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Sharp [Formula: see text] Law for the Minimizers of the Edge-Isoperimetric Problem on the Triangular Lattice
Elisa Davoli1, Paolo Piovano1, Ulisse Stefanelli1,2
1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
Journal of Nonlinear Science
|July 28, 2017
Summary
We studied the edge-isoperimetric problem for point sets in a triangular lattice, linking it to Wulff shape formation in crystallization. Minimizers achieve maximum area and minimum perimeter, closely resembling hexagonal shapes.
Area of Science:
- Discrete geometry
- Crystallography
- Mathematical physics
Background:
- The Wulff shape is fundamental in crystal growth, minimizing surface energy.
- The edge-isoperimetric problem (EIP) explores optimal shapes for discrete point sets.
Purpose of the Study:
- To investigate the EIP for n points in a triangular lattice.
- To connect EIP solutions to the Wulff shape in crystallization.
- To quantify maximal area and minimal perimeter for EIP minimizers.
Main Methods:
- Introducing novel definitions for perimeter and area in the triangular lattice.
- Characterizing EIP minimizers via an isoperimetric inequality.
- Analyzing the deviation of minimizers from hexagonal configurations.
Main Results:
- EIP minimizers attain maximal area and minimal perimeter for connected configurations.
- Maximal area and minimal perimeter are explicitly quantified in terms of n.
- Minimizers are hexagonal configurations with boundary deviations estimated to be at most a sharp, determined constant.
Conclusions:
- The EIP in the triangular lattice provides insights into Wulff shape formation.
- EIP minimizers offer a discrete analogue to continuous isoperimetric problems.
- The study precisely quantifies the deviation from ideal hexagonal shapes.
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