Poster 16

Rebecca Thompson-Flagg UT-Physics

The Geometry of Flowers in Three and Four Dimensions Buckling membranes are seen often in nature from kale leaves and daffodils to torn plastic sheets. These patterns are produced by imposing certain types of metrics on thin sheets. This poster looks specifically at patterns formed at the edge of trumpet shaped sheets which are forced to obey an exponentially decreasing metric. Using geometrical techniques a condition for cylindrical symmetry is found. Below this symmetry limit, the metric can be adopted everywhere by the sheet but it is not clear whether the buckling that occurs past the limit allows the sheet to adopt the metric everywhere. Equations developed by Nash are used to evolve a trumpet from below the limit past the limit. These equations are used to demonstrate that trumpets past this limit cannot fully adopt the exponential metric in three dimensions. The Whitney embedding theorem suggests that in four dimensions it should be possible for the sheets to adopt the metric everywhere. A molecular dynamics code is used to create a sheet with points connected by hookian springs. By changing the equilibrium distance between the springs a target metric can be imposed on the sheet. The energy of the sheet is minimized using the conjugate gradient algorithm. If the sheet fully adopts this target metric the energy of the sheet will be zero. The sheet is allowed to move into a fourth spacial dimension and the energy of the sheet in four dimensions, both below and above the symmetry limit, is compared to the minimum energy in three dimensions.