring in the natural world. Monica is eager to pursue interdisciplinary postgraduate research in mathematics and philosophy and really wants to know, “What does 0 divided by 0 actually mean?”