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Growth forms of grid tilings.
Daniel Demski1, Peter Hilgers2, Anton Shutov3
1301 W. Lincoln Avenue, Reed City, Michigan 49677, USA.
Summary
Quasiperiodic tilings generated by the grid method exhibit polygonal/polyhedral growth forms. These forms, derived from orthoplex projections, enhance understanding of tiling properties and topological densities.
Area of Science:
- Mathematics
- Crystallography
- Geometry
Background:
- Growth forms are key tiling invariants, well-understood for periodic tilings.
- Limited knowledge exists regarding growth forms in quasiperiodic tilings.
- The grid method provides a framework for constructing quasiperiodic tilings.
Purpose of the Study:
- To investigate and characterize the growth forms of quasiperiodic tilings generated by the grid method.
- To establish a connection between these growth forms and geometric projections.
- To explore the properties of these growth forms in two and three dimensions.
Main Methods:
- Utilizing the grid method to construct quasiperiodic tilings.
- Applying projection techniques from central sections of orthoplexes.
- Analyzing the geometric and topological properties of the resulting growth forms.
Main Results:
- Quasiperiodic tilings obtained via the grid method possess polygonal/polyhedral growth forms.
- These growth forms can be represented as projections of central sections of orthoplexes.
- Properties of these 2D and 3D growth forms were systematically studied.
Conclusions:
- The study successfully defines and characterizes growth forms for grid-based quasiperiodic tilings.
- This research provides a method for obtaining growth forms through orthoplex projections.
- Findings contribute to understanding coordination numbers and topological densities in quasicrystals.
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